Content-Type: multipart/mixed; boundary="-------------0001071639531" This is a multi-part message in MIME format. ---------------0001071639531 Content-Type: text/plain; name="00-8.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-8.keywords" Non-equilibrium statistical mechanics, Anharmonic chains, Low temperature asymptotics, Variational principle, Freidlin-Wentzell theory ---------------0001071639531 Content-Type: application/postscript; name="final.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="final.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: final.dvi %%Pages: 40 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips final -o %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 2000.01.07:1717 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S 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4939 y Fu(=)97 b Fn(p)1476 4953 y Fm(j)1528 4939 y Fn(;)1920 5225 y Fu(2)p eop %%Page: 3 3 3 2 bop 1103 573 a Fu(_)-42 b Fn(q)1127 587 y Fm(j)1262 573 y Fu(=)97 b Fk(\000r)1577 587 y Fm(q)1609 597 y Ff(j)1645 573 y Fn(V)20 b Fu(\()p Fn(q)s Fu(\))15 b Fn(;)107 b(j)31 b Fu(=)25 b(2)p Fn(;)15 b(:)g(:)g(:)i(;)e(n)20 b Fk(\000)g Fu(1)15 b Fn(;)1093 711 y Fu(_)-42 b Fn(q)1117 725 y Fm(n)1262 711 y Fu(=)97 b Fn(p)1476 725 y Fm(n)1538 711 y Fn(;)1093 848 y Fu(_)-42 b Fn(q)1117 862 y Fm(n)1262 848 y Fu(=)97 b Fk(\000r)1577 862 y Fm(q)1609 870 y Ff(n)1655 848 y Fn(V)20 b Fu(\()p Fn(q)s Fu(\))h(+)f Fn(r)1995 862 y Fm(n)2057 848 y Fn(;)1036 996 y(dr)1124 1010 y Ft(1)1262 996 y Fu(=)97 b Fk(\000)p Fn(\015)5 b Fu(\()p Fn(r)1629 1010 y Ft(1)1689 996 y Fk(\000)20 b Fn(\025)1833 958 y Ft(2)1873 996 y Fn(q)1914 1010 y Ft(1)1953 996 y Fu(\))p Fn(dt)g Fu(+)g(\(2)p Fn(\015)5 b(\025)2364 958 y Ft(2)2405 996 y Fn(T)2458 1010 y Ft(1)2497 996 y Fu(\))2532 958 y Ft(1)p Fm(=)p Ft(2)2643 996 y Fn(dw)2755 1010 y 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h(giv)m(en)f(b)m(y)1106 1941 y Fn(G)p Fu(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\))41 b(=)1642 1860 y Fh(X)1614 2042 y Fm(i)p Ft(=1)p Fm(;n)1806 1797 y Fh( )1909 1879 y Fn(r)1953 1846 y Ft(2)1950 1904 y Fm(i)p 1881 1920 138 4 v 1881 2003 a Fu(2)p Fn(\025)1979 1977 y Ft(2)2049 1941 y Fk(\000)20 b Fn(r)2181 1955 y Fm(i)2210 1941 y Fn(q)2251 1955 y Fm(i)2278 1797 y Fh(!)2364 1941 y Fu(+)g Fn(H)7 b Fu(\()p Fn(p;)15 b(q)s Fu(\))g Fn(;)543 b Fu(\(3\))448 2231 y(where)28 b Fn(H)7 b Fu(\()p Fn(p;)15 b(q)s Fu(\))28 b(is)f(the)i(Hamiltonian)d (of)i(the)g(isolated)g(systems)g(of)g(oscillators)f(giv)m(en)448 2344 y(b)m(y)35 b(\(1\).)53 b(W)-8 b(e)36 b(also)e(in)m(tro)s(duce)f (the)i(parameters)f Fn(")h Fu(\(the)g(mean)f(temp)s(erature)g(of)h(the) 448 2457 y(reserv)m(oirs\))30 b(and)g Fn(\021)k Fu(\(the)d(relativ)m(e) f(temp)s(erature)h(di\013erence\):)1329 2703 y Fn(")40 b Fu(=)1533 2642 y Fn(T)1586 2656 y Ft(1)1645 2642 y Fu(+)20 b Fn(T)1789 2656 y Fm(n)p 1533 2682 304 4 v 1662 2766 a Fu(2)1861 2703 y Fn(;)107 b(\021)44 b Fu(=)2203 2642 y Fn(T)2256 2656 y Ft(1)2315 2642 y Fk(\000)20 b Fn(T)2459 2656 y Fm(n)p 2203 2682 V 2203 2766 a Fn(T)2256 2780 y Ft(1)2315 2766 y Fu(+)g Fn(T)2459 2780 y Fm(n)2531 2703 y Fn(:)765 b Fu(\(4\))448 2947 y(Then)30 b(Eq.)g(\(2\))i(tak)m(es) f(the)g(form)1154 3151 y(_)-42 b Fn(q)101 b Fu(=)d Fk(r)1524 3165 y Fm(p)1563 3151 y Fn(G)15 b(;)1153 3289 y Fu(_)-43 b Fn(p)98 b Fu(=)g Fk(\000r)1595 3303 y Fm(q)1632 3289 y Fn(G)15 b(;)1090 3436 y(dr)101 b Fu(=)d Fk(\000)p Fn(\015)5 b(\025)1624 3399 y Ft(2)1663 3436 y Fk(r)1739 3450 y Fm(r)1777 3436 y Fn(Gdt)20 b Fu(+)f Fn(")2081 3399 y Ft(1)p Fm(=)p Ft(2)2192 3436 y Fu(\(2)p Fn(\015)5 b(\025)2377 3399 y Ft(2)2417 3436 y Fn(D)s Fu(\))2530 3399 y Ft(1)p Fm(=)p Ft(2)2640 3436 y Fn(dw)18 b(;)526 b Fu(\(5\))448 3640 y(where)38 b Fn(p)h Fu(=)g(\()p Fn(p)995 3654 y Ft(1)1034 3640 y Fn(;)15 b(:)g(:)g(:)i(;)e(p)1282 3654 y Fm(n)1329 3640 y Fu(\),)41 b Fn(q)h Fu(=)d(\()p Fn(q)1699 3654 y Ft(1)1738 3640 y Fn(;)15 b(:)g(:)g(:)i(;)e(q)1981 3654 y Fm(n)2028 3640 y Fu(\),)41 b Fn(r)g Fu(=)e(\()p Fn(r)2397 3654 y Ft(1)2437 3640 y Fn(;)15 b(r)2518 3654 y Fm(n)2565 3640 y Fu(\))39 b(and)f(where)g Fn(D)j Fu(is)d(the)448 3753 y(2)p Fn(d)21 b Fk(\002)f Fu(2)p Fn(d)31 b Fu(matrix)f(giv)m(en)g (b)m(y)1447 4018 y Fn(D)43 b Fu(=)1676 3874 y Fh( )1783 3961 y Fu(1)21 b(+)f Fn(\021)166 b Fu(0)1863 4074 y(0)d(1)21 b Fk(\000)f Fn(\021)2318 3874 y Fh(!)2414 4018 y Fn(:)882 b Fu(\(6\))448 4282 y(The)33 b(function)f Fn(G)h Fu(is)g(a)h(Liapuno)m (v)e(function,)h(non-increasing)f(in)g(time,)i(for)f(the)h(de-)448 4395 y(terministic)45 b(part)h(of)h(the)g(\015o)m(w)f(\(5\).)91 b(If)46 b(the)h(system)f(is)g(in)f(equilibrium,)h(i.e,)51 b(if)448 4508 y Fn(T)501 4522 y Ft(1)582 4508 y Fu(=)42 b Fn(T)748 4522 y Fm(n)836 4508 y Fu(=)f Fn(")g Fu(and)e Fn(\021)45 b Fu(=)c(0,)j(it)39 b(is)g(not)i(di\016cult)d(to)j(c)m(hec)m (k)g(that)g(the)f(generalized)448 4621 y(Gibbs)29 b(measure)1357 4734 y Fn(\026)1412 4748 y Fm(")1474 4734 y Fu(=)c Fn(Z)1639 4697 y Fl(\000)p Ft(1)1748 4734 y Fu(exp)15 b(\()p Fk(\000)p Fn(G)p Fu(\()p Fn(p;)g(q)s(;)g(r)s Fu(\))p Fn(=")p Fu(\))i Fn(;)793 b Fu(\(7\))448 4901 y(is)30 b(an)g(in)m(v)-5 b(arian)m(t)30 b(measure)g(for)g(the)h(Mark)m(o)m(v)h(pro)s(cess)e (solving)f(Eq.)h(\(5\).)1920 5225 y(3)p eop %%Page: 4 4 4 3 bop 589 573 a Fu(If)27 b(the)g(temp)s(erature)f(of)h(the)g(reserv)m (oirs)f(are)i(not)f(iden)m(tical,)f(no)h(explicit)e(form)m(ula)448 686 y(for)35 b(the)g(in)m(v)-5 b(arian)m(t)33 b(measure)i Fn(\026)1552 700 y Fm(T)1593 709 y Fj(1)1627 700 y Fm(;T)1688 708 y Ff(n)1769 686 y Fu(can)g(b)s(e)f(giv)m(en,)i(in)e(general.)53 b(It)35 b(is)f(the)h(goal)g(of)448 799 y(this)26 b(pap)s(er)f(to)j(pro) m(vide)d(a)i(v)-5 b(ariational)26 b(principle)d(for)j(the)h(leading)f (asymptotic)h(form)448 912 y(for)g Fn(\026)639 926 y Fm(T)680 935 y Fj(1)715 926 y Fm(;T)776 934 y Ff(n)822 912 y Fu(,)h(at)g(lo)m(w)f(temp)s(erature,)h Fn(")e Fk(!)f Fu(0.)40 b(T)-8 b(o)27 b(suggest)h(what)g Fn(\026)2705 926 y Fm(T)2746 935 y Fj(1)2780 926 y Fm(;T)2841 934 y Ff(n)2915 912 y Fu(lo)s(oks)e(lik)m(e,)i(w)m(e)448 1024 y(observ)m(e)i(that)f(a)g(t)m(ypical)g(con\014guration)f(of)h(a)g (reserv)m(oir)f(has)h(in\014nite)d(energy)-8 b(,)31 b(there-)448 1137 y(fore)26 b(the)g(reserv)m(oir)f(do)s(es)g(not)h(only)f(acts)i(as) f(a)g(sink)e(of)i(energy)g(but)f(true)g(\015uctuations)448 1250 y(can)33 b(tak)m(e)h(place.)46 b(The)32 b(ph)m(ysical)f(picture)h (is)f(as)h(follo)m(ws:)44 b(the)33 b(system)f(sp)s(ends)f(most)448 1363 y(of)25 b(the)g(time)f(v)m(ery)h(close)g(to)g(the)g(critical)f (set)h(of)g Fn(G)f Fu(\(in)f(fact)j(close)f(to)g(a)g(stable)g(equilib-) 448 1476 y(rium\))j(and)g(v)m(ery)h(rarely)f(\(t)m(ypically)h(after)g (an)g(exp)s(onen)m(tial)f(time\))h(an)f(excursion)g(far)448 1589 y(a)m(w)m(a)m(y)33 b(from)e(the)g(equilibria)c(o)s(ccurs.)42 b(This)30 b(picture)f(brings)h(us)g(in)m(to)h(the)g(framew)m(ork)448 1702 y(of)25 b(rare)f(ev)m(en)m(ts,)j(hence)d(in)m(to)g(the)g(theory)g (of)h(large)f(deviations)f(and)g(more)h(sp)s(eci\014cally)448 1815 y(the)i(F)-8 b(reidlin-W)g(en)m(tzell)25 b(theory)h([8)q(])g(of)g (small)f(random)g(p)s(erturbations)e(of)k(dynamical)448 1928 y(systems.)589 2041 y(In)j(the)h(follo)m(wing)e(w)m(e)i(emplo)m(y) f(notation)h(whic)m(h)e(is)g(essen)m(tially)h(that)h(of)g([8].)42 b(Let)448 2154 y Fk(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))32 b(denote)f(the)f(Banac)m(h)h(space)f(of)g(con)m(tin)m(uous)g(functions) e(\(paths\))j(with)d(v)-5 b(al-)448 2267 y(ues)31 b(in)f Fg(R)791 2234 y Ft(2)p Fm(d)p Ft(\()p Fm(n)p Ft(+1\))1087 2267 y Fu(equipp)s(ed)e(with)i(the)i(uniform)d(top)s(ology)-8 b(.)44 b(W)-8 b(e)33 b(in)m(tro)s(duce)d(the)h(fol-)448 2390 y(lo)m(wing)25 b(functional)g Fn(I)1199 2342 y Ft(\()p Fm(\021)r Ft(\))1192 2418 y Fm(x;T)1332 2390 y Fu(on)h(the)g(set)h(of)f (paths)g Fk(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\):)40 b(If)26 b Fn(\036)p Fu(\()p Fn(t)p Fu(\))g(=)f(\()p Fn(p)p Fu(\()p Fn(t)p Fu(\))p Fn(;)15 b(q)s Fu(\()p Fn(t)p Fu(\))p Fn(;)g(r)s Fu(\()p Fn(t)p Fu(\)\))448 2515 y(has)30 b(one)h Fn(L)839 2482 y Ft(2)878 2515 y Fu(-deriv)-5 b(ativ)m(e)31 b(with)e(resp)s(ect)h(to)h(time)f(and)g(satis\014es)g Fn(\036)p Fu(\(0\))c(=)f Fn(x)31 b Fu(w)m(e)g(set)844 2771 y Fn(I)891 2723 y Ft(\()p Fm(\021)r Ft(\))884 2798 y Fm(x;T)999 2771 y Fu(\()p Fn(\036)p Fu(\))41 b(=)1357 2709 y(1)p 1285 2750 191 4 v 1285 2833 a(4)p Fn(\015)5 b(\025)1435 2807 y Ft(2)1500 2656 y Fh(Z)1583 2682 y Fm(T)1546 2845 y Ft(0)1638 2771 y Fu(\()14 b(_)-39 b Fn(r)23 b Fu(+)d Fn(\015)5 b(\025)1933 2733 y Ft(2)1973 2771 y Fk(r)2049 2785 y Fm(r)2086 2771 y Fn(G)p Fu(\))p Fn(D)2271 2733 y Fl(\000)p Ft(1)2365 2771 y Fu(\()14 b(_)-39 b Fn(r)24 b Fu(+)19 b Fn(\015)5 b(\025)2660 2733 y Ft(2)2700 2771 y Fk(r)2776 2785 y Fm(r)2814 2771 y Fn(G)p Fu(\))p Fn(dt)15 b(;)280 b Fu(\(8\))448 3014 y(if)1147 3127 y(_)-42 b Fn(q)s Fu(\()p Fn(t)p Fu(\))26 b(=)e Fk(r)1474 3141 y Fm(p)1514 3127 y Fn(G)p Fu(\()p Fn(\036)p Fu(\()p Fn(t)p Fu(\)\))15 b Fn(;)125 b Fu(_)-43 b Fn(p)p Fu(\()p Fn(t)p Fu(\))26 b(=)f Fk(\000r)2378 3141 y Fm(q)2415 3127 y Fn(G)p Fu(\()p Fn(\036)p Fu(\()p Fn(t)p Fu(\)\))15 b Fn(;)567 b Fu(\(9\))448 3313 y(and)26 b Fn(I)668 3265 y Ft(\()p Fm(\021)r Ft(\))661 3341 y Fm(x;T)775 3313 y Fu(\()p Fn(\036)p Fu(\))g(=)f(+)p Fk(1)h Fu(otherwise.)39 b(Notice)27 b(that)g Fn(I)2169 3265 y Ft(\()p Fm(\021)r Ft(\))2162 3341 y Fm(x;T)2276 3313 y Fu(\()p Fn(\036)p Fu(\))f(=)f(0)i(if)e(and)h(only)f(if)g Fn(\036)p Fu(\()p Fn(t)p Fu(\))i(is)f(a)448 3426 y(solution)h(of)h(Eq.)g(\(5\))h(with)e (the)h(temp)s(erature)g Fn(")g Fu(set)h(equal)f(to)g(zero.)41 b(The)28 b(functional)448 3555 y Fn(I)495 3507 y Ft(\()p Fm(\021)r Ft(\))488 3582 y Fm(x;T)631 3555 y Fu(is)g(called)g(a)h(rate) h(function)d(and)i(it)f(describ)s(e,)g(in)f(the)i(sense)g(of)g(large)g (deviation,)448 3668 y(the)34 b(probabilit)m(y)e(of)i(the)g(path)f Fn(\036)p Fu(:)48 b(roughly)32 b(sp)s(eaking,)i(as)g Fn(")e Fk(!)e Fu(0,)36 b(the)e(asymptotic)448 3781 y(probabilit)m(y)28 b(of)j(the)f(path)g Fn(\036)h Fu(is)e(giv)m(en)i(b)m(y)1577 3996 y(exp)1731 3902 y Fh(\020)1781 3996 y Fk(\000)p Fn(I)1899 3948 y Ft(\()p Fm(\021)r Ft(\))1892 4023 y Fm(x;T)2006 3996 y Fu(\()p Fn(\036)p Fu(\))p Fn(=")2217 3902 y Fh(\021)2283 3996 y Fn(:)967 b Fu(\(10\))448 4230 y(F)-8 b(or)31 b Fn(x;)15 b(y)29 b Fk(2)c Fg(R)941 4197 y Ft(2)p Fm(d)p Ft(\()p Fm(n)p Ft(+1\))1235 4230 y Fu(w)m(e)31 b(de\014ne)f Fn(V)1705 4197 y Ft(\()p Fm(\021)r Ft(\))1802 4230 y Fu(\()p Fn(x;)15 b(y)s Fu(\))31 b(as)1280 4450 y Fn(V)1354 4413 y Ft(\()p Fm(\021)r Ft(\))1450 4450 y Fu(\()p Fn(x;)15 b(y)s Fu(\))42 b(=)55 b(inf)1813 4510 y Fm(T)10 b(>)p Ft(0)2064 4450 y Fu(inf)1969 4514 y Fm(\036)p Ft(:)p Fm(\036)p Ft(\()p Fm(T)g Ft(\)=)p Fm(y)2286 4450 y Fn(I)2333 4402 y Ft(\()p Fm(\021)r Ft(\))2326 4478 y Fm(x;T)2440 4450 y Fu(\()p Fn(\036)p Fu(\))15 b Fn(;)671 b Fu(\(11\))448 4721 y(and)30 b(for)g(an)m(y)h(sets)g Fn(B)5 b Fu(,)30 b Fn(C)i Fk(2)24 b Fg(R)1503 4688 y Ft(2)p Fm(d)p Ft(\()p Fm(n)p Ft(+1\))1798 4721 y Fu(w)m(e)31 b(set)1285 4926 y Fn(V)1358 4888 y Ft(\()p Fm(\021)r Ft(\))1455 4926 y Fu(\()p Fn(B)5 b(;)15 b(C)7 b Fu(\))40 b(=)136 b(inf)1862 4985 y Fm(x)p Fl(2)p Fm(B)s Ft(;)p Fm(y)r Fl(2)p Fm(C)2180 4926 y Fn(V)2253 4888 y Ft(\()p Fm(\021)r Ft(\))2350 4926 y Fu(\()p Fn(x;)15 b(y)s Fu(\))g Fn(:)675 b Fu(\(12\))1920 5225 y(4)p eop %%Page: 5 5 5 4 bop 448 573 a Fu(The)33 b(function)f Fn(V)1070 540 y Ft(\()p Fm(\021)r Ft(\))1167 573 y Fu(\()p Fn(x;)15 b(y)s Fu(\))35 b(represen)m(ts,)f(roughly)e(sp)s(eaking,)h(the)h(cost)g (to)g(bring)e(the)448 686 y(system)24 b(from)e Fn(x)h Fu(to)h Fn(y)i Fu(\(in)d(an)g(arbitrary)f(amoun)m(t)h(of)h(time\).)38 b(W)-8 b(e)24 b(in)m(tro)s(duce)f(an)g(equiv-)448 799 y(alence)37 b(relation)e(on)g(the)h(phase)g(space)g Fg(R)1944 766 y Ft(2)p Fm(d)p Ft(\()p Fm(n)p Ft(+1\))2208 799 y Fu(:)52 b(w)m(e)36 b(sa)m(y)h Fn(x)d Fk(\030)h Fn(y)j Fu(if)d Fn(V)3024 766 y Ft(\()p Fm(\021)r Ft(\))3121 799 y Fu(\()p Fn(x;)15 b(y)s Fu(\))35 b(=)448 912 y Fn(V)522 879 y Ft(\()p Fm(\021)r Ft(\))618 912 y Fu(\()p Fn(y)s(;)15 b(x)p Fu(\))27 b(=)d(0.)39 b(W)-8 b(e)23 b(divide)c(the)j(critical)f (set)h Fn(K)32 b Fu(=)25 b Fk(f)p Fn(x)15 b Fu(;)31 b Fk(r)p Fn(G)p Fu(\()p Fn(x)p Fu(\))25 b(=)g(0)p Fk(g)e Fu(\(ab)s(out)e(whic)m(h)448 1024 y(the)35 b(in)m(v)-5 b(arian)m(t)33 b(measure)i(concen)m(trates\))h(according)f(to)g(this)e (equiv)-5 b(alence)34 b(relation:)448 1137 y(w)m(e)26 b(ha)m(v)m(e)h Fn(K)32 b Fu(=)25 b Fk([)1048 1151 y Fm(i)1076 1137 y Fn(K)1153 1151 y Fm(i)1207 1137 y Fu(with)f Fn(x)h Fk(\030)g Fn(y)k Fu(if)24 b Fn(x)i Fk(2)e Fn(K)1974 1151 y Fm(i)2003 1137 y Fn(;)15 b(y)28 b Fk(2)d Fn(K)2279 1151 y Fm(i)2333 1137 y Fu(and)g Fn(x)g Fk(6\030)g Fn(y)k Fu(if)24 b Fn(x)i Fk(2)e Fn(K)3070 1151 y Fm(i)3099 1137 y Fn(;)15 b(y)28 b Fk(2)d Fn(K)3375 1151 y Fm(j)3412 1137 y Fu(,)448 1250 y Fn(i)h Fk(6)p Fu(=)f Fn(j)5 b Fu(.)589 1363 y(Our)40 b(\014rst)f(assumption)g(is)h(on)g(the)h (existence)g(of)g(an)f(in)m(v)-5 b(arian)m(t)40 b(measure,)j(the)448 1476 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Fu(\()p Fn(q)s Fu(\))i(is)e(strictly)g(con) m(v)m(ex.)448 4558 y Fg(Remark)35 b(1.3)46 b Fu(The)28 b(condition)g Fg(K2)h Fu(will)d(b)s(e)j(imp)s(ortan)m(t)f(to)i (establish)d(v)-5 b(arious)28 b(reg-)448 4671 y(ularit)m(y)h(prop)s (erties)f(of)i Fn(V)1344 4638 y Ft(\()p Fm(\021)r Ft(\))1441 4671 y Fu(\()p Fn(x;)15 b(y)s Fu(\).)41 b(It)30 b(will)d(allo)m(w)j(to) g(imply)e(sev)m(eral)i(con)m(trollabilit)m(y)448 4784 y(prop)s(erties)35 b(of)h(the)h(con)m(trol)f(system)h(asso)s(ciated)f (with)f(the)i(sto)s(c)m(hastic)g(di\013eren)m(tial)448 4897 y(equations)30 b(\(5\).)1920 5225 y(5)p eop %%Page: 6 6 6 5 bop 589 573 a Fu(F)-8 b(ollo)m(wing)37 b([8)q(],)j(w)m(e)d (consider)g(graphs)f(on)i(the)f(set)h Fk(f)p Fu(1)p Fn(;)15 b(:)g(:)g(:)j(;)d(L)p Fk(g)p Fu(.)62 b(A)38 b(graph)e(con-)448 686 y(sisting)30 b(of)h(arro)m(ws)g Fn(m)26 b Fk(!)g Fn(n)p Fu(,)31 b(\()p Fn(m)26 b Fk(2)g(f)p Fu(1)p Fn(;)15 b(:)g(:)g(:)j(;)d(L)p Fk(g)21 b(n)g(f)p Fn(i)p Fk(g)p Fu(,)33 b Fn(m)26 b Fk(2)g(f)p Fu(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(L)p Fk(g)p Fu(\),)32 b(is)e(called)h(a)448 799 y Fk(f)p Fn(i)p Fk(g)p Fu(-graph)h(if)559 971 y(1.)47 b(Ev)m(ery)30 b(p)s(oin)m(t)g Fn(j)5 b Fu(,)31 b Fn(j)g Fk(6)p Fu(=)25 b Fn(i)30 b Fu(is)g(the)g(initial)e(p)s(oin)m(t)h(of)i(exactly)g(one)g(arro)m(w.) 559 1153 y(2.)47 b(There)29 b(are)i(no)g(closed)f(cycles)g(in)f(the)i (graph.)448 1326 y(W)-8 b(e)35 b(denote)e Fn(G)p Fk(f)p Fn(i)p Fk(g)h Fu(the)g(set)f(of)h Fk(f)p Fn(i)p Fk(g)p Fu(-graphs.)50 b(The)32 b(w)m(eigh)m(t)i(of)f(the)h(set)g Fn(K)2999 1340 y Fm(i)3060 1326 y Fu(is)e(de\014ned)448 1439 y(b)m(y)1098 1552 y Fn(W)1197 1514 y Ft(\()p Fm(\021)r Ft(\))1293 1552 y Fu(\()p Fn(K)1405 1566 y Fm(i)1434 1552 y Fu(\))41 b(=)108 b(min)1621 1615 y Fm(g)r Fl(2)p Fm(G)p Ft(\()p Fl(f)p Fm(i)p Fl(g)p Ft(\))1993 1471 y Fh(X)1924 1647 y Fm(m)p Fl(!)p Fm(n)p Fl(2)p Fm(g)2198 1552 y Fn(V)2271 1514 y Ft(\()p Fm(\021)r Ft(\))2368 1552 y Fu(\()p Fn(K)2480 1566 y Fm(m)2547 1552 y Fn(;)15 b(K)2664 1566 y Fm(n)2711 1552 y Fu(\))g Fn(:)489 b Fu(\(13\))589 1784 y(Our)30 b(main)f(result)g(is)h(the)g(follo)m(wing:)448 1957 y Fg(Theorem)k(1.4)46 b Fe(Under)j(the)g(c)-5 b(onditions)51 b Fg(K1)d Fe(and)i Fg(K2)f Fe(the)g(invariant)h(me)-5 b(asur)g(e)448 2070 y Fn(\026)503 2084 y Fm(T)544 2093 y Fj(1)579 2084 y Fm(;T)640 2092 y Ff(n)729 2070 y Fu(=)42 b Fn(\026)897 2084 y Fm(";\021)1034 2070 y Fe(of)g(the)h(Markov)f(pr)-5 b(o)g(c)g(ess)45 b(\(5\))e(has)g(the)g(fol)5 b(lowing)43 b(asymptotic)i(b)-5 b(e-)448 2183 y(havior:)61 b(F)-7 b(or)42 b(any)g(op)-5 b(en)42 b(set)f Fn(D)j Fe(with)e(c)-5 b(omp)g(act)44 b(closur)-5 b(e)42 b(and)g(su\016ciently)g(r)-5 b(e)g(gular)448 2296 y(b)g(oundary)1292 2409 y Fu(lim)1286 2465 y Fm(")p Fl(!)p Ft(0)1440 2409 y Fn(")15 b Fu(log)i Fn(\026)1685 2423 y Fm(";\021)1779 2409 y Fu(\()p Fn(D)s Fu(\))41 b(=)58 b(inf)2079 2468 y Fm(x)p Fl(2)p Fm(D)2241 2409 y Fn(W)2340 2371 y Ft(\()p Fm(\021)r Ft(\))2436 2409 y Fu(\()p Fn(x)p Fu(\))15 b Fn(;)677 b Fu(\(14\))448 2597 y Fe(wher)-5 b(e)685 2783 y 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Fe(satis\014es)h(the)f(b)-5 b(ound,)33 b(for)h Fn(\021)28 b Fk(\025)d Fu(0)p Fe(,)454 3618 y Fu(\(1)c(+)f Fn(\021)s Fu(\))729 3581 y Fl(\000)p Ft(1)839 3524 y Fh(\020)889 3618 y Fn(G)p Fu(\()p Fn(x)p Fu(\))g Fk(\000)g Fu(min)1250 3668 y Fm(x)1361 3618 y Fn(G)p Fu(\()p Fn(x)p Fu(\))1555 3524 y Fh(\021)1645 3618 y Fk(\024)40 b Fn(W)1855 3581 y Ft(\()p Fm(\021)r Ft(\))1951 3618 y Fu(\()p Fn(x)p Fu(\))h Fk(\024)f Fu(\(1)21 b Fk(\000)f Fn(\021)s Fu(\))2500 3581 y Fl(\000)p Ft(1)2611 3524 y Fh(\020)2660 3618 y Fn(G)p Fu(\()p Fn(x)p Fu(\))h Fk(\000)e Fu(min)3021 3668 y Fm(x)3132 3618 y Fn(G)p Fu(\()p Fn(x)p Fu(\))3326 3524 y Fh(\021)3406 3618 y Fn(:)3275 3753 y Fu(\(17\))481 3866 y Fe(and)33 b(a)g(similar)h(b)-5 b(ound)34 b(for)f Fn(\021)c Fk(\024)c Fu(0)p Fe(.)448 4060 y Fg(Remark)35 b(1.5)46 b Fu(Eqs.)38 b(\(16\))27 b(and)e(\(17\))i(imply)c(that)j Fn(\026)2277 4074 y Fm(";\021)2397 4060 y Fu(reduces)f(to)h(the)g(Boltzmann-)448 4173 y(Gibbs)31 b(expression)g Fn(\026)1209 4187 y Fm(")1273 4173 y Fk(\030)d Fu(exp)15 b(\()p Fk(\000)p Fn(G=")p Fu(\))33 b(for)f Fn(\021)g Fk(!)c Fu(0)k(in)f(the)h(lo)m(w)g(temp)s(erature)g(limit.)448 4286 y(Of)k(course,)h(at)g Fn(\021)h Fu(=)c(0,)k(they)e(are)g(actually) f(equal)h(at)g(all)f(temp)s(eratures)h Fn(")p Fu(.)57 b(More-)448 4399 y(o)m(v)m(er)41 b(these)f(equations)f(imply)e(that)j (the)g(relativ)m(e)f(probabilit)m(y)e Fn(\026)2807 4413 y Fm(";\021)2901 4399 y Fu(\()p Fn(x)p Fu(\))p Fn(=\026)3123 4413 y Fm(";\021)3218 4399 y Fu(\()p Fn(y)s Fu(\))j(is)448 4512 y(\(asymptotically\))31 b(b)s(ounded)d(ab)s(o)m(v)m(e)j(and)f(b)s (elo)m(w)g(b)m(y)1365 4746 y(exp)15 b Fk(\000)1605 4627 y Fh(\024)1721 4685 y Fn(G)p Fu(\()p Fn(x)p Fu(\))p 1659 4725 319 4 v 1659 4809 a Fn(")p Fu(\(1)21 b Fk(\006)f Fn(\021)s Fu(\))2007 4746 y Fk(\000)2172 4685 y Fn(G)p Fu(\()p Fn(y)s Fu(\))p 2108 4725 V 2108 4809 a Fn(")p Fu(\(1)h Fk(\007)f Fn(\021)s Fu(\))2436 4627 y Fh(\025)2495 4746 y Fn(;)755 b Fu(\(18\))448 4976 y(so)31 b(that)g(no)f(esp)s (ecially)f(hot)h(or)h(cold)f(sp)s(ots)g(dev)m(elop)g(for)g Fn(\021)f Fk(6)p Fu(=)c(0.)1920 5225 y(6)p eop %%Page: 7 7 7 6 bop 448 573 a Fg(Remark)35 b(1.6)46 b Fu(The)32 b(theorem)h(dra)m (ws)f(hea)m(vily)g(from)g(the)h(large)g(deviations)f(theory)448 686 y(of)f(F)-8 b(reidlin-W)g(en)m(tzell)30 b([8].)42 b(But)31 b(the)g(theory)g(w)m(as)g(dev)m(elop)s(ed)f(for)g(sto)s(c)m (hastic)i(di\013er-)448 799 y(en)m(tial)h(equations)h(with)e(a)h (non-degenerate)i(\(elliptic\))d(generator,)k(but)c(for)h(Eq.)h(\(5\)) 448 912 y(this)c(is)g(not)h(the)g(case)g(since)f(the)h(random)f(force)i (acts)f(only)f(on)h(2)p Fn(d)g Fu(of)g(the)g(2)p Fn(d)p Fu(\()p Fn(n)21 b Fu(+)f(1\))448 1024 y(v)-5 b(ariables.)38 b(A)25 b(large)g(part)g(of)g(this)f(pap)s(er)g(is)g(dev)m(oted)i(to)f (simply)e(extending)h(F)-8 b(reidlin-)448 1137 y(W)g(en)m(tzell)36 b(theory)f(to)g(a)g(class)f(of)h(Mark)m(o)m(v)h(pro)s(cesses)f(con)m (taining)f(our)g(mo)s(del.)52 b(De-)448 1250 y(generate)31 b(di\013usions)c(ha)m(v)m(e)j(b)s(een)f(considered)f(in)g([3)q(])h(but) g(under)e(to)s(o)j(stringen)m(t)f(con-)448 1363 y(ditions)38 b(and)h(also)g(in)f([2)q(])i(but)e(with)h(conditions)f(quite)h (di\013eren)m(t)f(from)h(ours.)68 b(W)-8 b(e)448 1476 y(also)30 b(note)g(that)g(the)f(use)g(of)h(F)-8 b(reidlin-W)g(en)m (tzell)28 b(theory)h(in)f(non-equilibrium)d(statis-)448 1589 y(tical)34 b(mec)m(hanics)g(has)f(b)s(een)h(adv)m(o)s(cated)h(in)e (particular)f(b)m(y)i(Graham)g(\(see)g([10)r(])g(and)448 1702 y(references)e(therein\).)42 b(In)31 b(these)g(applications)f(to)i (non-equilibrium)26 b(statistical)31 b(me-)448 1815 y(c)m(hanics,)d(as) f(in)e([10)q(],)j(the)f(mo)s(dels)e(are)i(mostly)g(tak)m(en)g(as)g (mesoscopic:)40 b(the)26 b(v)-5 b(ariables)448 1928 y(of)34 b(the)f(system)h(describ)s(e)e(some)h(suitably)f(coarse-grained)i(quan) m(tities,)f(whic)m(h)f(\015uc-)448 2041 y(tuate)c(sligh)m(tly)e(around) g(their)g(a)m(v)m(erage)k(v)-5 b(alues.)39 b(In)26 b(con)m(trast)j(to)e (these)h(mo)s(dels,)f(ours)448 2154 y(is)j(en)m(tirely)g(microscopic)g (and)f(deriv)m(ed)h(from)g(\014rst)g(principles)d(and)j(the)h (small-noise)448 2267 y(limit)d(is)i(seen)g(as)h(a)g(lo)m(w-temp)s (erature)f(limit.)589 2441 y(Finally)43 b(w)m(e)i(relate)f(the)h(large) f(deviation)g(functional)f(to)i(a)f(kind)f(of)h(en)m(trop)m(y)448 2554 y(pro)s(duction.)39 b(As)30 b(in)f([7)q(])i(w)m(e)g(de\014ne)e (this)g(en)m(trop)m(y)j(pro)s(duction)c(\006)i(b)m(y)1601 2762 y(\006)40 b(=)h Fk(\000)1900 2701 y Fn(F)1958 2715 y Ft(1)p 1900 2741 98 4 v 1903 2825 a Fn(T)1956 2839 y Ft(1)2027 2762 y Fk(\000)2128 2701 y Fn(F)2186 2715 y Fm(n)p 2128 2741 106 4 v 2131 2825 a Fn(T)2184 2839 y Fm(n)2259 2762 y Fn(;)991 b Fu(\(19\))448 2963 y(where)907 3130 y Fn(F)965 3144 y Ft(1)1005 3130 y Fu(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\))98 b(=)g Fn(p)1602 3144 y Ft(1)1641 3130 y Fu(\()p Fn(r)1717 3144 y Ft(1)1777 3130 y Fk(\000)20 b Fn(\025)1921 3092 y Ft(2)1961 3130 y Fn(q)2002 3144 y Ft(1)2041 3130 y Fu(\))41 b(=)f Fn(\025)2281 3092 y Ft(2)2320 3130 y Fn(p)2366 3144 y Ft(1)2405 3130 y Fk(r)2481 3144 y Fm(r)2513 3153 y Fj(1)2552 3130 y Fn(G)p Fu(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\))g Fn(;)899 3268 y(F)957 3282 y Fm(n)1005 3268 y Fu(\()p Fn(p;)g(q)s(;)g(r)s Fu(\))98 b(=)g Fn(p)1602 3282 y Fm(n)1649 3268 y Fu(\()p Fn(r)1725 3282 y Fm(n)1793 3268 y Fk(\000)19 b Fn(\025)1936 3230 y Ft(2)1976 3268 y Fn(q)2017 3282 y Fm(n)2063 3268 y Fu(\))41 b(=)f Fn(\025)2303 3230 y Ft(2)2343 3268 y Fn(p)2389 3282 y Fm(n)2436 3268 y Fk(r)2512 3282 y Fm(r)2544 3290 y Ff(n)2590 3268 y Fn(G)p Fu(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\))g Fn(;)289 b Fu(\(20\))448 3434 y(are)39 b(the)g(energy)g(\015o)m (ws)f(from)g(the)h(c)m(hain)f(to)h(the)f(resp)s(ectiv)m(e)h(reserv)m (oirs.)64 b(In)38 b([6)q(])h(it)448 3547 y(is)i(sho)m(wn)g(that)h Fn(\026)1101 3561 y Fm(T)1142 3570 y Fj(1)1176 3561 y Fm(;T)1237 3569 y Ff(n)1284 3547 y Fu(\(\006\))i Fk(\025)g Fu(0)e(and)f Fn(\026)1909 3561 y Fm(T)1950 3570 y Fj(1)1984 3561 y Fm(;T)2045 3569 y Ff(n)2091 3547 y Fu(\(\006\))k(=)e(0)f(if)f (and)g(only)f(if)h Fn(T)3114 3561 y Ft(1)3197 3547 y Fu(=)j Fn(T)3365 3561 y Fm(n)3412 3547 y Fu(.)448 3660 y(This)29 b(implies)f(that,)j(in)e(the)i(stationary)f(state,)i(energy)f (is)e(\015o)m(wing)h(from)g(the)g(hotter)448 3773 y(reserv)m(oir)f (bath)f(to)i(the)f(colder)f(one.)40 b(With)29 b(the)f(parameters)i Fn(")f Fu(and)f Fn(\021)k Fu(as)d(de\014ned)e(in)448 3886 y(\(4\))32 b(w)m(e)f(de\014ne)e(\002)h(b)m(y)1255 4032 y(1)p 1255 4073 46 4 v 1257 4156 a Fn(")1311 4094 y Fu(\002)40 b(=)g(\006)g(=)1760 4032 y(1)p 1760 4073 V 1762 4156 a Fn(")1831 3975 y Fh(\022)1892 4094 y Fk(\000)2026 4032 y Fn(F)2084 4046 y Ft(1)p 1973 4073 205 4 v 1973 4156 a Fu(1)20 b(+)g Fn(\021)2208 4094 y Fk(\000)2358 4032 y Fn(F)2416 4046 y Fm(n)p 2308 4073 V 2308 4156 a Fu(1)h Fk(\000)f Fn(\021)2523 3975 y Fh(\023)2615 4094 y Fn(:)635 b Fu(\(21\))448 4299 y(In)25 b(order)h(to)g(sho)m(w)g(the)g (relation)f(b)s(et)m(w)m(een)i(the)f(rate)g(function)f Fn(I)2672 4313 y Fm(x;T)2812 4299 y Fu(and)g(the)h(en)m(trop)m(y)448 4412 y(pro)s(duction)31 b(\006,)i(w)m(e)f(in)m(tro)s(duce)g(the)g (time-rev)m(ersal)h Fn(J)9 b Fu(,)33 b(whic)m(h)e(is)h(the)g(in)m(v)m (olution)f(on)448 4525 y(the)44 b(phase)f(space)h Fg(R)1218 4492 y Ft(2)p Fm(d)p Ft(\()p Fm(n)p Ft(+1\))1525 4525 y Fu(giv)m(en)g(b)m(y)f Fn(J)9 b Fu(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\))48 b(=)f(\()p Fk(\000)p Fn(p;)15 b(q)s(;)g(r)s Fu(\).)80 b(The)43 b(follo)m(wing)448 4638 y(sho)m(ws)32 b(that)h(the)f(v)-5 b(alue)31 b(of)h(the)g(rate)h(function)e(of)h(a)g (path)g Fn(\036)g Fu(b)s(et)m(w)m(een)h Fn(x)f Fu(and)f Fn(y)k Fu(and)448 4751 y(is)27 b(equal)f(\(up)h(to)h(a)f(b)s(oundary)f (term\))h(to)h(the)g(v)-5 b(alue)26 b(of)i(the)f(rate)h(function)e(of)h (the)h(the)448 4863 y(time)f(rev)m(ersed)h(path)1224 4839 y(~)1213 4863 y Fn(\036)f Fu(b)s(et)m(w)m(een)h Fn(J)9 b(y)30 b Fu(and)c Fn(J)9 b(x)28 b Fu(min)m(us)d(the)j(en)m(trop) m(y)f(pro)s(duced)f(along)448 4976 y(this)k(path.)1920 5225 y(7)p eop %%Page: 8 8 8 7 bop 448 573 a Fg(Prop)s(osition)37 b(1.7)46 b Fe(L)-5 b(et)34 b Fn(\036)p Fu(\()p Fn(t)p Fu(\))c Fk(2)e(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))36 b Fe(with)f Fn(\036)p Fu(\(0\))30 b(=)f Fn(x)34 b Fe(and)h Fn(\036)p Fu(\()p Fn(T)13 b Fu(\))29 b(=)g Fn(y)s Fe(.)47 b(Either)448 702 y Fn(I)495 654 y Ft(\()p Fm(\021)r Ft(\))488 729 y Fm(x;T)603 702 y Fu(\()p Fn(\036)p Fu(\))41 b(=)f(+)p Fk(1)32 b Fe(or)h(we)g(have)625 928 y Fn(I)672 881 y Ft(\(0\))665 956 y Fm(x;T)780 928 y Fu(\()p Fn(\036)p Fu(\))99 b(=)e Fn(I)1218 881 y Ft(\(0\))1211 956 y Fm(J)6 b(y)r(;T)1368 928 y Fu(\()1415 901 y Fh(e)1403 928 y Fn(\036)q Fu(\))20 b(+)g Fn(G)p Fu(\()p Fn(y)s Fu(\))g Fk(\000)g Fn(G)p Fu(\()p Fn(x)p Fu(\))15 b Fn(;)234 b Fu(if)39 b Fn(\021)29 b Fu(=)c(0)15 b Fn(;)625 1136 y(I)672 1088 y Ft(\()p Fm(\021)r Ft(\))665 1163 y Fm(x;T)780 1136 y Fu(\()p Fn(\036)p Fu(\))99 b(=)e Fn(I)1218 1088 y Ft(\()p Fm(\021)r Ft(\))1211 1163 y Fm(J)6 b(y)r(;T)1368 1136 y Fu(\()1415 1109 y Fh(e)1403 1136 y 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(osition)f(1.7)k(are)f(an)f(asymptotic)g(v)m(ersion)g(of)g(iden)m (tities)448 1779 y(whic)m(h)f(app)s(ear)g(in)f(v)-5 b(arious)21 b(forms)g(in)f(the)i(literature.)37 b(These)21 b(iden)m(tities)g(are)h (the)g(basic)448 1892 y(ingredien)m(t)k(needed)h(for)g(the)h(pro)s(of)e (of)i(the)f(Galla)m(v)m(otti-Cohen)h(\015uctuation)f(theorem)448 2005 y([4)q(,)e(9)q(])g(for)g(sto)s(c)m(hastic)h(dynamics)e([17)q(,)i (18)q(,)f(20)q(])h(and)e(app)s(ear)h(as)g(w)m(ell)g(in)e(the)j (Jarsynski)448 2118 y(non-equilibrium)g(w)m(ork)k(relation)g([13)q(].) 589 2230 y(The)36 b(pap)s(er)e(is)h(organized)h(as)g(follo)m(ws:)50 b(In)35 b(Section)h(2)g(w)m(e)g(recall)f(the)h(large)g(de-)448 2343 y(viation)29 b(principle)d(for)k(the)g(paths)f(of)h(Mark)m(o)m (vian)h(sto)s(c)m(hastic)f(di\013eren)m(tial)e(equation)448 2456 y(and)37 b(using)f(metho)s(ds)g(from)h(con)m(trol)h(theory)f(w)m (e)h(pro)m(v)m(e)g(the)f(required)f(regularities)448 2569 y(prop)s(erties)23 b(of)h(the)g(function)f Fn(V)1540 2536 y Ft(\()p Fm(\021)r Ft(\))1637 2569 y Fu(\()p Fn(x;)15 b(y)s Fu(\))25 b(de\014ned)e(in)f(Eq.)i(\(11\).)41 b(Section)24 b(3)g(is)f(dev)m(oted)448 2682 y(to)38 b(an)f(extension)f(of)h(F)-8 b(reidlin-W)g(en)m(tzell)36 b(results)g(to)h(a)g(certain)g(class)g(of)g (di\013usions)448 2795 y(with)30 b(h)m(yp)s(o)s(elliptic)d(generators)k (\(Theorem)g(3.3\):)42 b(w)m(e)31 b(giv)m(e)g(a)g(set)g(of)g (conditions)e(un-)448 2908 y(der)h(whic)m(h)f(the)i(asymptotic)g(b)s (eha)m(vior)e(of)i(the)g(in)m(v)-5 b(arian)m(t)29 b(measure)i(is)e(pro) m(v)m(ed.)42 b(The)448 3021 y(result)28 b(of)g(Section)h(2)f(implies)e (that)j(our)f(mo)s(del,)g(under)f(Assumptions)f Fg(K1)j Fu(and)f Fg(K2)p Fu(,)448 3134 y(satis\014es)33 b(the)g(conditions)e (of)i(Theorem)f(3.3.)50 b(In)32 b(Section)g(4)i(w)m(e)f(pro)m(v)m(e)g (the)g(equalit)m(y)448 3247 y(\(16\))j(and)e(the)h(b)s(ound)e(\(17\))j (whic)m(h)d(dep)s(end)g(on)h(the)h(particular)e(prop)s(erties)g(of)i (our)448 3360 y(mo)s(del.)448 3644 y Fo(2)135 b(Large)46 b(deviations)g(and)f(Con)l(trol)h(Theory)448 3847 y Fu(In)36 b(this)g(section)h(w)m(e)g(\014rst)f(recall)g(a)h(certain)f(n)m(um)m(b) s(ers)f(of)i(concepts)h(and)e(theorems)448 3960 y(whic)m(h)d(will)e(b)s (e)i(cen)m(tral)h(in)e(our)h(analysis:)46 b(The)33 b(large)h(deviation) e(principle)e(for)k(the)448 4073 y(sample)h(path)h(of)g(di\013usions)d (in)m(tro)s(duced)i(b)m(y)h(Sc)m(hilder)d(for)j(the)g(Bro)m(wnian)f (motion)448 4186 y([22)q(])27 b(and)g(generalized)f(to)i(arbitrary)d (di\013usion)f(b)m(y)j([8)q(,)g(1,)g(25)q(])g(\(see)h(also)f([3]\),)i (and)d(the)448 4299 y(relationship)31 b(b)s(et)m(w)m(een)j(di\013usion) c(pro)s(cesses)j(and)g(con)m(trol)g(theory)-8 b(,)35 b(exempli\014ed)c(b)m(y)448 4412 y(the)36 b(Supp)s(ort)e(Theorem)h(of)h (Stro)s(o)s(c)m(k)g(and)f(V)-8 b(aradhan)36 b([24)q(].)58 b(With)35 b(these)h(to)s(ols)g(w)m(e)448 4525 y(then)28 b(pro)m(v)m(e)h(sev)m(eral)f(prop)s(erties)f(of)h(the)g(dynamics)e(for) i(our)g(mo)s(del.)38 b(W)-8 b(e)30 b(pro)m(v)m(e)e(that)448 4638 y(\\at)k(zero)g(temp)s(erature")g(the)f(\(deterministic\))e (dynamics)h(giv)m(en)h(b)m(y)g(is)f(dissipativ)m(e:)448 4751 y(the)38 b Fn(!)s Fu(-limit)d(set)j(is)f(the)g(set)h(of)g(the)g (critical)e(p)s(oin)m(t)h(of)g Fn(G)p Fu(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\).)62 b(W)-8 b(e)39 b(also)e(pro)m(v)m(e)448 4863 y(sev)m(eral)e(prop)s(erties)d(of)i(the)g(con)m(trol)h(system)f (asso)s(ciated)g(to)h(Eq.)f(\(5\):)49 b(a)34 b(lo)s(cal)f(con-)448 4976 y(trol)j(prop)s(ert)m(y)g(around)g(the)h(critical)e(p)s(oin)m(ts)h (of)g Fn(G)p Fu(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\))37 b(and)f(roughly)f(sp)s(eaking)1920 5225 y(8)p eop %%Page: 9 9 9 8 bop 448 573 a Fu(a)36 b(global)f(\\smo)s(othness")h(prop)s(ert)m(y) f(of)h(the)g(w)m(eigh)m(t)g(of)g(the)g(paths)f(b)s(et)m(w)m(een)h Fn(x)g Fu(and)448 686 y Fn(y)s Fu(,)g(when)d Fn(x)h Fu(and)g Fn(y)j Fu(v)-5 b(ary)d(.)53 b(The)33 b(cen)m(tral)i(h)m(yp)s(othesis)e (in)g(this)g(analysis)g(is)g(condition)448 799 y Fg(K2)p Fu(:)47 b(this)32 b(condition)g(implies)f(the)j(h)m(yp)s(o)s (ellipticit)m(y)-8 b(,)31 b([12)q(],)k(of)e(the)h(generator)g(of)g(the) 448 912 y(Mark)m(o)m(v)43 b(semigroup)c(asso)s(ciated)i(to)h(Eq.)e (\(5\),)45 b(but)40 b(it)g(implies)e(in)h(fact)j(a)f(kind)e(of)448 1024 y(global)34 b(h)m(yp)s(o)s(ellipticit)m(y)d(whic)m(h)i(will)f(b)s (e)i(used)f(here)i(to)g(pro)m(v)m(e)g(the)g(aforemen)m(tioned)448 1137 y(prop)s(erties)29 b(of)i(the)f(dynamics.)448 1381 y Fd(2.1)113 b(Sample)37 b(P)m(aths)g(Large)h(Deviation)e(and)j(Con)m (trol)d(Theory)448 1552 y Fu(Let)31 b(us)f(consider)f(the)i(sto)s(c)m (hastic)g(di\013eren)m(tial)e(equation)1278 1754 y Fn(dx)p Fu(\()p Fn(t)p Fu(\))41 b(=)f Fn(Y)20 b Fu(\()p Fn(x)p Fu(\))p Fn(dt)h Fu(+)f Fn(")2061 1717 y Ft(1)p Fm(=)p Ft(2)2171 1754 y Fn(\033)s Fu(\()p Fn(x)p Fu(\))p Fn(dw)r Fu(\()p Fn(t)p Fu(\))15 b Fn(;)670 b Fu(\(23\))448 1956 y(where)38 b Fn(x)f Fk(2)h Fn(X)45 b Fu(=)38 b Fg(R)1214 1923 y Fm(n)1261 1956 y Fu(,)i Fn(Y)20 b Fu(\()p Fn(x)p Fu(\))38 b(is)f(a)h Fk(C)1794 1923 y Fl(1)1907 1956 y Fu(v)m(ector)i(\014eld,)e Fn(w)r Fu(\()p Fn(t)p Fu(\))h(is)e(an)h (m-dimensional)448 2069 y(Wiener)f(pro)s(cess)g(and)g Fn(\033)s Fu(\()p Fn(x)p Fu(\))h(is)e(a)i Fk(C)1728 2036 y Fl(1)1840 2069 y Fu(map)f(from)g Fg(R)2349 2036 y Fm(m)2453 2069 y Fu(to)h Fg(R)2649 2036 y Fm(n)2696 2069 y Fu(.)62 b(Let)38 b Fk(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))39 b(de-)448 2182 y(note)33 b(the)e(Banac)m(h)i(space)f(of)g(con)m(tin)m (uous)g(functions)e(with)g(v)-5 b(alues)31 b(in)g Fg(R)3000 2149 y Fm(n)3079 2182 y Fu(equipp)s(ed)448 2295 y(with)24 b(the)h(uniform)d(top)s(ology)-8 b(.)40 b(Let)25 b Fn(L)1755 2262 y Ft(2)1795 2295 y Fu(\([0)p Fn(;)15 b(T)e Fu(]\))26 b(denote)g(the)f(set)g(of)g(square)f(in)m(tegrable)448 2408 y(functions)33 b(with)g(v)-5 b(alues)34 b(in)f Fg(R)1518 2375 y Fm(m)1620 2408 y Fu(and)h Fn(H)1877 2422 y Ft(1)1916 2408 y Fu(\([0)p Fn(;)15 b(T)e Fu(]\))36 b(denote)f(the)f(space)h(of)g (absolutely)448 2521 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b(ate)34 b(function)f Fn(I)1031 700 y Fm(x;T)1145 686 y Fu(\()p Fn(f)10 b Fu(\))p Fe(:)556 873 y(1.)46 b(F)-7 b(or)33 b(any)h(c)-5 b(omp)g(act)34 b Fn(K)e Fk(\032)25 b Fn(X)40 b Fe(and)34 b(any)f(close)-5 b(d)34 b Fn(F)k Fk(\032)25 b(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))p Fe(,)1105 1078 y Fu(lim)i(sup)1175 1152 y Fm(")p Fl(!)p Ft(0)1399 1078 y Fu(log)23 b(sup)1531 1155 y Fm(x)p Fl(2)p Fm(K)1697 1078 y Fn(P)1755 1092 y Fm(x)1799 1078 y Fu(\()p Fn(x)1886 1092 y Fm(")1949 1078 y Fk(2)i Fn(F)13 b Fu(\))40 b Fk(\024)h(\000)35 b Fu(inf)2379 1137 y Fm(x)p Fl(2)p Fm(K)2561 1078 y Fu(inf)2545 1138 y Fm(\036)p Fl(2)p Fm(F)2703 1078 y Fn(I)2743 1092 y Fm(x;T)2857 1078 y Fu(\()p Fn(\036)p Fu(\))p Fn(:)269 b Fu(\(25\))556 1372 y Fe(2.)46 b(F)-7 b(or)33 b(any)h(c)-5 b(omp)g(act)34 b Fn(K)e Fk(\032)25 b Fn(X)40 b Fe(and)34 b(any)f(op)-5 b(en)34 b Fn(G)25 b Fk(\032)f(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))p Fe(,)1118 1576 y Fu(lim)i(inf)1175 1633 y Fm(")p Fl(!)p Ft(0)1385 1576 y Fu(log)37 b(inf)1518 1636 y Fm(x)p Fl(2)p Fm(K)1684 1576 y Fn(P)1742 1590 y Fm(x)1786 1576 y Fu(\()p Fn(x)1873 1590 y Fm(")1935 1576 y Fk(2)25 b Fn(G)p Fu(\))40 b Fk(\025)g(\000)22 b Fu(sup)2365 1653 y Fm(x)p Fl(2)p Fm(K)2548 1576 y Fu(inf)2531 1636 y Fm(\036)p Fl(2)p Fm(G)2690 1576 y Fn(I)2730 1590 y Fm(x;T)2845 1576 y Fu(\()p Fn(\036)p Fu(\))p Fn(:)281 b Fu(\(26\))589 1871 y(Recall)34 b(that)g(for)g(our)f(mo)s(del)g(giv)m (en)h(b)m(y)g(Eq.)f(\(5\),)j(the)e(rate)h(function)e(tak)m(es)i(the)448 1984 y(form)41 b(giv)m(en)h(in)e(Eqs.)74 b(\(8\))42 b(and)f(\(9\).)75 b(W)-8 b(e)43 b(in)m(tro)s(duce)d(further)g(the)i(cost)g(function)448 2097 y Fn(V)501 2111 y Fm(T)556 2097 y Fu(\()p Fn(x;)15 b(y)s Fu(\))32 b(giv)m(en)e(b)m(y)1234 2301 y Fn(V)1287 2315 y Fm(T)1342 2301 y Fu(\()p Fn(x;)15 b(y)s Fu(\))41 b(=)291 b(inf)1704 2365 y Fm(\036)p Fl(2C)t Ft(\([0)p Fm(;T)10 b Ft(]\))i(:)f Fm(\036)p Ft(\()p Fm(T)f Ft(\)=)p Fm(y)2332 2301 y Fn(I)2372 2315 y Fm(x;T)2486 2301 y Fu(\()p Fn(\036)p Fu(\))15 b Fn(:)625 b 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b(the)h(equation)g(\(29\))h(a)f(con)m(trol)h(system.)60 b(W)-8 b(e)37 b(\014x)g(an)f(arbitrary)g(time)g Fn(T)49 b(>)35 b Fu(0.)448 4534 y(W)-8 b(e)41 b(denote)f(b)m(y)f Fn(')1110 4501 y Fm(u)1110 4556 y(x)1196 4534 y Fu(:)h([0)p Fn(;)15 b(T)e Fu(])42 b Fk(!)e Fg(R)1713 4501 y Fm(n)1800 4534 y Fu(the)f(solution)f(of)i(the)g(di\013eren)m(tial)d(equations)448 4647 y(\(29\))k(with)d(con)m(trol)h Fn(u)g Fu(and)f(initial)e (condition)i Fn(x)p Fu(.)66 b(The)39 b(corresp)s(ondence)f(b)s(et)m(w)m (een)448 4759 y(the)45 b(sto)s(c)m(hastic)g(system)g(Eq.)f(\(23\))i (and)e(the)g(deterministic)e(system)j(Eq.)f(\(29\))i(is)448 4872 y(exempli\014ed)37 b(b)m(y)i(the)g(Supp)s(ort)d(Theorem)j(of)g (Stro)s(o)s(c)m(k)g(and)f(V)-8 b(aradhan)39 b([24)q(].)67 b(The)1897 5225 y(10)p eop %%Page: 11 11 11 10 bop 448 573 a Fu(supp)s(ort)27 b(of)i(the)f(di\013usion)e(pro)s (cess)i Fn(x)p Fu(\()p Fn(t)p Fu(\))h(with)e(initial)e(condition)i Fn(x)i Fu(on)f([0)p Fn(;)15 b(T)e Fu(],)30 b(is,)e(b)m(y)448 686 y(de\014nition,)h(the)h(smallest)g(closed)g(subset)g Fk(S)1986 700 y Fm(x)2060 686 y Fu(of)g Fk(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))33 b(suc)m(h)d(that)1557 854 y Fn(P)1615 868 y Fm(x)1660 854 y Fu([)p Fn(x)p Fu(\()p Fn(t)p Fu(\))c Fk(2)e(S)2006 868 y Fm(x)2050 854 y Fu(])41 b(=)f(1)31 b Fn(:)947 b Fu(\(30\))448 1023 y(The)34 b(Supp)s(ort)e(Theorem)h (asserts)i(that)f(the)g(supp)s(ort)e(of)i(the)h(di\013usion)c(is)i (equal)g(to)448 1136 y(the)e(set)g(of)f(solutions)f(of)i(Eq.)f(\(29\))i (as)f(the)f(con)m(trol)h Fn(u)f Fu(is)g(v)-5 b(aried:)1355 1304 y Fk(S)1410 1318 y Fm(x)1495 1304 y Fu(=)p 1606 1225 870 4 v 40 w Fk(f)p Fn(')1710 1278 y Fm(u)1710 1327 y(x)1796 1304 y Fu(:)41 b Fn(u)25 b Fk(2)g Fn(L)2087 1278 y Ft(2)2126 1304 y Fu(\([0)p Fn(;)15 b(T)e Fu(]\))32 b Fk(g)f Fn(;)745 b Fu(\(31\))448 1473 y(for)38 b(all)e Fn(x)h Fk(2)g Fg(R)993 1440 y Fm(k)1036 1473 y Fu(.)63 b(The)37 b(con)m(trol)h(system)g(\(29\))h(is)d(said)h(to)h(b)s(e)f Fe(str)-5 b(ongly)41 b(c)-5 b(ompletely)448 1586 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y Fu(\))g(+)g Fn(H)i Fu(\()p Fn(p;)15 b(q)s Fu(\))p Fe(,)448 4704 y(i.e.,)1226 4827 y Fn(A)40 b Fu(=)1446 4733 y Fh(n)1501 4827 y Fn(x)25 b Fk(2)g Fg(R)1742 4790 y Ft(2)p Fm(d)p Ft(\()p Fm(n)p Ft(+1\))2046 4827 y Fu(:)41 b Fk(r)p Fn(G)p Fu(\()p Fn(x)p Fu(\))25 b(=)g(0)2548 4733 y Fh(o)2634 4827 y Fn(:)616 b Fu(\(33\))1897 5225 y(11)p eop %%Page: 12 12 12 11 bop 448 573 a Fe(Pr)-5 b(o)g(of:)40 b Fu(As)25 b(noted)g(in)f(the)i(in)m(tro)s(duction)d Fn(G)p Fu(\()p Fn(x)p Fu(\))i(is)g(a)g(Liapuno)m(v)g(function)f(for)h(the)g(\015o)m(w) 448 686 y(giv)m(en)31 b(b)m(y)f(\(32\).)42 b(A)31 b(simple)d (computation)i(sho)m(ws)g(that)1044 860 y Fn(d)p 1028 900 81 4 v 1028 983 a(dt)1118 921 y(G)p Fu(\()p Fn(x)p Fu(\()p Fn(t)p Fu(\)\))99 b(=)e Fk(\000)p Fn(\015)5 b(\025)1858 884 y Ft(2)1941 840 y Fh(X)1913 1022 y Fm(i)p Ft(=1)p Fm(;n)2090 921 y Fu(\()p Fn(\025)2178 884 y Fl(\000)p Ft(2)2272 921 y Fn(r)2313 935 y Fm(i)2341 921 y Fu(\()p Fn(t)p Fu(\))21 b Fk(\000)f Fn(q)2597 935 y Fm(i)2625 921 y Fu(\()p Fn(t)p Fu(\)\))2763 884 y Ft(2)1514 1166 y Fu(=)97 b Fk(\000)p Fn(\015)5 b(\025)1858 1129 y Ft(2)1941 1085 y Fh(X)1913 1267 y Fm(i)p Ft(=1)p Fm(;n)2105 1166 y Fk(jr)2206 1180 y Fm(r)2238 1190 y Ff(i)2268 1166 y Fn(G)p Fu(\()p Fn(x)p Fu(\()p Fn(t)p Fu(\)\))p Fk(j)2590 1129 y Ft(2)2671 1166 y Fk(\024)40 b Fu(0)15 b Fn(:)408 b Fu(\(34\))448 1440 y(Therefore)34 b(it)f(is)g(enough)g(to)i(sho)m(w)e (that)h(the)g(\015o)m(w)g(do)s(es)f(not)h(get)h(\\stuc)m(k")g(at)g (some)448 1553 y(p)s(oin)m(t)e(of)h(the)g(h)m(yp)s(er-surfaces)g(\()p Fn(\025)1646 1520 y Fl(\000)p Ft(2)1740 1553 y Fn(r)1781 1567 y Fm(i)1809 1553 y Fu(\()p Fn(t)p Fu(\))24 b Fk(\000)e Fn(q)2070 1567 y Fm(i)2098 1553 y Fu(\()p Fn(t)p Fu(\)\))2236 1520 y Ft(2)2307 1553 y Fu(=)31 b(0)p Fn(;)15 b(i)32 b Fu(=)f(1)p Fn(;)15 b(n)35 b Fu(whic)m(h)d(do)s(es)i(not)448 1666 y(b)s(elong)c(to)h(the)f(set)h Fn(A)p Fu(.)589 1778 y(Let)24 b(us)e(assume)g(the)h(con)m(trary)-8 b(,)26 b(i.e.,)e(that,)i(for)c(some)h(tra)5 b(jectory)24 b(and)e(some)h(times) 448 1891 y Fn(T)501 1905 y Ft(1)566 1891 y Fn(<)i(T)715 1905 y Ft(2)785 1891 y Fu(w)m(e)31 b(ha)m(v)m(e)1217 2084 y Fn(G)p Fu(\()p Fn(x)p Fu(\()p Fn(t)p Fu(\)\))41 b(=)f Fn(G)p Fu(\()p Fn(x)p Fu(\()p Fn(T)1913 2098 y Ft(1)1953 2084 y Fu(\)\))46 b(for)30 b Fn(t)25 b Fk(2)g Fu([)p Fn(T)2430 2098 y Ft(1)2470 2084 y Fn(;)15 b(T)2563 2098 y Ft(2)2603 2084 y Fu(])g Fn(:)607 b Fu(\(35\))448 2276 y(W)-8 b(e)37 b(sho)m(w)f(that)h(this)e(implies)e(that)j Fn(x)p Fu(\()p Fn(t)p Fu(\))f Fk(2)f Fn(A)p Fu(,)k(for)e Fn(t)e Fk(2)g Fu([)p Fn(T)2544 2290 y Ft(1)2584 2276 y Fn(;)15 b(T)2677 2290 y Ft(2)2717 2276 y Fu(].)57 b(F)-8 b(rom)37 b(Eqs.)e(\(34\))448 2389 y(and)30 b(\(35\))i(w)m(e)f(ha)m(v)m (e,)h(for)e Fn(t)25 b Fk(2)g Fu([)p Fn(T)1547 2403 y Ft(1)1587 2389 y Fn(;)15 b(T)1680 2403 y Ft(2)1720 2389 y Fu(],)30 b(the)h(iden)m(tit)m(y)1213 2582 y Fn(\025)1266 2544 y Fl(\000)p Ft(2)1361 2582 y Fn(r)1402 2596 y Ft(1)1441 2582 y Fu(\()p Fn(t)p Fu(\))21 b Fk(\000)f Fn(q)1697 2596 y Ft(1)1736 2582 y Fu(\()p Fn(t)p Fu(\))41 b(=)f Fk(r)2067 2596 y Fm(r)2099 2605 y Fj(1)2137 2582 y Fn(G)p Fu(\()p Fn(x)p Fu(\()p Fn(t)p Fu(\)\))h(=)f(0)15 b Fn(:)604 b Fu(\(36\))448 2774 y(F)-8 b(urther,)31 b(using)e(Eq.\(32\),)j(w)m(e)f (obtain)504 3010 y(0)41 b(=)727 2948 y Fn(d)p 711 2989 V 711 3072 a(dt)801 3010 y Fu(\()p Fn(\025)889 2972 y Fl(\000)p Ft(2)984 3010 y Fn(r)1025 3024 y Ft(1)1064 3010 y Fu(\()p Fn(t)p Fu(\))21 b Fk(\000)f Fn(q)1320 3024 y Ft(1)1359 3010 y Fu(\()p Fn(t)p Fu(\)\))41 b(=)f Fk(\000)p Fn(\015)5 b Fu(\()p Fn(\025)1860 2972 y Fl(\000)p Ft(2)1955 3010 y Fn(r)1996 3024 y Ft(1)2035 3010 y Fu(\()p Fn(t)p Fu(\))21 b Fk(\000)f Fn(q)2291 3024 y Ft(1)2330 3010 y Fu(\()p Fn(t)p Fu(\)\))h Fk(\000)f Fn(p)2626 3024 y Ft(1)2665 3010 y Fu(\()p Fn(t)p Fu(\))41 b(=)f Fk(\000)p Fn(p)3037 3024 y Ft(1)3076 3010 y Fu(\()p Fn(t)p Fu(\))15 b Fn(:)56 b Fu(\(37\))448 3223 y(Th)m(us)30 b(w)m(e)g(get)1430 3336 y Fn(p)1476 3350 y Ft(1)1515 3336 y Fu(\()p Fn(t)p Fu(\))41 b(=)f Fk(r)1846 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b(=)f(0)31 b(w)m(e)g(can)f(rewrite)g (Eq.)g(\(39\))i(as)f(follo)m(ws:)861 4349 y(0)41 b(=)f Fk(r)1134 4363 y Fm(q)1166 4372 y Fj(1)1204 4349 y Fn(U)1276 4311 y Ft(\(1\))1370 4349 y Fu(\()p Fn(q)1446 4363 y Ft(1)1486 4349 y Fu(\()p Fn(t)p Fu(\)\))21 b Fk(\000)f Fn(\025)1789 4311 y Ft(2)1828 4349 y Fn(q)1869 4363 y Ft(1)1908 4349 y Fu(\()p Fn(t)p Fu(\))h(+)f Fk(r)2199 4363 y Fm(q)2231 4372 y Fj(1)2269 4349 y Fn(U)2341 4311 y Ft(\(2\))2435 4349 y Fu(\()p Fn(q)2511 4363 y Ft(1)2550 4349 y Fu(\()p Fn(t)p Fu(\))h Fk(\000)f Fn(q)2806 4363 y Ft(2)2845 4349 y Fu(\()p Fn(t)p Fu(\)\))15 b Fn(:)252 b Fu(\(41\))448 4541 y(By)34 b(the)g(con)m(v)m(exit)m(y)h(condition)d (on)h Fn(U)1764 4508 y Ft(\(2\))1858 4541 y Fu(,)i Fg(K2)p Fu(,)f Fk(r)p Fn(U)2259 4508 y Ft(\(2\))2384 4541 y Fu(:)c Fg(R)2517 4508 y Fm(d)2589 4541 y Fk(!)g Fg(R)2788 4508 y Fm(d)2862 4541 y Fu(is)j(a)g(di\013eomor-)448 4654 y(phism)c(with)h(an)h(in)m(v)m(erse)g(whic)m(h)f(w)m(e)i(denote)f Fn(W)13 b Fu(.)43 b(W)-8 b(e)32 b(can)f(therefore)h(solv)m(e)f (Eq.\(41\))448 4767 y(in)e(terms)i(of)f Fn(q)952 4781 y Ft(2)1022 4767 y Fu(and)f(w)m(e)i(obtain)786 4970 y Fn(q)827 4984 y Ft(2)866 4970 y Fu(\()p Fn(t)p Fu(\))41 b(=)f Fn(q)1162 4984 y Ft(1)1201 4970 y Fu(\()p Fn(t)p Fu(\))21 b Fk(\000)f Fn(W)1529 4876 y Fh(\020)1579 4970 y Fn(\025)1632 4932 y Ft(2)1671 4970 y Fn(q)1712 4984 y Ft(1)1751 4970 y Fu(\()p Fn(t)p Fu(\))h Fk(\000)f(r)2042 4984 y Fm(q)2074 4993 y Fj(1)2112 4970 y Fn(U)2184 4932 y Ft(\(1\))2278 4970 y Fu(\()p Fn(q)2354 4984 y Ft(1)2394 4970 y Fu(\()p Fn(t)p Fu(\)\))2532 4876 y Fh(\021)2623 4970 y Fk(\021)40 b Fn(F)13 b Fu(\()p Fn(q)2881 4984 y Ft(1)2920 4970 y Fu(\()p Fn(t)p Fu(\)\))i Fn(:)177 b Fu(\(42\))1897 5225 y(12)p eop %%Page: 13 13 13 12 bop 448 573 a Fu(F)-8 b(rom)31 b(this)e(w)m(e)i(conclude)f(that) 672 777 y Fn(p)718 791 y Ft(2)758 777 y Fu(\()p Fn(t)p Fu(\))40 b(=)57 b(_)-41 b Fn(q)1054 791 y Ft(2)1093 777 y Fu(\()p Fn(t)p Fu(\))41 b(=)f Fk(r)1424 791 y Fm(q)1456 800 y Fj(1)1494 777 y Fn(F)13 b Fu(\()p Fn(q)1641 791 y Ft(1)1680 777 y Fu(\()p Fn(t)p Fu(\)\))18 b(_)-43 b Fn(q)1859 791 y Ft(1)1899 777 y Fu(\()p Fn(t)p Fu(\))41 b(=)f Fk(r)2230 791 y Fm(q)2262 800 y Fj(1)2300 777 y Fn(F)13 b Fu(\()p Fn(q)2447 791 y Ft(1)2487 777 y Fu(\()p Fn(t)p Fu(\)\))p Fn(p)2671 791 y Ft(1)2711 777 y Fu(\()p Fn(t)p Fu(\))41 b(=)f(0)15 b Fn(;)224 b Fu(\(43\))448 981 y(and)30 b(th)m(us)1430 1094 y Fn(p)1476 1108 y Ft(2)1515 1094 y Fu(\()p Fn(t)p Fu(\))41 b(=)f Fk(r)1846 1108 y Fm(p)1882 1117 y Fj(2)1920 1094 y Fn(G)p Fu(\()p Fn(x)p Fu(\()p Fn(t)p Fu(\)\))h(=)g(0)15 b Fn(:)820 b Fu(\(44\))448 1261 y(Pro)s(ceeding)33 b(b)m(y)g(induction)e(along)i(the)h(c)m(hain)f (it)g(is)f(easy)i(to)g(see)g(that)g(if)e Fn(G)p Fu(\()p Fn(x)p Fu(\()p Fn(t)p Fu(\)\))i(is)448 1374 y(constan)m(t)e(on)e(the)h (in)m(terv)-5 b(al)29 b([)p Fn(T)1504 1388 y Ft(1)1544 1374 y Fn(;)15 b(T)1637 1388 y Ft(2)1677 1374 y Fu(],)31 b(then)f(one)h(has)1637 1578 y Fk(r)p Fn(G)p Fu(\()p Fn(x)p Fu(\()p Fn(t)p Fu(\)\))41 b(=)f(0)15 b Fn(;)1028 b Fu(\(45\))448 1782 y(and)30 b(therefore)h Fn(x)p Fu(\()p Fn(t)p Fu(\))26 b Fk(2)f Fn(A)p Fu(.)40 b(This)29 b(concludes)h(the)g (pro)s(of)g(of)g(Lemma)h(2.2.)p 3078 1790 50 67 v 448 2141 a Fd(2.3)113 b(Con)m(tin)m(uit)m(y)35 b(prop)s(erties)i(of)g Fc(V)2004 2090 y Ft(\()p Fm(\021)r Ft(\))1982 2165 y Fm(T)2101 2141 y Fb(\()p Fc(x;)17 b(y)t Fb(\))448 2313 y Fu(In)34 b(the)h(analysis)e(of)h(the)h(asymptotic)g(b)s(eha)m(vior)e (of)i(the)g(in)m(v)-5 b(arian)m(t)33 b(measure)i(it)f(will)448 2426 y(b)s(e)e(imp)s(ortan)m(t)f(to)i(establish)d(certain)i(con)m(tin)m (uit)m(y)g(prop)s(erties)e(of)i(the)g(cost)h(function)448 2549 y Fn(V)522 2501 y Ft(\()p Fm(\021)r Ft(\))501 2577 y Fm(T)618 2549 y Fu(\()p Fn(x;)15 b(y)s Fu(\).)74 b(W)-8 b(e)42 b(pro)m(v)m(e)g(\014rst)e(a)h(global)f(prop)s(ert)m(y:)62 b(w)m(e)41 b(sho)m(w)g(that)g(for)g(an)m(y)g(time)448 2683 y Fn(T)13 b Fu(,)40 b Fn(V)652 2635 y Ft(\()p Fm(\021)r Ft(\))632 2710 y Fm(T)749 2683 y Fu(\()p Fn(x;)15 b(y)s Fu(\))39 b(as)g(a)f(map)g(from)f Fn(X)c Fk(\002)25 b Fn(X)46 b Fk(!)38 b Fg(R)g Fu(is)f(ev)m(erywhere)i(\014nite)e(and)g (upp)s(er)448 2816 y(semicon)m(tin)m(uous.)67 b(F)-8 b(urthermore)39 b(w)m(e)h(need)f(a)h(lo)s(cal)f(prop)s(ert)m(y)f(of)i Fn(V)2918 2768 y Ft(\()p Fm(\021)r Ft(\))2898 2843 y Fm(T)3015 2816 y Fu(\()p Fn(x;)15 b(y)s Fu(\))40 b(near)448 2929 y(the)k Fn(!)s Fu(-limit)e(set)i(of)g(the)g(zero-temp)s(erature)h (dynamics)d(\(see)j(Lemma)f(2.2\).)82 b(W)-8 b(e)448 3058 y(pro)m(v)m(e)30 b(that)f(if)f Fn(x)g Fu(and)g Fn(y)k Fu(are)d(su\016cien)m(tly)e(close)i(to)g(this)f Fn(!)s Fu(-limit)f(set)i(then)f Fn(V)3129 3010 y Ft(\()p Fm(\021)r Ft(\))3109 3085 y Fm(T)3226 3058 y Fu(\()p Fn(x;)15 b(y)s Fu(\))448 3171 y(is)29 b(small.)38 b(Both)30 b(results)e(are)i (obtained)e(using)g(con)m(trol)i(theory)f(and)g(h)m(yp)s(o)s (ellipticit)m(y)-8 b(.)448 3394 y Fg(Prop)s(osition)37 b(2.3)46 b Fe(Assume)37 b(c)-5 b(ondition)39 b Fg(K2)p Fe(.)55 b(Then)38 b(the)g(functions)f Fn(V)3027 3346 y Ft(\()p Fm(\021)r Ft(\))3006 3421 y Fm(T)3123 3394 y Fe(,)i(for)f(al)5 b(l)448 3507 y Fn(T)38 b(>)25 b Fu(0)33 b Fe(and)h Fn(V)963 3474 y Ft(\()p Fm(\021)r Ft(\))1092 3507 y Fe(ar)-5 b(e)34 b(upp)-5 b(er)34 b(semic)-5 b(ontinuous)34 b(maps)g(:)41 b Fn(X)28 b Fk(\002)20 b Fn(X)32 b Fk(!)26 b Fg(R)p Fe(.)448 3730 y(Pr)-5 b(o)g(of:)42 b Fu(By)31 b(de\014nition)d Fn(V)1348 3682 y Ft(\()p Fm(\021)r Ft(\))1328 3758 y Fm(T)1445 3730 y Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))31 b(is)f(giv)m(en)g(b)m(y)1198 3991 y Fn(V)1271 3943 y Ft(\()p Fm(\021)r Ft(\))1251 4019 y Fm(T)1368 3991 y Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))42 b(=)e(inf)1861 3930 y(1)p 1861 3970 46 4 v 1861 4054 a(2)1931 3876 y Fh(Z)2014 3903 y Fm(T)1977 4065 y Ft(0)2117 3910 y Fh(X)2085 4093 y Fm(j)t Ft(=1)p Fm(;n)2285 3991 y Fk(j)p Fn(u)2362 4005 y Fm(j)2398 3991 y Fu(\()p Fn(t)p Fu(\))p Fk(j)2526 3954 y Ft(2)2567 3991 y Fn(dt)15 b(;)588 b Fu(\(46\))448 4289 y(where)36 b(the)g(in\014m)m(um)d(in)i(\(46\))i(is)e(tak)m(en)i(o)m(v)m (er)g(all)d Fn(u)h Fu(=)f(\()p Fn(u)2509 4303 y Ft(1)2549 4289 y Fn(;)15 b(u)2641 4303 y Fm(n)2688 4289 y Fu(\))35 b Fk(2)f Fn(L)2915 4256 y Ft(2)2954 4289 y Fu(\([0)p Fn(;)15 b(T)e Fu(]\))37 b(suc)m(h)448 4402 y(that)1279 4607 y(_)-42 b Fn(q)101 b Fu(=)d Fk(r)1649 4621 y Fm(p)1688 4607 y Fn(G)15 b(;)1278 4744 y Fu(_)-43 b Fn(p)98 b Fu(=)g Fk(\000r)1720 4758 y Fm(q)1757 4744 y Fn(G)15 b(;)1276 4891 y Fu(_)-39 b Fn(r)101 b Fu(=)d Fk(\000)p Fn(\015)5 b(\025)1749 4854 y Ft(2)1788 4891 y Fk(r)1864 4905 y Fm(r)1902 4891 y Fn(G)20 b Fu(+)f(\(2)p Fn(\015)5 b(\025)2269 4854 y Ft(2)2310 4891 y Fn(D)s Fu(\))2423 4854 y Ft(1)p Fm(=)p Ft(2)2533 4891 y Fn(u)15 b(;)650 b Fu(\(47\))1897 5225 y(13)p eop %%Page: 14 14 14 13 bop 448 573 a Fu(with)29 b(b)s(oundary)g(conditions)972 744 y(\()p Fn(p)p Fu(\(0\))p Fn(;)15 b(q)s Fu(\(0\))p Fn(;)g(r)s Fu(\(0\)\))29 b(=)c Fn(y)18 b(;)198 b Fu(\()p Fn(p)p Fu(\()p Fn(T)13 b Fu(\))p Fn(;)i(q)s Fu(\()p Fn(T)e Fu(\))p Fn(;)i(r)s Fu(\()p Fn(T)e Fu(\)\))27 b(=)e Fn(z)19 b(:)362 b Fu(\(48\))448 915 y(In)31 b(other)g(w)m(ords,)g(the)g (in\014m)m(um)e(in)g(\(46\))k(is)d(tak)m(en)i(o)m(v)m(er)h(all)c(con)m (trols)j Fn(u)f Fu(whic)m(h)e(steer)448 1028 y Fn(y)i Fu(to)d Fn(z)t Fu(.)40 b(F)-8 b(or)28 b(notational)f(simplicit)m(y)e(w) m(e)j(let)g Fn(r)2047 1042 y Ft(1)2111 1028 y Fu(=)d Fn(q)2248 1042 y Ft(0)2315 1028 y Fu(and)i Fn(r)2530 1042 y Fm(n)2602 1028 y Fu(=)e Fn(q)2739 1042 y Fm(n)p Ft(+1)2875 1028 y Fu(.)40 b(F)-8 b(urthermore)448 1141 y(w)m(e)33 b(set)f Fn(Q)27 b Fu(=)h(\()p Fn(q)1001 1155 y Ft(0)1040 1141 y Fn(;)15 b(q)1121 1155 y Ft(1)1161 1141 y Fn(;)g(:)g(:)g(:)h(q)1363 1155 y Fm(n)1410 1141 y Fn(;)f(q)1491 1155 y Fm(n)p Ft(+1)1628 1141 y Fu(\))28 b Fk(2)f Fg(R)1857 1108 y Fm(d)p Ft(\()p Fm(n)p Ft(+2\))2118 1141 y Fu(and)k Fn(P)40 b Fu(=)28 b(\()17 b(_)-42 b Fn(q)2569 1155 y Ft(1)2608 1141 y Fn(;)15 b(:)g(:)g(:)i(;)32 b Fu(_)-42 b Fn(q)2851 1155 y Fm(n)2898 1141 y Fu(\))28 b Fk(2)f Fg(R)3127 1108 y Fm(dn)3211 1141 y Fu(.)45 b(The)448 1254 y(equations)30 b(\(47\))i(tak)m(e)g(the)f(form)939 1425 y(_)-41 b Fn(q)964 1439 y Ft(0)1101 1425 y Fu(=)98 b Fk(\000)p Fn(\015)5 b(\025)1446 1387 y Ft(2)1485 1425 y Fk(r)1561 1439 y Fm(q)1593 1448 y Fj(0)1631 1425 y Fn(G)p Fu(\()p Fn(q)1779 1439 y Ft(0)1818 1425 y Fn(;)15 b(q)1899 1439 y Ft(1)1938 1425 y Fu(\))21 b(+)f(\(2)p Fn(\015)5 b(\025)2270 1387 y Ft(2)2310 1425 y Fu(\(1)21 b(+)f Fn(\021)s Fu(\)\))2620 1387 y Ft(1)p Fm(=)p Ft(2)2731 1425 y Fn(u)2783 1439 y Ft(1)2838 1425 y Fn(;)943 1562 y Fu(\177)-52 b Fn(q)977 1577 y Fm(l)1101 1562 y Fu(=)98 b Fk(\000r)1417 1576 y Fm(q)1449 1586 y Ff(j)1484 1562 y Fn(G)p Fu(\()p Fn(q)1632 1577 y Fm(l)q Fl(\000)p Ft(1)1748 1562 y Fn(;)15 b(q)1829 1577 y Fm(l)1855 1562 y Fn(;)g(q)1936 1577 y Fm(l)q Ft(+1)2052 1562 y Fu(\))91 b Fn(l)27 b Fu(=)e(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(n)g(;)842 1716 y Fu(_)-42 b Fn(q)866 1730 y Fm(n)p Ft(+1)1101 1716 y Fu(=)98 b Fk(\000)p Fn(\015)5 b(\025)1446 1679 y Ft(2)1485 1716 y Fk(r)1561 1730 y Fm(q)1593 1739 y Ff(n)p Fj(+1)1716 1716 y Fn(G)p Fu(\()p Fn(q)1864 1730 y Fm(n)1910 1716 y Fn(;)15 b(q)1991 1730 y Fm(n)p Ft(+1)2128 1716 y Fu(\))21 b(+)f(\(2)p Fn(\015)5 b(\025)2460 1679 y Ft(2)2500 1716 y Fu(\(1)21 b Fk(\000)f Fn(\021)s Fu(\)\))2810 1679 y Ft(1)p Fm(=)p Ft(2)2921 1716 y Fn(u)2973 1730 y Fm(n)3035 1716 y Fn(;)215 b Fu(\(49\))448 1887 y(with)29 b(b)s(oundary)g (conditions)1144 2058 y(\()p Fn(P)13 b Fu(\(0\))p Fn(;)i(Q)p Fu(\(0\)\))44 b(=)c Fn(y)18 b(;)106 b Fu(\()q Fn(P)13 b Fu(\()p Fn(T)g Fu(\))p Fn(;)i(Q)p Fu(\()p Fn(T)e Fu(\)\))42 b(=)e Fn(z)19 b(:)534 b Fu(\(50\))448 2229 y(By)42 b(condition)f Fg(K2)p Fu(,)k Fk(r)1293 2243 y Fm(q)1331 2229 y Fn(U)1403 2196 y Ft(\(2\))1497 2229 y Fu(\()p Fn(q)s Fu(\))e(is)d(a)j (di\013eomorphism.)72 b(As)42 b(a)g(consequence)h(the)448 2342 y(iden)m(tit)m(y)1436 2455 y(\177)-52 b Fn(q)1470 2470 y Fm(l)1536 2455 y Fu(=)40 b Fk(\000r)1794 2469 y Fm(q)1826 2481 y Ff(l)1853 2455 y Fn(G)p Fu(\()p Fn(q)2001 2470 y Fm(l)q Fl(\000)p Ft(1)2117 2455 y Fn(;)15 b(q)2198 2470 y Fm(l)2224 2455 y Fn(;)g(q)2305 2470 y Fm(l)q Ft(+1)2421 2455 y Fu(\))819 b(\(51\))448 2606 y(can)31 b(b)s(e)f(solv)m(ed)h(for)f (either)g Fn(q)1451 2621 y Fm(l)q Fl(\000)p Ft(1)1597 2606 y Fu(or)h Fn(q)1750 2621 y Fm(l)q Ft(+1)1866 2606 y Fu(:)41 b(there)31 b(are)g(smo)s(oth)f(functions)f Fn(G)3102 2621 y Fm(l)3159 2606 y Fu(and)h Fn(H)3412 2621 y Fm(l)448 2718 y Fu(suc)m(h)g(that)1434 2889 y Fn(q)1475 2904 y Fm(l)q Fl(\000)p Ft(1)1689 2889 y Fu(=)97 b Fn(G)1929 2904 y Fm(l)1955 2889 y Fu(\()p Fn(q)2031 2904 y Fm(l)2057 2889 y Fn(;)22 b Fu(\177)-52 b Fn(q)2138 2904 y Fm(l)2164 2889 y Fn(;)15 b(q)2245 2904 y Fm(l)q Ft(+1)2361 2889 y Fu(\))g Fn(;)839 b Fu(\(52\))1434 3027 y Fn(q)1475 3042 y Fm(l)q Ft(+1)1689 3027 y Fu(=)97 b Fn(H)1933 3042 y Fm(l)1959 3027 y Fu(\()p Fn(q)2035 3042 y Fm(l)q Fl(\000)p Ft(1)2151 3027 y Fn(;)15 b(q)2232 3042 y Fm(l)2258 3027 y Fn(;)22 b Fu(\177)-52 b Fn(q)2339 3042 y Fm(l)2365 3027 y Fu(\))15 b Fn(:)835 b Fu(\(53\))448 3198 y(Using)32 b(this)f(w)m(e)h(rewrite)g(no)m(w)g(the)g(equations)g (in)f(the)h(follo)m(wing)f(form:)44 b(W)-8 b(e)33 b(assume)448 3311 y(for)i(simplicit)m(y)d Fn(n)j Fu(is)e(an)i(ev)m(en)h(n)m(um)m(b)s (er)d(and)i(w)m(e)g(set)h Fn(j)i Fu(=)32 b Fn(n=)p Fu(2.)55 b(\(If)35 b Fn(n)f Fu(is)g(o)s(dd,)h(tak)m(e)448 3424 y Fn(j)k Fu(=)33 b(\()p Fn(n)23 b Fu(+)g(1\))p Fn(=)p Fu(2)37 b(and)d(up)g(to)i(minor)e(mo)s(di\014cations)f(the)i(argumen)m (t)h(go)s(es)g(as)f(in)f(the)448 3537 y(ev)m(en)d(case\).)589 3650 y(W)-8 b(e)44 b(rewrite)e(Eq.)h(\(49\))h(as)f(follo)m(ws:)64 b(F)-8 b(or)43 b(the)g(\014rst)f Fn(j)34 b Fu(+)28 b(1)43 b(equations)f(w)m(e)h(use)448 3763 y(Eq.)31 b(\(52\))h(and)d(\014nd)628 3972 y Fn(u)680 3986 y Ft(1)818 3972 y Fu(=)1297 3911 y(1)p 996 3951 647 4 v 996 4037 a(\(2)p Fn(\015)5 b(\025)1181 4011 y Ft(2)1222 4037 y Fu(\(1)21 b(+)f Fn(\021)s Fu(\)\))1532 4011 y Ft(1)p Fm(=)p Ft(2)1653 3972 y Fu(\()d(_)-42 b Fn(q)1729 3986 y Ft(0)1788 3972 y Fu(+)20 b Fn(\015)5 b(\025)1984 3934 y Ft(2)2024 3972 y Fk(r)2100 3986 y Fm(q)2132 3995 y Fj(0)2170 3972 y Fn(G)p Fu(\()p Fn(q)2318 3986 y Ft(0)2357 3972 y Fn(;)15 b(q)2438 3986 y Ft(1)2477 3972 y Fu(\)\))41 b Fk(\021)f Fn(G)2771 3986 y Ft(0)2810 3972 y Fu(\()p Fn(q)2886 3986 y Ft(0)2926 3972 y Fn(;)32 b Fu(_)-42 b Fn(q)3007 3986 y Ft(0)3046 3972 y Fn(;)15 b(q)3127 3986 y Ft(1)3167 3972 y Fu(\))g Fn(;)653 4161 y(q)694 4176 y Fm(l)818 4161 y Fu(=)98 b Fn(G)1059 4176 y Fm(l)q Ft(+1)1174 4161 y Fu(\()p Fn(q)1250 4176 y Fm(l)q Ft(+1)1366 4161 y Fn(;)22 b Fu(\177)-52 b Fn(q)1447 4176 y Fm(l)q Ft(+1)1563 4161 y Fn(;)15 b(q)1644 4176 y Fm(l)q Ft(+2)1761 4161 y Fu(\))91 b Fn(l)27 b Fu(=)e(0)p Fn(;)15 b Fu(1)p Fn(;)g(:)g(:)g(:)j(;)d(j)26 b Fk(\000)20 b Fu(1)15 b Fn(:)666 b Fu(\(54\))448 4332 y(F)-8 b(or)34 b(the)f(remaining)e Fn(n)22 b Fu(+)f(1)i Fk(\000)e Fn(j)35 b Fu(=)30 b Fn(j)d Fu(+)22 b(1)33 b(equations)g(w)m(e)g(use)g(Eq.)g(\(53\))h(and)f(obtain) 448 4445 y(the)e(equiv)-5 b(alen)m(t)30 b(equations)835 4649 y Fn(u)887 4663 y Fm(n)1032 4649 y Fu(=)1512 4588 y(1)p 1211 4628 V 1211 4714 a(\(2)p Fn(\015)5 b(\025)1396 4688 y Ft(2)1436 4714 y Fu(\(1)21 b Fk(\000)f Fn(\021)s Fu(\)\))1746 4688 y Ft(1)p Fm(=)p Ft(2)1867 4649 y Fu(\()e(_)-43 b Fn(q)1943 4663 y Fm(n)p Ft(+1)2101 4649 y Fu(+)20 b Fn(\015)5 b(\025)2297 4612 y Ft(2)2336 4649 y Fk(r)2412 4663 y Fm(q)2444 4672 y Ff(n)p Fj(+1)2567 4649 y Fn(G)p Fu(\()p Fn(q)2715 4663 y Fm(n)2762 4649 y Fn(;)15 b(q)2843 4663 y Fm(n)p Ft(+1)2980 4649 y Fu(\)\))1032 4839 y Fk(\021)98 b Fn(H)1277 4853 y Fm(n)p Ft(+1)1414 4839 y Fu(\()p Fn(q)1490 4853 y Fm(n)1537 4839 y Fn(;)15 b(q)1618 4853 y Fm(n)p Ft(+1)1755 4839 y Fn(;)32 b Fu(_)-42 b Fn(q)1836 4853 y Fm(n)p Ft(+1)1973 4839 y Fu(\))15 b Fn(;)883 4976 y(q)924 4991 y Fm(l)1032 4976 y Fu(=)83 b Fn(H)1262 4991 y Fm(l)q Fl(\000)p Ft(1)1378 4976 y Fu(\()p Fn(q)1454 4991 y Fm(l)q Fl(\000)p Ft(2)1570 4976 y Fn(;)15 b(q)1651 4991 y Fm(l)q Fl(\000)p Ft(1)1767 4976 y Fn(;)22 b Fu(\177)-52 b Fn(q)1848 4991 y Fm(l)q Fl(\000)p Ft(1)1964 4976 y Fu(\))91 b Fn(l)28 b Fu(=)d Fn(j)g Fu(+)20 b(2)p Fn(;)15 b(:)g(:)g(:)i(;)e(n)21 b Fu(+)e(1)c Fn(:)383 b Fu(\(55\))1897 5225 y(14)p eop %%Page: 15 15 15 14 bop 448 573 a Fu(Ob)m(viously)30 b(b)s(oth)i(sets)g(of)h (equations)f(\(54\))h(and)f(\(55\))h(can)g(b)s(e)e(solv)m(ed)h (iterativ)m(ely)g(to)448 686 y(express)37 b Fn(u)824 700 y Ft(1)863 686 y Fn(;)15 b(q)944 700 y Ft(0)984 686 y Fn(;)g(:)g(:)g(:)h(;)f(q)1226 700 y Fm(j)t Fl(\000)p Ft(1)1390 686 y Fu(and)36 b Fn(q)1614 700 y Fm(j)t Ft(+2)1740 686 y Fn(;)15 b(:)g(:)g(:)i(;)e(q)1983 700 y Fm(n)p Ft(+1)2120 686 y Fn(;)g(u)2212 700 y Fm(n)2296 686 y Fu(as)37 b(functions)f(of)h (only)f Fn(q)3170 700 y Fm(j)3206 686 y Fu(,)j Fn(q)3311 700 y Fm(j)t Ft(+1)448 799 y Fu(and)33 b(a)g(certain)g(n)m(um)m(b)s(er) f(of)h(their)f(deriv)-5 b(ativ)m(es.)48 b(W)-8 b(e)34 b(note)f Fn(q)2575 766 y Ft([)p Fm(\013)p Ft(])2693 799 y Fu(=)d(\()p Fn(q)s(;)15 b(q)2957 766 y Ft(\(1\))3052 799 y Fn(;)g(:)g(:)g(:)h(;)f(q)3297 766 y Ft(\()p Fm(\013)p Ft(\))3402 799 y Fu(\))448 912 y(where)31 b Fn(q)756 879 y Ft(\()p Fm(k)r Ft(\))879 912 y Fu(=)26 b Fn(d)1023 879 y Fm(k)1066 912 y Fn(q)s(=dt)1235 879 y Fm(k)1278 912 y Fu(.)42 b(F)-8 b(rom)32 b(Eq.)e(\(54\))j(w)m(e)e(obtain,)g(for)g (some)g(smo)s(oth)f(functions)448 1024 y Fn(I)488 1038 y Ft(0)528 1024 y Fn(;)15 b(:)g(:)g(:)h(I)729 1038 y Fm(j)766 1024 y Fu(,)30 b(the)h(set)g(of)f(equations)798 1240 y Fn(u)850 1254 y Ft(1)988 1240 y Fu(=)98 b Fn(I)1197 1254 y Ft(0)1252 1146 y Fh(\020)1301 1240 y Fn(q)1345 1192 y Ft([2)p Fm(j)t Ft(+1])1342 1266 y Fm(j)1546 1240 y Fn(;)15 b(q)1630 1192 y Ft([2)p Fm(j)t Fl(\000)p Ft(1])1627 1266 y Fm(j)t Ft(+1)1832 1146 y Fh(\021)1911 1240 y Fn(;)1339 b Fu(\(56\))807 1423 y Fn(q)848 1438 y Fm(k)988 1423 y Fu(=)98 b Fn(I)1197 1438 y Fm(k)r Ft(+1)1345 1329 y Fh(\020)1395 1423 y Fn(q)1439 1375 y Ft([2\()p Fm(j)t Fl(\000)p Fm(k)r Ft(\)])1436 1449 y Fm(j)1698 1423 y Fn(;)15 b(q)1782 1375 y Ft([2\()p Fm(j)t Fl(\000)p Ft(1)p Fl(\000)p Fm(k)r Ft(\)])1779 1449 y 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Fm(j)1454 2033 y Fn(;)15 b(q)1538 1985 y Ft([2)p Fm(j)t Ft(+1])1535 2059 y Fm(j)t Ft(+1)1740 1939 y Fh(\021)1819 2033 y Fn(:)1431 b Fu(\(59\))448 2240 y(So)23 b(far)f(w)m(e)i(ha)m(v)m(e)g(simply)c(rewritten)i(the)h (di\013eren)m(tial)e(equations)h(of)h(motion)g(in)e(an)i(im-)448 2353 y(plicit)i(form.)39 b(F)-8 b(rom)26 b(this)g(w)m(e)g(can)h(dra)m (w)f(the)h(follo)m(wing)d(conclusions.)38 b(If)26 b(\()p Fn(P)13 b Fu(\()p Fn(t)p Fu(\))p Fn(;)i(Q)p Fu(\()p Fn(t)p Fu(\)\))448 2466 y(is)22 b(a)h(solution)e(of)i(Eq.)g(\(49\))h(with)e (giv)m(en)g(con)m(trol)h Fn(u)j Fu(=)f(\()p Fn(u)2362 2480 y Ft(1)2401 2466 y Fn(;)15 b(u)2493 2480 y Fm(n)2541 2466 y Fu(\),)25 b(then,)f(using)d(Eqs.)h(\(56\))448 2579 y(and)j(\(59\))i Fn(u)f Fu(can)g(b)s(e)f(written)g(as)g(follo)m (ws:)38 b(there)26 b(is)e(a)i(smo)s(oth)g(function)e Fn(B)30 b Fu(suc)m(h)25 b(that)1299 2791 y Fn(u)p Fu(\()p Fn(t)p Fu(\))41 b(=)f Fn(B)1694 2696 y Fh(\020)1744 2791 y Fn(q)1788 2743 y Ft([2)p Fm(j)t Ft(+1])1785 2816 y Fm(j)1989 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Ft(2)p Fm(d)p Ft(\()p Fm(n)p Ft(+1\))3346 3336 y Fk(!)448 3459 y Fg(R)526 3426 y Ft(2)p Fm(d)p Ft(\()p Fm(n)p Ft(+1\))790 3459 y Fu(,)e(where)f(\()p Fn(P)13 b Fu(\()p Fn(t)p Fu(\))p Fn(;)i(Q)p Fu(\()p Fn(t)p Fu(\)\))27 b(=)e Fn(N)1789 3365 y Fh(\020)1839 3459 y Fn(q)1883 3412 y Ft([2)p Fm(j)t Ft(+1])1880 3485 y Fm(j)2084 3459 y Fn(;)15 b(q)2168 3412 y Ft([2)p Fm(j)t Ft(+1])2165 3485 y Fm(j)t Ft(+1)2369 3365 y Fh(\021)2449 3459 y Fu(is)29 b(giv)m(en)i(b)m(y)664 3701 y Fn(q)705 3716 y Fm(k)846 3701 y Fu(=)98 b Fn(I)1055 3716 y Fm(k)r Ft(+1)1203 3607 y Fh(\020)1252 3701 y Fn(q)1296 3653 y Ft([2\()p Fm(j)t Fl(\000)p Fm(k)r Ft(\)])1293 3727 y Fm(j)1555 3701 y Fn(;)15 b(q)1639 3653 y Ft([2\()p Fm(j)t Fl(\000)p Ft(1)p Fl(\000)p Fm(k)r Ft(\)])1636 3727 y Fm(j)t Ft(+1)1989 3607 y Fh(\021)2069 3701 y Fn(;)106 b(k)29 b Fu(=)c(0)p Fn(;)15 b Fu(1)p Fn(;)g(:)g(:)g(:)i(;)e(j)27 b Fk(\000)20 b Fu(1)15 b Fn(;)681 3884 y Fu(_)-42 b Fn(q)705 3899 y Fm(k)846 3884 y Fu(=)1036 3861 y(_)1015 3884 y Fn(I)1055 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y Fm(j)t Ft(+1)2187 4224 y Fh(\021)2267 4318 y Fn(;)106 b(k)29 b Fu(=)c Fn(j)h Fu(+)20 b(2)15 b Fn(:)g(:)g(:)h(;)f(n)21 b Fu(+)f(1)15 b Fn(;)681 4501 y Fu(_)-42 b Fn(q)705 4516 y Fm(k)846 4501 y Fu(=)1047 4478 y(_)1015 4501 y Fn(J)1065 4516 y Fm(n)p Ft(+2)p Fl(\000)p Fm(k)1311 4407 y Fh(\020)1360 4501 y Fn(q)1404 4453 y Ft([2\()p Fm(k)r Fl(\000)p Fm(j)t Fl(\000)p Ft(2\)+1])1401 4527 y Fm(j)1844 4501 y Fn(;)15 b(q)1928 4453 y Ft([2\()p Fm(k)r Fl(\000)p Fm(j)t Fl(\000)p Ft(1\)+1])1925 4527 y Fm(j)t Ft(+1)2368 4407 y Fh(\021)2448 4501 y Fn(;)106 b(k)28 b Fu(=)d Fn(j)h Fu(+)20 b(2)15 b Fn(:)g(:)g(:)i(;)e(n)g(:)54 b Fu(\(61\))448 4708 y(W)-8 b(e)40 b(no)m(w)f(sho)m(w)g(that)g Fn(N)49 b Fu(is)38 b(a)h(homeomorphism,)g(b)m(y)g(constructing)f(explicitly)f(its)448 4832 y(in)m(v)m(erse.)60 b(W)-8 b(e)39 b(use)d(the)h(equations)g(of)g (motion)f(\(49\))j(to)f(deriv)m(e)e(equations)h(for)f Fn(q)3326 4784 y Ft([2)p Fm(j)t Ft(])3323 4857 y Fm(j)448 4976 y Fu(and)29 b Fn(q)668 4928 y Ft([2)p Fm(j)t Ft(])665 5002 y Fm(j)t Ft(+1)792 4976 y Fu(.)40 b(Di\013eren)m(tiating)29 b(rep)s(eatedly)g(the)h(equations)f(with)g(resp)s(ect)g(to)h(time)g (one)1897 5225 y(15)p eop %%Page: 16 16 16 15 bop 448 573 a Fu(inductiv)m(ely)24 b(\014nds)g(functions)g(smo)s (oth)i(functions)e Fn(K)2292 587 y Ft(0)2332 573 y Fn(;)15 b(:)g(:)g(:)h(K)2570 587 y Ft(2)p Fm(j)2668 573 y Fu(and)25 b Fn(L)2902 587 y Ft(0)2941 573 y Fn(;)15 b(:)g(:)g(:)i(L)3165 587 y Ft(2)p Fm(j)3262 573 y Fu(suc)m(h)448 686 y(that)633 885 y Fn(q)677 837 y Ft(\()p Fm(k)r Ft(\))674 911 y Fm(j)873 885 y Fu(=)98 b Fn(K)1119 900 y Fm(k)1177 791 y Fh(\020)1226 885 y Fn(q)1270 837 y Ft([0])1267 910 y(0)1349 885 y Fn(;)15 b(q)1433 837 y Ft([1])1430 910 y(1)1512 885 y Fn(;)g(:)g(:)g(:)h(;)f(q)1757 837 y Ft([1])1754 911 y Fm(j)t Fl(\000)p Ft(1)1881 885 y Fn(;)g(q)1965 837 y Ft([)p Fm(k)r Fl(\000)p Ft(2])1962 911 y Fm(j)2137 885 y Fn(;)g(q)2221 837 y Ft([)p Fm(k)r Fl(\000)p Ft(2])2218 911 y Fm(j)t Ft(+1)2394 791 y Fh(\021)2474 885 y Fn(;)106 b(k)28 b Fu(=)d(0)p Fn(;)15 b(:)g(:)g(:)i(;)e Fu(2)p Fn(j)22 b(;)123 b Fu(\(62\))607 1068 y Fn(q)651 1020 y Ft(\()p Fm(k)r Ft(\))648 1094 y Fm(j)t Ft(+1)873 1068 y Fu(=)98 b Fn(L)1104 1083 y Fm(k)1161 974 y Fh(\020)1211 1068 y Fn(q)1255 1020 y Ft([)p Fm(k)r Fl(\000)p Ft(2])1252 1094 y Fm(j)1427 1068 y Fn(;)15 b(q)1511 1020 y Ft([)p Fm(k)r Fl(\000)p Ft(2])1508 1094 y Fm(j)t Ft(+1)1683 1068 y Fn(;)g(q)1767 1020 y Ft([1])1764 1094 y Fm(j)t Ft(+2)1891 1068 y Fn(;)g(:)g(:)g(:)i(;)e(q)2137 1030 y Ft([1])2134 1090 y Fm(n)2215 1068 y Fn(;)g(q)2299 1020 y Ft([0])2296 1092 y Fm(n)p Ft(+1)2434 974 y Fh(\021)2513 1068 y Fn(;)107 b(k)28 b Fu(=)d(0)p Fn(;)15 b Fu(1)p Fn(;)g(:)g(:)g(:)j(;)d Fu(2)p Fn(j)22 b(:)m Fu(\(63\))448 1297 y(Eqs.)39 b(\(62\))h(and)f(\(63\))h(de\014ne)e(inductiv)m(ely)f(a) i(smo)s(oth)g(map)f Fn(M)49 b Fu(from)38 b Fg(R)3053 1264 y Ft(2)p Fm(d)p Ft(\()p Fm(n)p Ft(+1\))3356 1297 y Fu(to)448 1410 y Fg(R)526 1377 y Ft(2)p Fm(d)p Ft(\()p Fm(n)p Ft(+1\))821 1410 y Fu(giv)m(en)30 b(b)m(y)1445 1439 y Fh(\020)1495 1534 y Fn(q)1539 1486 y Ft([2)p Fm(j)t Ft(])1536 1559 y Fm(j)1649 1534 y Fn(;)15 b(q)1733 1486 y Ft([2)p Fm(j)t Ft(])1730 1559 y Fm(j)t Ft(+1)1857 1439 y Fh(\021)1947 1534 y Fu(=)40 b Fn(M)10 b Fu(\()p Fn(P)s(;)15 b(Q)p Fu(\))g Fn(:)836 b Fu(\(64\))448 1713 y(W)-8 b(e)32 b(ha)m(v)m(e)f(sho)m(wn)f(that)h(if)f(\()p Fn(P)13 b Fu(\()p Fn(t)p Fu(\))p Fn(;)i(Q)p Fu(\()p Fn(t)p Fu(\)\))32 b(is)d(a)i(solution)e(of)i(Eq.)f(\(49\),)i(then)1233 1834 y Fh(\020)1282 1928 y Fn(q)1326 1880 y Ft([2)p Fm(j)t Ft(])1323 1954 y Fm(j)1437 1928 y Fu(\()p Fn(t)p Fu(\))p Fn(;)15 b(q)1624 1880 y Ft([2)p Fm(j)t Ft(])1621 1954 y Fm(j)t Ft(+1)1748 1928 y Fu(\()p Fn(t)p Fu(\))1851 1834 y Fh(\021)1927 1928 y Fu(=)25 b Fn(M)g Fu(\()p Fn(P)13 b Fu(\()p Fn(t)p Fu(\))p Fn(;)i(Q)p Fu(\()p Fn(t)p Fu(\)\))32 b Fn(:)623 b Fu(\(65\))448 2143 y(Since)43 b(the)g(solution)f(of)i (\(49\))h(is)d(unique)g(this)g(sho)m(ws)h(that)h Fn(M)54 b Fu(is)42 b(the)i(in)m(v)m(erse)f(of)448 2256 y(the)37 b(map)f Fn(N)46 b Fu(giv)m(en)37 b(b)m(y)f(Eq.)g(\(61\))i(and)e(th)m (us)g Fn(N)46 b Fu(is)36 b(a)g(homeomorphism)f(\(in)g(fact)j(a)448 2369 y(di\013eomorphism\).)589 2481 y(W)-8 b(e)26 b(ha)m(v)m(e)f(pro)m (v)m(en)f(the)g(follo)m(wing:)36 b(The)23 b(system)h(of)g(equations)g (\(49\))h(with)e(b)s(ound-)448 2594 y(ary)31 b(data)g(\(50\))h(is)d (equiv)-5 b(alen)m(t)30 b(to)h(equation)f(\(60\))i(with)d(the)i(b)s (oundary)d(data)685 2715 y Fh(\020)734 2809 y Fn(q)778 2761 y Ft([2)p Fm(j)t Ft(])775 2835 y Fm(j)889 2809 y Fu(\(0\))p Fn(;)15 b(q)1088 2761 y Ft([2)p Fm(j)t Ft(])1085 2835 y Fm(j)t Ft(+1)1213 2809 y Fu(\(0\))1328 2715 y Fh(\021)1419 2809 y Fu(=)40 b Fn(M)10 b Fu(\()p Fn(y)s Fu(\))15 b Fn(;)1893 2715 y Fh(\020)1943 2809 y Fn(q)1987 2761 y Ft([2)p Fm(j)t Ft(])1984 2835 y Fm(j)2098 2809 y Fu(\()p Fn(T)e Fu(\))p Fn(;)i(q)2318 2761 y Ft([2)p Fm(j)t Ft(])2315 2835 y Fm(j)t Ft(+1)2442 2809 y Fu(\()p Fn(T)e Fu(\))2578 2715 y Fh(\021)2668 2809 y Fu(=)40 b Fn(M)10 b Fu(\()p Fn(z)t Fu(\))15 b Fn(:)242 b Fu(\(66\))448 3019 y(F)-8 b(rom)44 b(this)e(the)i(assertion)f(of)g(the)h(theorem)g (follo)m(ws)e(easily:)66 b(First)43 b(w)m(e)h(see)f(that)448 3142 y Fn(V)522 3095 y Ft(\()p Fm(\021)r Ft(\))501 3170 y Fm(T)618 3142 y Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))27 b Fn(<)e Fk(1)p Fu(,)30 b(for)g(all)e Fn(T)38 b(>)25 b Fu(0)31 b(and)e(for)g(all)g Fn(y)s(;)15 b(z)t Fu(.)41 b(Indeed)29 b(c)m(ho)s(ose)i(an)m(y)f(su\016cien)m(tly)448 3255 y(smo)s(oth)46 b(curv)m(es)f Fn(q)1119 3269 y Fm(j)1155 3255 y Fu(\()p Fn(t)p Fu(\))h(and)f Fn(q)1537 3269 y Fm(j)t Ft(+1)1663 3255 y Fu(\()p Fn(t)p Fu(\))h(whic)m(h)e(satis\014es) h(the)g(b)s(oundary)f(conditions)448 3368 y(\(66\))29 b(and)e(consider)f(the)h Fn(u)g Fu(giv)m(en)g(b)m(y)g(Eq.)h(\(60\).)41 b(Then)26 b(the)h(function)f(\()p Fn(P)13 b Fu(\()p Fn(t)p Fu(\))p Fn(;)i(Q)p Fu(\()p Fn(t)p Fu(\)\))27 b(=)448 3497 y Fn(M)562 3403 y Fh(\020)611 3497 y Fn(q)655 3449 y Ft([2)p Fm(j)t Ft(])652 3523 y Fm(j)766 3497 y Fu(\()p Fn(t)p Fu(\))p Fn(;)15 b(q)953 3449 y Ft([2)p Fm(j)t Ft(])950 3523 y Fm(j)t Ft(+1)1077 3497 y Fu(\()p Fn(t)p Fu(\))1180 3403 y Fh(\021)1265 3497 y Fu(is)33 b(a)h(solution)f(of)i (Eq.)f(\(49\))i(with)d(b)s(oundary)f(data)j(\(50\))h(and)448 3626 y(with)29 b(a)i(con)m(trol)g Fn(u)p Fu(\()p Fn(t)p Fu(\))g(giv)m(en)f(b)m(y)g(\(60\))i(whic)m(h)d(steers)i Fn(y)i Fu(to)f Fn(z)t Fu(.)589 3754 y(In)37 b(order)f(to)i(pro)m(v)m(e) g(the)g(upp)s(er)d(semicon)m(tin)m(uit)m(y)h(of)h Fn(V)2552 3706 y Ft(\()p Fm(\021)r Ft(\))2531 3782 y Fm(T)2648 3754 y Fu(\()p Fn(y)s(;)15 b(z)t Fu(\),)41 b(let)c(us)f(c)m(ho)s(ose) 448 3888 y(some)g Fn(\017)e(>)f Fu(0.)57 b(By)36 b(de\014nition)d(of)i Fn(V)1722 3840 y Ft(\()p Fm(\021)r Ft(\))1701 3915 y Fm(T)1854 3888 y Fu(there)h(is)e(a)i(con)m(trol)g Fn(u)f Fu(whic)m(h)f(steers)i Fn(y)j Fu(to)d Fn(z)448 4001 y Fu(along)31 b(a)g(path)f Fn(\036)25 b Fu(=)g Fn(\036)1208 3968 y Fm(u)1283 4001 y Fu(suc)m(h)31 b(that)1379 4210 y Fn(I)1419 4224 y Fm(y)r(;T)1531 4210 y Fu(\()p Fn(\036)1620 4173 y Fm(u)1665 4210 y Fu(\))41 b Fk(\024)f Fn(V)1925 4162 y Ft(\()p Fm(\021)r Ft(\))1905 4238 y Fm(T)2022 4210 y Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))22 b(+)d Fn(\017=)p Fu(2)c Fn(;)770 b Fu(\(67\))448 4409 y(and)1299 4522 y Fn(u)p Fu(\()p Fn(t)p Fu(\))41 b(=)f Fn(B)1694 4428 y Fh(\020)1744 4522 y Fn(q)1788 4474 y Ft([2)p Fm(j)t Ft(+1])1785 4548 y Fm(j)1989 4522 y Fu(\()p Fn(t)p Fu(\))p Fn(;)15 b(q)2176 4474 y Ft([2)p Fm(j)t Ft(+1])2173 4548 y Fm(j)t Ft(+1)2378 4522 y Fu(\()p Fn(t)p Fu(\))2481 4428 y Fh(\021)2561 4522 y Fn(;)689 b Fu(\(68\))448 4697 y(Using)30 b(the)g(smo)s(othness)g(of)h Fn(B)5 b Fu(,)30 b(w)m(e)h(c)m(ho)s(ose)g(curv)m(es)37 b(~)-51 b Fn(q)2319 4711 y Fm(j)2385 4697 y Fu(and)37 b(~)-52 b Fn(q)2603 4711 y Fm(j)t Ft(+1)2759 4697 y Fu(suc)m(h)30 b(that)1042 4916 y(sup)1001 4997 y Fm(t)p Fl(2)p Ft([0)p Fm(;T)10 b Ft(])1234 4916 y Fk(j)p Fn(q)1303 4868 y Ft([2)p Fm(j)t Ft(+1])1300 4942 y Fm(j)1524 4916 y Fk(\000)27 b Fu(~)-52 b Fn(q)1659 4868 y Ft([2)p Fm(j)t Ft(+1])1656 4942 y Fm(j)1860 4916 y Fk(j)21 b Fu(+)f Fk(j)p Fn(q)2066 4868 y Ft([2)p Fm(j)t Ft(+1])2063 4942 y Fm(j)t Ft(+1)2287 4916 y Fk(\000)27 b Fu(~)-52 b Fn(q)2422 4868 y Ft([2)p Fm(j)t Ft(+1])2419 4942 y Fm(j)t Ft(+1)2623 4916 y Fk(j)41 b(\024)f Fn(\016)19 b(;)391 b Fu(\(69\))1897 5225 y(16)p eop %%Page: 17 17 17 16 bop 448 573 a Fu(and)30 b Fn(\016)k Fu(is)c(so)g(small)f(that) 1354 747 y(~)-50 b Fn(u)p Fu(\()p Fn(t)p Fu(\))41 b(=)f Fn(B)5 b Fu(\()i(~)-52 b Fn(q)1809 700 y Ft([2)p Fm(j)t Ft(+1])1806 773 y Fm(j)2010 747 y Fn(;)22 b Fu(~)-52 b Fn(q)2094 700 y Ft([2)p Fm(j)t Ft(+1])2091 773 y Fm(j)t Ft(+1)2295 747 y Fu(\))15 b Fn(;)905 b Fu(\(70\))448 922 y(satis\014es)1458 1061 y(sup)1417 1142 y Fm(t)p Fl(2)p Ft([0)p Fm(;T)10 b Ft(])1650 1061 y Fk(j)p Fn(u)p Fu(\()p Fn(t)p Fu(\))21 b Fk(\000)26 b Fu(~)-51 b Fn(u)p Fu(\()p Fn(t)p Fu(\))p Fk(j)41 b(\024)2259 937 y Fh(r)p 2342 937 86 4 v 2366 999 a Fn(\017)p 2352 1040 66 4 v 2352 1123 a(T)2443 1061 y(:)807 b Fu(\(71\))448 1312 y(W)-8 b(e)37 b(note)43 b(~)-52 b Fn(y)37 b Fu(=)d Fn(M)10 b Fu(\()d(~)-52 b Fn(q)1181 1264 y Ft([2)p Fm(j)t Ft(])1178 1338 y Fm(j)1292 1312 y Fu(\(0\))p Fn(;)23 b Fu(~)-53 b Fn(q)1491 1264 y Ft([2)p Fm(j)t Ft(])1488 1338 y Fm(j)t Ft(+1)1616 1312 y Fu(\(0\)\))37 b(and)j(~)-50 b Fn(z)38 b Fu(=)c Fn(N)10 b Fu(\()d(~)-52 b Fn(q)2332 1264 y Ft([2)p Fm(j)t Ft(])2329 1338 y Fm(j)2443 1312 y Fu(\()p Fn(T)13 b Fu(\))p Fn(;)22 b Fu(~)-52 b Fn(q)2663 1264 y Ft([2)p Fm(j)t Ft(])2660 1338 y Fm(j)t Ft(+1)2787 1312 y Fu(\()p Fn(T)13 b Fu(\)\))36 b(and)f Fn(\036)3230 1279 y Fm(u)3311 1312 y Fu(the)448 1425 y(path)30 b(along)h(whic)m(h)e(the)i(con)m(trol) f Fn(u)h Fu(steers)f(the)h(system,)g(one)f(obtains)1412 1613 y Fk(j)p Fn(I)1477 1627 y Fm(y)r(;T)1589 1613 y Fu(\()p Fn(\036)1678 1576 y Fm(u)1723 1613 y Fu(\))21 b Fk(\000)f Fn(I)1915 1627 y Ft(~)-40 b Fm(y)r(;T)2021 1613 y Fu(\()p Fn(\036)2115 1576 y Ft(~)g Fm(u)2156 1613 y Fu(\))p Fk(j)41 b(\024)2382 1552 y Fn(\017)p 2378 1592 46 4 v 2378 1676 a Fu(2)2449 1613 y Fn(:)801 b Fu(\(72\))448 1815 y(By)31 b(the)f(con)m(tin)m(uit)m(y)g(of)g(the)h(map)f Fn(N)10 b Fu(,)30 b(w)m(e)h(can)f(c)m(ho)s(ose)h Fn(\016)2407 1782 y Fl(0)2461 1815 y Fu(so)g(small)d(that)j(if)e Fk(j)p Fn(y)23 b Fk(\000)j Fu(~)-51 b Fn(y)r Fk(j)20 b Fu(+)448 1928 y Fk(j)p Fn(z)25 b Fk(\000)g Fu(~)-50 b Fn(z)t Fk(j)26 b(\024)f Fn(\016)867 1895 y Fl(0)891 1928 y Fu(,)30 b(then)1075 2119 y Fk(j)p Fn(q)1144 2071 y Ft([2)p Fm(j)t Ft(])1141 2145 y Fm(j)1256 2119 y Fu(\(0\))21 b Fk(\000)27 b Fu(~)-52 b Fn(q)1527 2071 y Ft([2)p Fm(j)t Ft(])1524 2145 y Fm(j)1638 2119 y Fu(\(0\))p Fk(j)21 b Fu(+)f Fk(j)p Fn(q)1959 2071 y Ft([2)p Fm(j)t Ft(])1956 2145 y Fm(j)t Ft(+1)2083 2119 y Fu(\(0\))h Fk(\000)27 b Fu(~)-52 b Fn(q)2354 2071 y Ft([2)p Fm(j)t Ft(])2351 2145 y Fm(j)t Ft(+1)2477 2119 y Fu(\(0\))p Fk(j)1075 2288 y Fu(+)p Fk(j)p Fn(q)1215 2241 y Ft([2)p Fm(j)t Ft(])1212 2314 y Fm(j)1326 2288 y Fu(\()p Fn(T)13 b Fu(\))21 b Fk(\000)27 b Fu(~)-52 b Fn(q)1618 2241 y Ft([2)p Fm(j)t Ft(])1615 2314 y Fm(j)1729 2288 y Fu(\()p Fn(T)13 b Fu(\))p Fk(j)21 b Fu(+)e Fk(j)p Fn(q)2070 2241 y Ft([2)p Fm(j)t Ft(])2067 2314 y Fm(j)t Ft(+1)2194 2288 y Fu(\()p Fn(T)13 b Fu(\))21 b Fk(\000)26 b Fu(~)-51 b Fn(q)2486 2241 y Ft([2)p Fm(j)t Ft(])2483 2314 y Fm(j)t Ft(+1)2609 2288 y Fu(\()p Fn(T)13 b Fu(\))p Fk(j)26 b(\024)f Fn(\016)19 b(:)448 2463 y Fu(Therefore)30 b(for)h(all)e(suc)m(h)36 b(~)-51 b Fn(y)s(;)21 b Fu(~)-51 b Fn(z)35 b Fu(w)m(e)c(ha)m(v)m(e)1161 2648 y Fn(V)1234 2601 y Ft(\()p Fm(\021)r Ft(\))1214 2676 y Fm(T)1331 2648 y Fu(\()6 b(~)-51 b Fn(y)s(;)21 b Fu(~)-51 b Fn(z)5 b Fu(\))41 b Fk(\024)f Fn(I)1733 2662 y Ft(~)-40 b Fm(y)r(;T)1839 2648 y Fu(\()p Fn(\036)1933 2611 y Ft(~)g Fm(u)1974 2648 y Fu(\))41 b Fk(\024)f Fn(V)2234 2601 y Ft(\()p Fm(\021)r Ft(\))2214 2676 y Fm(T)2331 2648 y Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))21 b(+)f Fn(\017)15 b(:)551 b Fu(\(73\))448 2843 y(This)34 b(sho)m(ws)h(the)h(upp)s(er)d(semicon)m(tin)m(uit)m(y)h (of)i Fn(V)2147 2796 y Ft(\()p Fm(\021)r Ft(\))2127 2871 y Fm(T)2244 2843 y Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))37 b(and)d(the)i(upp)s(er)d(semicon-)448 2956 y(tin)m(uit)m(y)39 b(of)h Fn(V)939 2923 y Ft(\()p Fm(\021)r Ft(\))1036 2956 y Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))41 b(follo)m(ws)e(easily)g(from)h (this.)68 b(This)38 b(concludes)h(the)h(pro)s(of)f(of)448 3069 y(Lemma)31 b(2.3.)p 1004 3077 50 67 v 589 3262 a(An)21 b(immediate)g(consequence)h(of)g(this)e(Lemma)i(is)e(a)i(b)s(ound)d(on) i(the)h(cost)g(function)448 3375 y(around)30 b(critical)f(p)s(oin)m(ts) g(of)i(the)f(generalized)g(Hamiltonian)f Fn(G)p Fu(.)448 3558 y Fg(Corollary)35 b(2.4)46 b Fe(F)-7 b(or)36 b(any)f Fn(x)28 b Fk(2)g Fn(A)h Fu(=)f Fk(f)p Fn(y)47 b Fu(:)d Fk(r)p Fn(G)p Fu(\()p Fn(y)s Fu(\))28 b(=)g(0)p Fk(g)36 b Fe(and)f(any)g Fn(h)29 b(>)f Fu(0)35 b Fe(ther)-5 b(e)35 b(is)448 3671 y Fn(\016)29 b(>)c Fu(0)33 b Fe(such)g(that,)h(if)e Fk(j)p Fn(y)23 b Fk(\000)d Fn(x)p Fk(j)h Fu(+)f Fk(j)p Fn(z)k Fk(\000)c Fn(x)p Fk(j)26 b(\024)f Fn(\016)s Fe(,)33 b(then)g(one)g(has)1633 3845 y Fn(V)1706 3808 y Ft(\()p Fm(\021)r Ft(\))1803 3845 y Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))42 b Fk(\024)e Fn(h)15 b(:)1023 b Fu(\(74\))448 4028 y Fe(Pr)-5 b(o)g(of:)42 b Fu(If)30 b Fn(x)25 b Fk(2)g Fn(A)p Fu(,)31 b Fn(x)f Fu(is)g(a)g(stationary)h(p)s(oin)m(t)e(of)i (the)f(equation)1585 4203 y(_)-42 b Fn(q)101 b Fu(=)d Fk(r)1955 4217 y Fm(p)1994 4203 y Fn(G)15 b(;)1584 4341 y Fu(_)-43 b Fn(p)98 b Fu(=)g Fk(\000r)2026 4355 y Fm(q)2063 4341 y Fn(G)15 b(;)1582 4478 y Fu(_)-39 b Fn(r)101 b Fu(=)d Fk(\000)p Fn(\015)5 b(\025)2055 4441 y Ft(2)2094 4478 y Fk(r)2170 4492 y Fm(r)2208 4478 y Fn(G)15 b(:)955 b Fu(\(75\))448 4653 y(As)32 b(a)h(consequence)g(the)f(con)m(trol)g Fn(u)c Fk(\021)g Fu(0)33 b(steers)f(0)h(to)f(0)h(and)e(hence)h Fn(V)2928 4620 y Ft(\()p Fm(\021)r Ft(\))3025 4653 y Fu(\()p Fn(x;)15 b(x)p Fu(\))29 b(=)e(0.)448 4766 y(The)37 b(upp)s(er)d(semicon)m(tin)m(uit)m(y)i(of)h Fn(V)1704 4733 y Ft(\()p Fm(\021)r Ft(\))1801 4766 y Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))38 b(immediately)d(implies)f(the)j(statemen)m(t)448 4879 y(of)31 b(the)f(corollary)-8 b(.)p 1163 4887 V 1897 5225 a(17)p eop %%Page: 18 18 18 17 bop 448 573 a Fg(Remark)35 b(2.5)46 b Fu(This)28 b(corollary)i(sligh)m(tly)e(falls)h(short)g(of)i(what)f(is)f(needed)h (to)h(obtain)448 686 y(the)f(asymptotics)h(of)f(the)g(in)m(v)-5 b(arian)m(t)29 b(measure.)40 b(More)31 b(detailed)e(information)f(ab)s (out)448 799 y(the)39 b(geometry)g(of)g(the)f(con)m(trol)h(paths)f (around)f(the)h(stationary)h(p)s(oin)m(ts)e(is)g(needed)448 912 y(and)30 b(will)e(b)s(e)i(pro)m(v)m(ed)g(in)f(the)i(next)f (subsection.)448 1155 y Fd(2.4)113 b(Geometry)36 b(of)i(the)f(paths)h (around)h(the)e(stationary)g(p)s(oin)m(ts)448 1327 y Fu(Let)31 b(us)f(consider)f(a)i(con)m(trol)g(system)f(of)h(the)f(form) 1483 1586 y(_)-41 b Fn(x)40 b Fu(=)g Fn(Y)20 b Fu(\()p Fn(x)p Fu(\))h(+)2006 1480 y Fm(m)1977 1505 y Fh(X)1980 1687 y Fm(i)p Ft(=1)2112 1586 y Fn(X)2187 1600 y Fm(i)2216 1586 y Fu(\()p Fn(x)p Fu(\))p Fn(u)2390 1600 y Fm(i)3275 1586 y Fu(\(76\))448 1862 y(where)47 b Fn(x)52 b Fk(2)g Fg(R)1023 1829 y Fm(n)1070 1862 y Fu(,)f Fn(Y)20 b Fu(\()p Fn(x)p Fu(\))p Fn(;)15 b(X)1456 1876 y Fm(i)1486 1862 y Fu(\()p Fn(x)p Fu(\))47 b(are)g(smo)s(oth)g(v)m(ector)h(\014elds.)88 b(W)-8 b(e)48 b(assume)e(that)448 1975 y Fn(Y)20 b Fu(\()p Fn(x)p Fu(\))p Fn(;)15 b(X)758 1989 y Fm(i)788 1975 y Fu(\()p Fn(x)p Fu(\))31 b(are)h(suc)m(h)f(that)g(Eq.)g(\(76\))i(has)e (a)g(unique)e(solution)h(for)h(all)e(time)i Fn(t)26 b(>)g Fu(0.)448 2088 y(W)-8 b(e)44 b(w)m(an)m(t)g(to)g(in)m(v)m(estigate)g (prop)s(erties)d(of)i(the)g(set)h(whic)m(h)d(can)j(b)s(e)e(reac)m(hed)i (from)448 2201 y(a)e(giv)m(en)g(p)s(oin)m(t)e(b)m(y)i(allo)m(wing)e (only)g(con)m(trols)i(with)e(b)s(ounded)g(size.)74 b(The)41 b(class)g(of)448 2314 y(con)m(trols)31 b Fn(u)f Fu(w)m(e)h(consider)e (is)h(giv)m(en)g(b)m(y)630 2517 y Fk(U)687 2531 y Fm(M)807 2517 y Fu(=)40 b Fk(f)p Fn(u)30 b Fu(piecewise)g(smo)s(oth)p Fn(;)46 b Fu(with)29 b Fk(j)p Fn(u)2086 2531 y Fm(i)2114 2517 y Fu(\()p Fn(t)p Fu(\))p Fk(j)d(\024)f Fn(M)56 b(;)45 b Fu(1)26 b Fk(\024)f Fn(i)h Fk(\024)f Fn(m)p Fk(g)45 b Fn(:)182 b Fu(\(77\))448 2721 y(W)-8 b(e)27 b(denote)f Fn(Y)961 2688 y Fm(M)941 2745 y Fl(\024)p Fm(\034)1040 2721 y Fu(\()p Fn(x)p Fu(\))h(the)f(set)g(of)g(p)s(oin)m(ts)e(whic)m(h) h(can)h(b)s(e)f(reac)m(hed)h(from)f Fn(x)h Fu(in)e(time)i(less)448 2834 y(than)h Fn(\034)37 b Fu(with)26 b(a)i(con)m(trol)g Fn(u)d Fk(2)g(U)1534 2848 y Fm(M)1613 2834 y Fu(.)39 b(W)-8 b(e)29 b(sa)m(y)f(that)g(the)f(con)m(trol)h(system)f(is)f Fe(smal)5 b(l-time)448 2947 y(lo)-5 b(c)g(al)5 b(ly)36 b(c)-5 b(ontr)g(ol)5 b(lable)34 b Fu(\(STLC\))d(at)h Fn(x)f Fu(if)g Fn(Y)1901 2914 y Fm(M)1881 2971 y Fl(\024)p Fm(\034)1980 2947 y Fu(\()p Fn(x)p Fu(\))h(con)m(tains)f(a)h(neigh)m(b) s(orho)s(o)s(d)d(of)j Fn(x)f Fu(for)448 3060 y(ev)m(ery)g Fn(\034)36 b(>)25 b Fu(0.)589 3172 y(The)h(follo)m(wing)f(result)g(is)g (standard)h(in)f(con)m(trol)i(theory)-8 b(,)28 b(see)e(e.g.)41 b([23)q(])26 b(or)h([19)q(])f(for)448 3285 y(a)31 b(pro)s(of.)448 3497 y Fg(Prop)s(osition)37 b(2.6)46 b Fe(Consider)37 b(the)g(c)-5 b(ontr)g(ol)39 b(system)f(Eq.)d(\(76\))j(with)g Fn(u)32 b Fk(2)g(U)3152 3511 y Fm(M)3231 3497 y Fe(.)54 b(L)-5 b(et)448 3610 y Fn(x)500 3624 y Ft(0)570 3610 y Fe(b)g(e)29 b(an)i(e)-5 b(quilibrium)29 b(p)-5 b(oint)32 b(of)e Fn(Y)20 b Fu(\()p Fn(x)p Fu(\))p Fe(,)31 b(i.e.,)e Fn(Y)20 b Fu(\()p Fn(x)2202 3624 y Ft(0)2242 3610 y Fu(\))26 b(=)f(0)p Fe(.)41 b(If)30 b(the)g(line)-5 b(ar)31 b(sp)-5 b(an)31 b(of)g(the)448 3723 y(br)-5 b(ackets)997 3836 y Fu(ad)1093 3798 y Fm(k)1136 3836 y Fu(\()p Fn(Y)20 b Fu(\)\()p Fn(X)1389 3850 y Fm(i)1418 3836 y Fu(\)\()p Fn(x)p Fu(\))94 b Fn(i)26 b Fu(=)f(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(m)g(;) 109 b(k)28 b Fu(=)d(0)p Fn(;)15 b Fu(1)p Fn(;)g Fu(2)p Fn(;)g(:)g(:)g(:)34 b(;)387 b Fu(\(78\))448 4002 y Fe(has)34 b(r)-5 b(ank)34 b Fn(n)e Fe(at)h Fn(x)1065 4016 y Ft(0)1137 4002 y Fe(then)g(Eq.)f(\(76\))i(is)e(STLC)h(at)g Fn(x)2232 4016 y Ft(0)2271 4002 y Fe(.)448 4214 y(Pr)-5 b(o)g(of:)46 b Fu(One)32 b(pro)m(v)m(es)h(Lemma)f(2.6)h(b)m(y)f(linearizing)e (around)h Fn(X)2638 4228 y Ft(0)2710 4214 y Fu(and)h(using)e(e.g.)48 b(the)448 4327 y(implicit)28 b(function)h(theorem,)i(see)g(e.g.)41 b([19)r(],)30 b(Chapter)g(6,)h(Theorem)f(1.)p 3037 4335 50 67 v 589 4539 a(As)h(a)g(consequence)g(of)f(Lemma)h(2.6)g(and)f (results)f(obtained)h(in)f([6)q(])h(one)h(gets)448 4751 y Fg(Lemma)i(2.7)46 b Fe(Consider)35 b(the)f(c)-5 b(ontr)g(ol)35 b(system)g(given)e(by)g(Eqs.)44 b(\(47\))35 b(with)f Fn(u)27 b Fk(2)f(U)3330 4765 y Fm(M)3409 4751 y Fe(.)448 4863 y(L)-5 b(et)43 b Fn(x)667 4877 y Ft(0)749 4863 y Fe(b)-5 b(e)42 b(a)h(critic)-5 b(al)43 b(p)-5 b(oint)44 b(of)f Fn(G)p Fu(\()p Fn(x)p Fu(\))p Fe(.)71 b(If)42 b(c)-5 b(ondition)44 b Fg(K2)f Fe(is)f(satis\014e)-5 b(d,)47 b(then)c(the)448 4976 y(system)34 b(\(47\))f(is)g(STLC)g(at)g Fn(x)1475 4990 y Ft(0)1514 4976 y Fe(.)1897 5225 y Fu(18)p eop %%Page: 19 19 19 18 bop 448 573 a Fe(Pr)-5 b(o)g(of:)40 b Fu(The)26 b(prop)s(ert)m(y)f(of)h(small)f(time)g(lo)s(cal)h(con)m(trollabilit)m (y)e(is)h(expressed)h(as)g(a)g(con-)448 686 y(dition)g(that)j(certain)e (brac)m(k)m(ets)j(generate)f(the)f(whole)f(tangen)m(t)i(space)f(at)h (some)f(p)s(oin)m(t)448 799 y Fn(x)500 813 y Ft(0)540 799 y Fu(.)39 b(This)24 b(prop)s(ert)m(y)h(is)g(ob)m(viously)f(related) i(to)h(the)e(h)m(yp)s(o)s(ellipticit)m(y)e(of)j(the)g(generator)448 912 y(of)34 b(the)g(Mark)m(o)m(v)h(pro)s(cess)e(\(5\))i(asso)s(ciated)f (to)h(the)e(con)m(trol)h(system)g(\(47\).)52 b(The)33 b(gen-)448 1024 y(erator)i(of)f(the)g(Mark)m(o)m(v)i(pro)s(cess)d(whic) m(h)g(solv)m(es)h Fn(dy)s Fu(\()p Fn(t)p Fu(\))d(=)g Fn(Y)20 b Fu(\()p Fn(x)p Fu(\))j(+)2811 960 y Fh(P)2898 1047 y Fm(i)2942 1024 y Fn(X)3017 1038 y Fm(i)3045 1024 y Fu(\()p Fn(x)p Fu(\))p Fn(dw)3279 1038 y Fm(i)3308 1024 y Fu(\()p Fn(t)p Fu(\),)448 1137 y(where)31 b Fn(w)777 1151 y Fm(i)805 1137 y Fu(\()p Fn(t)p Fu(\))h(is)e(a)h(1-dimensional)e (pro)s(cess)i(is)f(giv)m(en)h(on)g(su\016cien)m(tly)f(smo)s(oth)h (func-)448 1250 y(tions)j(b)m(y)f(the)h(di\013eren)m(tial)f(op)s (erator)h Fn(L)d Fu(=)f(\(1)p Fn(=)p Fu(2\))2207 1186 y Fh(P)2297 1273 y Fm(i)2325 1250 y Fu(\()p Fk(r)23 b(\001)g Fn(X)2582 1264 y Fm(i)2610 1250 y Fu(\)\()p Fn(X)2755 1264 y Fm(i)2807 1250 y Fk(\001)g(r)p Fu(\))f(+)g Fn(Y)43 b Fk(\001)23 b(r)p Fu(.)51 b(If)448 1363 y Fn(Y)20 b Fu(\()p Fn(x)p Fu(\))32 b(and)f Fn(X)928 1377 y Fm(i)957 1363 y Fu(\()p Fn(x)p Fu(\))h(are)f Fk(C)1316 1330 y Fl(1)1391 1363 y Fu(,)h(then)f Fn(L)g Fu(is)g(h)m(yp)s(o)s(elliptic)d (if)i(the)i(Lie)f(algebra)g(generated)448 1476 y(b)m(y)24 b Fn(Y)c Fu(\()p Fn(X)7 b Fu(\))25 b(and)e Fn(X)1063 1490 y Fm(i)1091 1476 y Fu(\()p Fn(x)p Fu(\))i(generates)g(the)f (tangen)m(t)h(space)f(at)g(eac)m(h)h(p)s(oin)m(t)e Fn(x)h Fu([12)q(].)38 b(F)-8 b(or)25 b(the)448 1589 y(system)30 b(of)g(equations)f(\(5\),)i(it)e(is)g(pro)m(v)m(ed)h(in)e([6)q(],)i (that)g(if)f(condition)f Fg(K2)i Fu(is)f(satis\014ed,)448 1702 y(then)h(the)h(brac)m(k)m(ets)1027 1900 y(ad)1123 1862 y Fm(k)1166 1900 y Fu(\()p Fn(Y)20 b Fu(\)\()p Fn(X)1419 1914 y Fm(i)1448 1900 y Fu(\)\()p Fn(x)p Fu(\))92 b Fn(i)26 b Fu(=)f(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(m)g(;)106 b(k)29 b Fu(=)c(0)p Fn(;)15 b Fu(1)p Fn(;)g Fu(2)p Fn(;)g(:)g(:)g(:)420 b Fu(\(79\))448 2098 y(generates)36 b(the)f(tangen)m(t)g(space)g(at)g (eac)m(h)h(p)s(oin)m(t)d Fn(x)p Fu(,)i(in)e(particular)g(at)i(ev)m(ery) g(critical)448 2211 y(p)s(oin)m(t)27 b Fn(x)735 2225 y Ft(0)775 2211 y Fu(,)i(and)e(therefore)i(b)m(y)f(Lemma)g(2.6,)i(the)e (con)m(trol)h(system)f(Eq.)h(\(47\))g(is)e(STLC)448 2324 y(at)k Fn(x)611 2338 y Ft(0)651 2324 y Fu(.)p 766 2332 50 67 v 589 2532 a(With)j(these)h(results)e(w)m(e)h(can)h(deriv)m(e)f (the)g(basic)f(fact)j(on)e(the)g(geometry)i(of)e(the)448 2645 y(con)m(trol)d(paths)f(around)f(equilibrium)d(p)s(oin)m(ts)j(of)i Fn(G)p Fu(\()p Fn(x)p Fu(\).)448 2852 y Fg(Prop)s(osition)37 b(2.8)46 b Fe(Consider)40 b(the)f(c)-5 b(ontr)g(ol)42 b(system)d(given)g(by)g(\(47\).)62 b(L)-5 b(et)39 b Fn(x)3194 2866 y Ft(0)3273 2852 y Fe(b)-5 b(e)39 b(a)448 2964 y(critic)-5 b(al)30 b(p)-5 b(oint)30 b(of)f Fn(G)p Fu(\()p Fn(x)p Fu(\))g Fe(and)h Fn(B)5 b Fu(\()p Fn(\032)p Fu(\))29 b Fe(the)h(b)-5 b(al)5 b(l)29 b(of)g(r)-5 b(adius)30 b Fn(\032)f Fe(c)-5 b(enter)g(e)g(d)31 b(at)e Fn(X)2983 2978 y Ft(0)3023 2964 y Fe(.)40 b(Then)29 b(for)448 3077 y(any)34 b Fn(h)26 b(>)f Fu(0)p Fe(,)33 b(ther)-5 b(e)34 b(ar)-5 b(e)34 b Fn(\032)1332 3044 y Fl(0)1381 3077 y Fn(>)25 b Fu(0)33 b Fe(and)h Fn(\032)26 b(>)f Fu(0)33 b Fe(with)h Fn(\032)25 b(<)h(\032)2393 3044 y Fl(0)2416 3077 y Fn(=)p Fu(3)34 b Fe(such)f(that)h(the)f(fol)5 b(lowing)448 3190 y(hold:)44 b(F)-7 b(or)33 b(any)h Fn(x;)15 b(y)28 b Fk(2)d Fn(B)5 b Fu(\()p Fn(\032)p Fu(\))p Fe(,)33 b(ther)-5 b(e)33 b(is)g Fn(T)38 b(>)25 b Fu(0)33 b Fe(and)g Fn(u)25 b Fk(2)g(U)2513 3204 y Fm(M)2625 3190 y Fe(with)1425 3388 y Fn(\036)1479 3351 y Fm(u)1525 3388 y Fu(\(0\))h(=)f Fn(x)201 b(\036)2069 3351 y Fm(u)2114 3388 y Fu(\()p Fn(T)13 b Fu(\))26 b(=)f Fn(y)18 b(;)815 b Fu(\(80\))1296 3586 y Fn(\036)1350 3549 y Fm(u)1396 3586 y Fu(\()p Fn(t)p Fu(\))25 b Fk(2)g Fn(B)5 b Fu(\(2)p Fn(\032)1811 3549 y Fl(0)1835 3586 y Fn(=)p Fu(3\))15 b Fn(;)203 b(t)25 b Fk(2)f Fu([0)p Fn(;)15 b(T)e Fu(])i Fn(;)688 b Fu(\(81\))448 3750 y Fe(and)1658 3863 y Fn(I)1698 3877 y Fm(x;T)1813 3863 y Fu(\()p Fn(\036)1902 3826 y Fm(u)1947 3863 y Fu(\))41 b Fk(\024)f Fn(h)15 b(:)1049 b Fu(\(82\))448 4069 y Fe(Pr)-5 b(o)g(of:)40 b Fu(T)-8 b(ogether)28 b(with)d(the)i(con)m(trol)f(system) h(\(47\),)i(w)m(e)e(consider)e(the)i(time-rev)m(ersed)448 4182 y(system)1307 4359 y(_)1304 4380 y(~)-52 b Fn(q)101 b Fu(=)d Fk(\000r)1755 4394 y Fm(p)1794 4380 y Fn(G)15 b(;)1306 4497 y Fu(_)1303 4518 y(~)-53 b Fn(p)98 b Fu(=)g Fk(r)1684 4532 y Fm(q)1722 4518 y Fn(G)15 b(;)1307 4644 y Fu(_)1302 4665 y(~)-49 b Fn(r)100 b Fu(=)e Fn(\015)5 b(\025)1713 4628 y Ft(2)1753 4665 y Fk(r)1829 4679 y Fm(r)1866 4665 y Fn(G)20 b Fu(+)g(\(2)p Fn(\015)5 b(\025)2234 4628 y Ft(2)2274 4665 y Fn(D)s Fu(\))2387 4628 y Ft(1)p Fm(=)p Ft(2)2498 4665 y Fn(u)15 b(:)685 b Fu(\(83\))448 4863 y(Lemma)29 b(2.7)g(implies)c(the)k(STLC)d(of)j(the)f(con)m(trol)h (system)f(\(47\).)41 b(F)-8 b(urthermore)29 b(from)448 4976 y(Lemma)24 b(2.6)i(it)d(is)g(easy)i(to)g(see)f(the)g(con)m(trol)h (system)f(\(83\))i(is)d(STLC)f(if)h(and)h(only)f(if)g(the)1897 5225 y(19)p eop %%Page: 20 20 20 19 bop 448 573 a Fu(con)m(trol)32 b(system)g(\(47\))g(is.)43 b(W)-8 b(e)33 b(note)f Fn(\036)1795 540 y Fm(u)1871 573 y Fu(\()1918 549 y(~)1906 573 y Fn(\036)1960 540 y Fm(u)2006 573 y Fu(\))f(the)h(solution)e(of)h(Eq.)h(\(47\))g(\(Eq.)g(\(83\)\))448 686 y(and)d Fn(Y)697 653 y Fm(M)677 711 y(T)776 686 y Fu(\()p Fn(x)p Fu(\))h(\()977 663 y(~)963 686 y Fn(Y)1037 653 y Fm(M)1016 711 y(T)1116 686 y Fu(\()p Fn(x)p Fu(\)\))g(the)g(set)g (of)g(reac)m(hable)f(p)s(oin)m(ts)g(for)g(the)h(con)m(trol)g(system)f (\(47\))448 799 y(\(\(83\)\).)45 b(Using)30 b(the)h(con)m(v)m(exit)m(y) i(of)e(the)g(set)h(of)f(v)-5 b(alues)30 b(the)h(con)m(trol)h(can)f (assume)g(and)448 912 y(the)k(con)m(tin)m(uous)f(dep)s(endence)g(of)h Fn(\036)1722 879 y Fm(u)1801 912 y Fu(on)g Fn(u)p Fu(,)g(it)g(is)e (easy)i(to)h(see)f(\([23)q(],)h(Prop.)53 b(2.3.1\))448 1024 y(that)31 b Fn(Y)718 992 y Fm(M)698 1050 y(T)797 1024 y Fu(\()p Fn(x)p Fu(\))g(is)f(a)g(compact)i(set.)589 1137 y(W)-8 b(e)30 b(c)m(ho)s(ose)f(no)m(w)f Fn(M)38 b Fu(and)27 b Fn(T)41 b Fu(suc)m(h)28 b(that)h Fn(M)2107 1104 y Ft(2)2146 1137 y Fn(T)38 b Fk(\024)25 b Fn(h)p Fu(.)40 b(Since)27 b Fn(Y)2758 1104 y Fm(M)2738 1163 y(T)2837 1137 y Fu(\()p Fn(x)p Fu(\))i(and)3176 1114 y(~)3162 1137 y Fn(Y)3235 1104 y Fm(M)3215 1163 y(T)3314 1137 y Fu(\()p Fn(x)p Fu(\))448 1250 y(are)i(compact,)h(there)f(is)e Fn(\032)1360 1217 y Fl(0)1409 1250 y Fn(>)24 b Fu(0)31 b(suc)m(h)f(that)1392 1455 y Fn(Y)1465 1417 y Fm(M)1445 1477 y(T)1544 1455 y Fu(\()p Fn(x)p Fu(\))p Fn(;)1720 1432 y Fu(~)1706 1455 y Fn(Y)1780 1417 y Fm(M)1759 1477 y(T)1859 1455 y Fu(\()p Fn(x)p Fu(\))c Fk(\032)f Fn(B)5 b Fu(\(2)p Fn(\032)2304 1417 y Fl(0)2327 1455 y Fn(=)p Fu(3\))15 b Fn(:)783 b Fu(\(84\))448 1659 y(F)-8 b(urthermore)37 b(w)m(e)h(ma)m(y)f(c)m(ho)s(ose)h Fn(\032)1668 1626 y Fl(0)1729 1659 y Fu(arbitrarily)c(small)i(b)m(y)g(c)m(ho)s(osing)h Fn(M)47 b Fu(and/or)37 b Fn(T)448 1772 y Fu(su\016cien)m(tly)24 b(small.)38 b(By)25 b(Lemma)g(2.7,)j(b)s(oth)c(systems)h(\(47\))i(and)e (\(83\))h(are)g(STLC)e(and)448 1885 y(th)m(us)30 b(there)h(is)e Fn(\032)d(>)f Fu(0)30 b(with)f Fn(\032)d(<)f(\032)1639 1852 y Fl(0)1662 1885 y Fn(=)p Fu(3)31 b(and)1471 2089 y Fn(B)5 b Fu(\()p Fn(\032)p Fu(\))26 b Fk(\032)f Fn(Y)1857 2051 y Fm(M)1837 2111 y(T)1936 2089 y Fu(\()p Fn(x)p Fu(\))p Fn(;)2113 2066 y Fu(~)2098 2089 y Fn(Y)2172 2051 y Fm(M)2151 2111 y(T)2251 2089 y Fu(\()p Fn(x)p Fu(\))15 b Fn(:)862 b Fu(\(85\))448 2293 y(Therefore)30 b(there)h(are)g(con)m (trols)f Fn(u)1638 2307 y Ft(1)1678 2293 y Fn(;)15 b(u)1770 2307 y Ft(2)1835 2293 y Fk(2)25 b(U)1978 2307 y Fm(M)2087 2293 y Fu(suc)m(h)30 b(that)1285 2497 y Fn(\036)1339 2460 y Fm(u)1380 2469 y Fj(1)1419 2497 y Fu(\(0\))41 b(=)f Fn(x)1738 2511 y Ft(0)1793 2497 y Fn(;)257 b(\036)2129 2460 y Fm(u)2170 2469 y Fj(1)2209 2497 y Fu(\()p Fn(T)13 b Fu(\))41 b(=)f Fn(y)18 b(;)690 b Fu(\(86\))1297 2608 y Fh(e)1285 2635 y Fn(\036)1339 2598 y Fm(u)1380 2607 y Fj(2)1419 2635 y Fu(\(0\))41 b(=)f Fn(x)1738 2649 y Ft(0)1793 2635 y Fn(;)2087 2608 y Fh(e)2075 2635 y Fn(\036)2129 2598 y Fm(u)2170 2607 y Fj(2)2209 2635 y Fu(\()p Fn(T)13 b Fu(\))41 b(=)f Fn(x)15 b(:)686 b Fu(\(87\))448 2839 y(By)33 b(rev)m(ersing)f(the)h(time,)f(the)h(tra)5 b(jectory)1970 2812 y Fh(e)1959 2839 y Fn(\036)2013 2806 y Fm(u)2054 2815 y Fj(2)2093 2839 y Fu(\()p Fn(t)p Fu(\))32 b(yields)f(a)i(tra)5 b(jectory)34 b Fn(\036)3045 2806 y Fm(u)3086 2815 y Fj(2)3124 2839 y Fu(\()p Fn(t)p Fu(\))f(with)448 2952 y Fn(\036)502 2919 y Fm(u)543 2928 y Fj(2)582 2952 y Fu(\(0\))43 b(=)e Fn(x)g Fu(and)e Fn(\036)1185 2919 y Fm(u)1226 2928 y Fj(2)1265 2952 y Fu(\()p Fn(T)13 b Fu(\))42 b(=)g Fn(x)1608 2966 y Ft(0)1647 2952 y Fu(.)71 b(Concatenating)40 b(the)h(tra)5 b(jectories)41 b Fn(\036)3067 2919 y Fm(u)3108 2928 y Fj(2)3147 2952 y Fu(\()p Fn(t)p Fu(\))g(and)448 3065 y Fn(\036)502 3032 y Fm(u)543 3041 y Fj(1)582 3065 y Fu(\()p Fn(t)p Fu(\))36 b(yields)d(a)i(path)g Fn(\036)h Fu(from)e Fn(x)h Fu(to)h Fn(y)i Fu(whic)m(h)c(do)s(es)h(not)g(lea)m(v)m (e)h(the)g(ball)d Fn(B)5 b Fu(\(2)p Fn(\032)3287 3032 y Fl(0)3311 3065 y Fn(=)p Fu(3\))448 3178 y(and)30 b(for)g(whic)m(h)f (w)m(e)i(ha)m(v)m(e)h(the)e(estimate)1113 3412 y Fn(I)1153 3426 y Fm(x;)p Ft(2)p Fm(T)1302 3412 y Fu(\()p Fn(\036)p Fu(\))42 b(=)1589 3350 y(1)p 1589 3391 46 4 v 1589 3474 a(2)1659 3297 y Fh(Z)1742 3323 y Ft(2)p Fm(T)1705 3485 y Ft(0)1848 3412 y Fn(dt)p Fk(j)p Fn(u)p Fu(\()p Fn(t)p Fu(\))p Fk(j)2133 3374 y Ft(2)2214 3412 y Fk(\024)e Fn(M)2423 3374 y Ft(2)2463 3412 y Fn(T)53 b Fk(\024)40 b Fn(h)15 b(;)503 b Fu(\(88\))448 3648 y(and)30 b(this)f(concludes)h(the)h(pro)s (of)e(of)i(Corollary)e(2.8.)p 2343 3656 50 67 v 448 4034 a Fo(3)135 b(Asymptotics)45 b(of)h(the)f(in)l(v)-7 b(arian)l(t)46 b(measure)448 4237 y Fu(In)37 b(this)f(section)h(w)m(e)h(pro)m(v)m(e)g (an)f(extension)f(of)i(F)-8 b(reidlin-W)g(en)m(tzell)35 b(theory)j([8])g(for)f(a)448 4350 y(certain)f(class)f(of)h(di\013usion) d(pro)s(cesses)i(with)g(h)m(yp)s(o)s(elliptic)d(generators)37 b(concerning)448 4463 y(the)c(in)m(v)-5 b(arian)m(t)32 b(measure.)46 b(Suc)m(h)32 b(extensions,)h(for)f(the)g(problem)f(of)i (the)f(exit)h(from)f(a)448 4576 y(domain,)37 b(exist,)g(see)f([2)q(],)i (where)d(a)h(strong)g(h)m(yp)s(o)s(ellipticit)m(y)c(condition)j(is)g (assumed)448 4689 y(whic)m(h)22 b(is)g(not)h(satis\014ed)f(in)f(our)i (mo)s(del)e(and)i(see)g(also)g([3)q(],)h(where)f(their)f(assumption)f (of)448 4802 y(small-time)27 b(lo)s(cal)g(con)m(trollabilit)m(y)f(on)h (the)h(b)s(oundary)e(of)i(the)g(domain)f(is)f(to)s(o)j(strong)448 4915 y(for)i(our)g(purp)s(oses.)41 b(Once)31 b(the)h(con)m(trol)f (theory)h(estimates)g(ha)m(v)m(e)g(b)s(een)f(established,)1897 5225 y(20)p eop %%Page: 21 21 21 20 bop 448 573 a Fu(our)39 b(pro)s(of)f(follo)m(ws)g(rather)h (closely)f(the)h(pro)s(of)f(of)h(F)-8 b(reidlin-W)g(en)m(tzell)38 b([8)q(])h(and)f(the)448 686 y(presen)m(tation)31 b(of)f(it)g(giv)m(en) h(in)e([3])i(with)e(a)i(n)m(um)m(b)s(er)e(of)h(tec)m(hnical)h(mo)s (di\014cations.)589 799 y(W)-8 b(e)32 b(consider)d(a)i(sto)s(c)m (hastic)g(di\013eren)m(tial)e(equation)h(of)h(the)f(form)1366 1003 y Fn(dx)1465 1017 y Fm(")1542 1003 y Fu(=)41 b Fn(Y)20 b Fu(\()p Fn(x)1814 1017 y Fm(")1851 1003 y Fu(\))g(+)g Fn(")2039 965 y Ft(1)p Fm(=)p Ft(2)2149 1003 y Fn(\033)s Fu(\()p Fn(x)2291 1017 y Fm(")2329 1003 y Fu(\))p Fn(dw)e(;)756 b Fu(\(89\))448 1207 y(where)21 b Fn(x)k Fk(2)g Fn(X)32 b Fu(=)25 b Fg(R)1146 1174 y Fm(n)1194 1207 y Fu(,)e Fn(Y)d Fu(\()p Fn(x)p Fu(\))h(is)f(a)h Fk(C)1659 1174 y Fl(1)1755 1207 y Fu(v)m(ector)h(\014eld,)g Fn(\033)s Fu(\()p Fn(x)p Fu(\))g(a)f Fk(C)2551 1174 y Fl(1)2647 1207 y Fu(map)f(from)g Fg(R)3122 1174 y Fm(m)3210 1207 y Fu(to)i Fg(R)3390 1174 y Fm(n)448 1320 y Fu(and)k Fn(w)r Fu(\()p Fn(t)p Fu(\))i(a)f(standard)e Fn(m)p Fu(-dimensional)f(Wiener)i (pro)s(cess.)39 b(W)-8 b(e)28 b(view)e(the)g(sto)s(c)m(hastic)448 1433 y(pro)s(cess)k(giv)m(en)h(b)m(y)f(Eq.)g(\(89\))i(as)f(a)g(small)d (p)s(erturbation)h(of)h(the)h(dynamical)e(system)1739 1637 y(_)-41 b Fn(x)40 b Fu(=)g Fn(Y)20 b Fu(\()p Fn(x)p Fu(\))15 b Fn(:)1114 b Fu(\(90\))448 1841 y(W)-8 b(e)32 b(denote)f Fn(I)938 1855 y Fm(x;T)1053 1841 y Fu(\()p Fk(\001)p Fu(\))g(the)g(large)g(deviation)f(functional)f(asso)s(ciated) i(to)g(Eq.)g(\(89\))h(\(see)448 1954 y(Eq.)h(\(24\)\))h(and)e(denote)h Fn(V)1372 1968 y Fm(T)1427 1954 y Fu(\()p Fn(x;)15 b(y)s Fu(\))33 b(and)f Fn(V)20 b Fu(\()p Fn(x;)15 b(y)s Fu(\))34 b(the)e(cost)i(functions)d(giv)m(en)h(b)m(y)g(\(27\))448 2067 y(and)e(\(28\).)44 b(As)31 b(in)e([8)q(])i(w)m(e)g(in)m(tro)s (duce)e(an)i(equiv)-5 b(alence)30 b(relation)g Fk(\030)h Fu(on)f Fn(X)38 b Fu(de\014ned)30 b(as)448 2180 y(follo)m(ws:)40 b Fn(x)25 b Fk(\030)g Fn(y)33 b Fu(if)d Fn(V)20 b Fu(\()p Fn(x;)15 b(y)s Fu(\))26 b(=)f Fn(V)20 b Fu(\()p Fn(y)s(;)15 b(x)p Fu(\))27 b(=)d(0.)589 2293 y(Our)30 b(assumptions)e(on)j(the)f (di\013usion)e(pro)s(cess)i Fn(x)2312 2307 y Fm(")2348 2293 y Fu(\()p Fn(t)p Fu(\))h(are)g(the)g(follo)m(wing)585 2481 y Fk(\017)46 b Fg(L0)30 b Fu(The)g(pro)s(cess)g Fn(x)1377 2495 y Fm(")1414 2481 y Fu(\()p Fn(t)p Fu(\))g(has)g(an)h(in) m(v)-5 b(arian)m(t)29 b(measure)i Fn(\026)2627 2495 y Fm(")2663 2481 y Fu(.)585 2668 y Fk(\017)46 b Fg(L1)30 b Fu(There)g(is)f(a)i(\014nite)e(n)m(um)m(b)s(er)g(of)i(compact)h(sets) e Fn(K)2536 2682 y Ft(1)2576 2668 y Fn(;)15 b(K)2693 2682 y Ft(2)2733 2668 y Fn(;)g(:)g(:)g(:)i(K)2972 2682 y Fm(L)3054 2668 y Fu(suc)m(h)30 b(that)759 2856 y(1.)47 b(F)-8 b(or)29 b(an)m(y)g(t)m(w)m(o)h(p)s(oin)m(ts)d Fn(x;)15 b(y)32 b Fu(b)s(elonging)27 b(to)i(the)g(same)g Fn(K)2794 2870 y Fm(i)2851 2856 y Fu(w)m(e)g(ha)m(v)m(e)h Fn(x)25 b Fk(\030)g Fn(y)s Fu(.)759 3002 y(2.)47 b(If)30 b Fn(x)25 b Fk(2)g Fn(K)1207 3016 y Fm(i)1265 3002 y Fu(,)31 b Fn(y)d Fk(2)d Fn(K)1557 3016 y Fm(j)1593 3002 y Fu(,)31 b(with)e Fn(i)d Fk(6)p Fu(=)f Fn(j)5 b Fu(,)31 b(then)f Fn(x)25 b Fk(6\030)g Fn(y)s Fu(.)759 3148 y(3.)47 b(Ev)m(ery)38 b Fn(!)s Fu(-limit)f(set)i(of)g(the)f(dynamical)f(system) i(\(90\))h(is)e(con)m(tained)g(in)876 3261 y Fn(K)953 3275 y Fm(i)981 3261 y Fu(.)676 3449 y(W)-8 b(e)33 b(let)e Fn(B)5 b Fu(\()p Fn(\032)p Fu(\))32 b(denote)h(the)f Fn(\032)f Fu(neigh)m(b)s(orho)s(o)s(d)f(of)i Fk([)2459 3463 y Fm(i)2486 3449 y Fn(K)2563 3463 y Fm(i)2623 3449 y Fu(and)g Fn(\034)2842 3463 y Fm(\032)2913 3449 y Fu(the)g(\014rst)f (time)676 3562 y(the)i(di\013usion)e Fn(x)1257 3576 y Fm(")1294 3562 y Fu(\()p Fn(t)p Fu(\))j(hits)f(the)g(set)i Fn(B)5 b Fu(\()p Fn(\032)p Fu(\).)50 b(W)-8 b(e)35 b(assume)e(that)i (for)e(an)m(y)h Fn(")3220 3576 y Ft(0)3290 3562 y Fn(>)d Fu(0)676 3675 y(the)c(exp)s(ected)g(hitting)e(time)i Fn(E)1769 3689 y Fm(x)1813 3675 y Fu(\()p Fn(\034)1888 3689 y Fm(\032)1928 3675 y Fu(\))g(of)g(the)g(di\013usion)d(with)h (initial)f(condition)676 3787 y Fn(x)30 b Fu(is)f(b)s(ounded)f (uniformly)g(for)i(0)c Fk(\024)f Fn(")g Fk(\024)g Fn(")2146 3801 y Ft(0)2186 3787 y Fu(.)585 3975 y Fk(\017)46 b Fg(L2)36 b Fu(The)g(di\013usion)d(pro)s(cess)j Fn(x)1767 3989 y Fm(")1804 3975 y Fu(\()p Fn(t)p Fu(\))g(has)g(an)h(h)m(yp)s(o)s (elliptic)32 b(generator.)60 b(More-)676 4088 y(o)m(v)m(er,)49 b(for)44 b(an)m(y)h Fn(x)k Fk(2)f Fn(K)1539 4102 y Fm(i)1612 4088 y Fu(the)d(con)m(trol)g(system)f(asso)s(ciated)h(to)g(Eq.)g (\(89\))h(is)676 4201 y(small-time)29 b(lo)s(cally)f(con)m(trollable.) 585 4389 y Fk(\017)46 b Fg(L3)22 b Fu(The)f(di\013usion)f(pro)s(cess)h (is)h(strongly)f(completely)h(con)m(trollable,)h(i.e.,)h(for)e(all)676 4501 y Fn(T)38 b(>)25 b Fu(0,)k Fn(V)1015 4515 y Fm(T)1070 4501 y Fu(\()p Fn(x;)15 b(y)s Fu(\))27 b Fn(<)d Fk(1)29 b Fu(and,)g(moreo)m(v)m(er,)i Fn(V)2192 4515 y Fm(T)2247 4501 y Fu(\()p Fn(x;)15 b(y)s Fu(\))29 b(is)f(upp)s(er)f(semicon)m(tin) m(uous)676 4614 y(as)j(a)h(map)f(from)g Fn(X)e Fk(\002)19 b Fn(X)38 b Fu(to)31 b Fg(R)p Fu(.)448 4827 y Fg(Remark)k(3.1)46 b Fu(F)-8 b(or)33 b(the)f(mo)s(del)g(w)m(e)g(consider,)g(condition)f (whic)m(h)h(ensures)f(that)i Fg(L0)448 4940 y Fu(holds)27 b(are)h(giv)m(en)g(in)f([6,)i(5].)40 b(F)-8 b(or)29 b(condition)e Fg(L1)h Fu(w)m(e)g(assume)g(that)g(the)g(set)h(of)f(critical)1897 5225 y(21)p eop %%Page: 22 22 22 21 bop 448 573 a Fu(p)s(oin)m(ts)31 b(of)g Fn(G)p Fu(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\))31 b(is)g(a)h(compact)g(set)g(and) f(item)g(\(iii\))f(follo)m(ws)g(from)h(Lemma)h(2.2.)448 686 y(The)h(b)s(ound)f(on)h(the)h(exp)s(ected)g(hitting)e(time)h(will)e (b)s(e)i(pro)m(v)m(ed)h(in)e(a)i(separate)g(pub-)448 799 y(lication)c([21)q(].)41 b(F)-8 b(or)31 b(condition)e Fg(L2)p Fu(,)i(the)f(h)m(yp)s(o)s(ellipticit)m(y)d(of)k(the)f (generator)i(and)e(the)448 912 y(small-time)36 b(lo)s(cal)g(con)m (trollabilit)m(y)g(follo)m(ws)g(from)g Fg(K2)p Fu(,)k(see)d(Lemmas)g (2.6,)j(2.7)f(and)448 1024 y(2.8.)j(Condition)28 b Fg(L3)j Fu(is)e(a)i(consequence)g(of)g(condition)e Fg(K2)p Fu(,)h(see)h(Prop)s (osition)e(2.3.)448 1228 y Fg(Remark)35 b(3.2)46 b Fu(Condition)26 b Fg(L2)i Fu(is)f(a)i Fe(lo)-5 b(c)g(al)30 b Fu(prop)s(ert)m(y)e(of)g (the)g(dynamics)f(and)h(as)g(suc)m(h)448 1341 y(su\016cien)m(t)d (conditions)e(can)i(b)s(e)f(giv)m(en)h(in)e(terms)i(of)g(adequate)h (Lie)e(algebra.)39 b(A)25 b(simple)448 1454 y(su\016cien)m(t)30 b(condition)e(for)i(small-time)f(lo)s(cal)g(con)m(trollabilit)m(y)f(w)m (as)i(quoted)g(and)g(used)448 1566 y(in)e(Section)g(2.4.)42 b(More)29 b(general)g(su\016cien)m(t)f(conditions)g(ha)m(v)m(e)i(b)s (een)e(pro)m(v)m(ed,)h(see)h([23)q(])448 1679 y(and)j(references)g (therein.)49 b(condition)31 b Fg(L3)j Fu(is)e(a)i Fe(glob)-5 b(al)34 b Fu(condition)e(on)h(the)g(dynamics)448 1792 y(and)23 b(w)m(e)g(are)g(not)h(a)m(w)m(are)g(of)f(an)m(y)g(general)g (condition)f(whic)m(h)f(w)m(ould)h(imply)f Fg(L3)i Fu(\(except)448 1905 y(of)31 b(course)f(ellipticit)m(y)e(of)j(the)g(generator\).)589 2108 y(T)-8 b(o)29 b(describ)s(e)d(the)i(asymptotic)h(b)s(eha)m(vior)e (of)h(the)g(in)m(v)-5 b(arian)m(t)27 b(measure)h Fn(\026)3100 2122 y Fm(")3164 2108 y Fu(w)m(e)h(will)448 2221 y(need)h(the)h(follo)m (wing)e(quan)m(tities.)40 b(W)-8 b(e)31 b(let)905 2416 y Fn(V)20 b Fu(\()p Fn(K)1090 2430 y Fm(i)1119 2416 y Fn(;)15 b(K)1236 2430 y Fm(j)1273 2416 y Fu(\))99 b(=)224 b(inf)1575 2476 y Fm(y)r Fl(2)p Fm(K)1719 2486 y Ff(i)1746 2476 y Fm(;z)s Fl(2)p Fm(K)1909 2486 y Ff(j)1955 2416 y Fn(V)20 b Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))g Fn(;)108 b(i;)15 b(j)32 b Fu(=)25 b(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(L)g(;)310 b Fu(\(91\))972 2605 y Fn(V)21 b Fu(\()p Fn(K)1158 2619 y Fm(i)1186 2605 y Fn(;)15 b(z)t Fu(\))100 b(=)127 b(inf)1575 2664 y Fm(y)r Fl(2)p Fm(K)1719 2674 y Ff(i)1761 2605 y Fn(V)20 b Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))g Fn(;)108 b(i)25 b Fu(=)g(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(L)g(:)588 b Fu(\(92\))448 2838 y(W)-8 b(e)32 b(set)1195 2951 y Fn(W)13 b Fu(\()p Fn(K)1406 2965 y Fm(i)1434 2951 y Fu(\))41 b(=)81 b(min)1621 3015 y Fm(g)r Fl(2)p Fm(G)p Fl(f)p Fm(i)p Fl(g)1966 2870 y Fh(X)1869 3058 y Ft(\()p Fm(m)p Fl(!)p Fm(n)p Ft(\))p Fl(2)p Fm(g)2198 2951 y Fn(V)20 b Fu(\()p Fn(K)2383 2965 y Fm(m)2450 2951 y Fn(;)15 b(K)2567 2965 y Fm(n)2615 2951 y Fu(\))g Fn(;)585 b Fu(\(93\))448 3209 y(where)25 b(the)h(set)g(of)f Fk(f)p Fn(i)p Fk(g)p Fu(-graphs)i Fn(G)p Fk(f)p Fn(i)p Fk(g)f Fu(is)e(de\014ned)g(in)g(the)i (paragraph)f(ab)s(o)m(v)m(e)h(Theorem)448 3322 y(1.4.)40 b(The)24 b(asymptotics)g(of)h(the)g(in)m(v)-5 b(arian)m(t)23 b(measure)h(is)g(giv)m(en)g(b)m(y)g(the)h(function)e Fn(W)13 b Fu(\()p Fn(x)p Fu(\))448 3435 y(giv)m(en)31 b(b)m(y)973 3548 y Fn(W)13 b Fu(\()p Fn(x)p Fu(\))41 b(=)f(min)1410 3605 y Fm(i)1513 3548 y Fu(\()p Fn(W)13 b Fu(\()p Fn(K)1759 3562 y Fm(i)1787 3548 y Fu(\))21 b(+)f Fn(V)g Fu(\()p Fn(K)2119 3562 y Fm(i)2148 3548 y Fn(;)15 b(x)p Fu(\)\))21 b Fk(\000)f Fu(min)2482 3605 y Fm(j)2589 3548 y Fn(W)13 b Fu(\()p Fn(K)2800 3562 y Fm(j)2836 3548 y Fu(\))i Fn(:)364 b Fu(\(94\))589 3747 y(W)-8 b(e)25 b(call)d(a)i(domain)e Fn(D)28 b Fk(\032)d Fn(X)30 b Fu(regular,)24 b(if)e(the)h(b)s(oundary)e(of)j Fn(D)s Fu(,)g Fn(@)5 b(D)26 b Fu(is)c(a)i(piecewise)448 3860 y(smo)s(oth)31 b(manifold.)38 b(Our)29 b(main)h(result)f(is)g(the) i(follo)m(wing:)448 4064 y Fg(Theorem)j(3.3)46 b Fe(Assume)33 b(c)-5 b(onditions)35 b Fg(L0-L3)p Fe(.)44 b(L)-5 b(et)34 b Fn(D)i Fe(b)-5 b(e)33 b(a)h(r)-5 b(e)g(gular)34 b(domain)h(with)448 4177 y(c)-5 b(omp)g(act)39 b(closur)-5 b(e)38 b(such)f(that)h Fu(dist)o(\()p Fn(D)s(;)15 b Fk([)1875 4191 y Fm(i)1903 4177 y Fn(K)1980 4191 y Fm(i)2009 4177 y Fu(\))33 b Fn(>)f Fu(0)p Fe(.)55 b(Then)37 b(the)h(\(unique\))e(invariant)448 4289 y(me)-5 b(asur)g(e)34 b Fn(\026)855 4303 y Fm(")924 4289 y Fe(of)f(the)g(pr)-5 b(o)g(c)g(ess)35 b Fn(x)1542 4303 y Fm(")1579 4289 y Fu(\()p Fn(t)p Fu(\))e Fe(satis\014es)1352 4484 y Fu(lim)1346 4541 y Fm(")p Fl(!)p Ft(0)1500 4484 y Fn(")15 b Fu(ln)f Fn(\026)1703 4498 y Fm(")1740 4484 y Fu(\()p Fn(D)s Fu(\))41 b(=)f Fk(\000)31 b Fu(inf)2126 4544 y Fm(z)s Fl(2)p Fm(D)2283 4484 y Fn(W)13 b Fu(\()p Fn(z)t Fu(\))i Fn(:)737 b Fu(\(95\))448 4710 y Fe(In)33 b(p)-5 b(articular)35 b(if)d(ther)-5 b(e)34 b(is)e(a)h(single)g(critic) -5 b(al)33 b(set)g Fn(K)39 b Fe(one)33 b(has)1305 4905 y Fu(lim)1299 4961 y Fm(")p Fl(!)p Ft(0)1453 4905 y Fn(")15 b Fu(ln)f Fn(\026)1656 4919 y Fm(")1693 4905 y Fu(\()p Fn(D)s Fu(\))41 b(=)f Fk(\000)31 b Fu(inf)2079 4964 y Fm(z)s Fl(2)p Fm(D)2236 4905 y Fn(V)21 b Fu(\()p Fn(K)q(;)15 b(z)t Fu(\))g Fn(:)691 b Fu(\(96\))1897 5225 y(22)p eop %%Page: 23 23 23 22 bop 589 573 a Fu(W)-8 b(e)28 b(\014rst)d(recall)g(some)i(general) f(results)f(on)h(h)m(yp)s(o)s(elliptic)d(di\013usions)g(obtained)i(in) 448 686 y([15)q(],)37 b(in)d(particular)f(a)i(v)m(ery)h(useful)d (represen)m(tation)i(of)g(the)h(in)m(v)-5 b(arian)m(t)34 b(measure)h Fn(\026)3405 700 y Fm(\017)448 799 y Fu(in)e(terms)h(of)g (em)m(b)s(edded)f(Mark)m(o)m(v)j(c)m(hains,)e(see)h(Prop)s(osition)d (3.4)j(b)s(elo)m(w.)51 b(Then)33 b(w)m(e)448 912 y(pro)m(v)m(e)27 b(the)f(large)f(deviations)g(estimates.)39 b(Let)27 b Fn(U)35 b Fu(and)25 b Fn(V)46 b Fu(b)s(e)25 b(op)s(en)g(subset)g(of)g Fn(X)33 b Fu(with)448 1024 y(compact)f(closure)e(with)p 1323 952 72 4 v 29 w Fn(U)36 b Fk(\032)25 b Fn(V)c Fu(.)41 b(Belo)m(w,)31 b Fn(U)41 b Fu(and)30 b Fn(V)51 b Fu(will)27 b(b)s(e)j(the)h(disjoin)m(t)e(union)g(of)448 1137 y(small)e(neigh)m(b)s (orho)s(o)s(ds)e(of)j(the)g(sets)g Fn(K)1795 1151 y Fm(i)1823 1137 y Fu(.)40 b(W)-8 b(e)29 b(in)m(tro)s(duce)e(an)g(increasing)g (sequence)h(of)448 1250 y(Mark)m(o)m(v)k(times)e Fn(\034)1062 1264 y Ft(0)1101 1250 y Fn(;)15 b(\033)1193 1264 y Ft(0)1233 1250 y Fn(;)g(\034)1313 1264 y Ft(1)1353 1250 y Fn(;)g(:)g(:)g(:)32 b Fu(de\014ned)d(as)h(follo)m(ws.)40 b(W)-8 b(e)32 b(set)f Fn(\034)2627 1264 y Ft(0)2691 1250 y Fu(=)25 b(0)31 b(and)1220 1428 y Fn(\033)1272 1442 y Fm(n)1417 1428 y Fu(=)98 b(inf)6 b Fk(f)p Fn(t)25 b(>)g(\034)1936 1442 y Fm(n)2023 1428 y Fu(:)41 b Fn(x)2141 1442 y Fm(")2178 1428 y Fu(\()p Fn(t)p Fu(\))25 b Fk(2)g Fn(@)5 b(V)21 b Fk(g)711 b Fu(\(97\))1232 1565 y Fn(\034)1272 1579 y Fm(n)1417 1565 y Fu(=)98 b(inf)6 b Fk(f)p Fn(t)25 b(>)g(\033)1948 1579 y Fm(n)p Fl(\000)p Ft(1)2126 1565 y Fu(:)40 b Fn(x)2243 1579 y Fm(")2280 1565 y Fu(\()p Fn(t)p Fu(\))26 b Fk(2)f Fn(@)5 b(U)10 b Fk(g)610 b Fu(\(98\))448 1743 y(As)32 b(a)g(consequence)h(of)e(h)m (yp)s(o)s(ellipticit)m(y)e(and)i(the)h(strong)g(complete)g(con)m (trollabilit)m(y)448 1856 y(of)k(the)f(con)m(trol)h(problem)d(asso)s (ciated)j(to)g(the)f(di\013usion)e Fn(x)2526 1870 y Fm(")2563 1856 y Fu(\()p Fn(t)p Fu(\))i(\(condition)f Fg(L2)i Fu(and)448 1969 y Fg(L3)p Fu(\))d(w)m(e)g(ha)m(v)m(e)g(the)g(follo)m(wing)d (result)i(pro)m(v)m(en)g(in)f([15)q(])i(whic)m(h)e(extends)h(to)h (di\013usions)448 2081 y(with)39 b(h)m(yp)s(o)s(elliptic)d(generators) 41 b(the)e(c)m(haracterization)i(of)f(in)m(v)-5 b(arian)m(t)39 b(measures)g(in)448 2194 y(terms)26 b(of)g(recurrence)g(prop)s(erties)f (of)h(the)g(pro)s(cess)g Fn(x)2272 2208 y Fm(")2308 2194 y Fu(\()p Fn(t)p Fu(\))h(and)e(whic)m(h)g(is)g(standard)g(for)448 2307 y(di\013usions)j(with)h(elliptic)f(generators.)589 2420 y(W)-8 b(e)32 b(recall)e(that)h(the)f(di\013usion)e Fn(x)1761 2434 y Fm(")1798 2420 y Fu(\()p Fn(t)p Fu(\))i(is)g(p)s (ositiv)m(e)f(recurren)m(t)h(if)559 2585 y(1.)47 b(It)30 b(is)f(recurren)m(t,)i(i.e.,)g(for)f(all)f Fn(x)c Fk(2)g Fn(X)38 b Fu(for)30 b(all)f(op)s(en)h(set)h Fn(U)k Fk(\032)25 b Fn(X)38 b Fu(one)30 b(has)1801 2763 y Fn(P)1872 2725 y Fm(")1859 2785 y(x)1908 2763 y Fu(\()p Fk(R)2020 2777 y Fm(U)2080 2763 y Fu(\))40 b(=)h(1)963 b(\(99\))676 2940 y(where)29 b Fk(R)1015 2954 y Fm(U)1105 2940 y Fu(is)g(the)i(ev)m (en)m(t)h(giv)m(en)e(b)m(y)842 3117 y Fk(R)919 3131 y Fm(U)1003 3117 y Fu(=)25 b Fk(f)p Fn(x)1196 3131 y Fm(")1233 3117 y Fu(\()p Fn(t)1301 3131 y Fm(n)1348 3117 y Fu(\))h Fk(2)f Fn(U)40 b Fu(for)30 b(an)h(increasing)f(sequence)h Fn(t)2699 3131 y Fm(n)2771 3117 y Fk(!)25 b(1g)15 b Fn(:)167 b Fu(\(100\))559 3327 y(2.)47 b(F)-8 b(or)31 b(all)e Fn(x)c Fk(2)g Fn(X)38 b Fu(and)30 b(for)g(all)f(op)s(en)h(sets)g Fn(U)36 b Fk(\032)25 b Fn(X)7 b Fu(,)31 b(one)f(has)1789 3505 y Fn(E)1861 3467 y Fm(")1856 3527 y(x)1900 3505 y Fu(\()p Fn(\033)1987 3519 y Fm(U)2046 3505 y Fu(\))41 b Fn(<)f Fk(1)906 b Fu(\(101\))676 3682 y(where)29 b Fn(\033)990 3696 y Fm(U)1075 3682 y Fu(=)c(inf)5 b Fk(f)p Fn(t;)15 b(x)1451 3696 y Fm(")1489 3682 y Fu(\()p Fn(t)p Fu(\))25 b Fk(2)g Fn(U)10 b Fk(g)p Fu(.)448 3847 y(It)35 b(is)f(pro)m(v)m(en)i(in)d([15)r(],)j(Theorem)f(4.1,)i(that)f(if)e(the) h(di\013usion)d Fn(x)2701 3861 y Fm(")2738 3847 y Fu(\()p Fn(t)p Fu(\))j(is)f(h)m(yp)s(o)s(elliptic)448 3960 y(and)k(strongly)f (completely)h(con)m(trollable)g(then)g(the)g(di\013usion)d(admits)j(a)g (\(unique\))448 4073 y(in)m(v)-5 b(arian)m(t)39 b(measure)h Fn(\026)1258 4087 y Fm(")1335 4073 y Fu(if)f(and)g(only)g(if)g Fn(x)1968 4087 y Fm(")2005 4073 y Fu(\()p Fn(t)p Fu(\))h(is)f(p)s (ositiv)m(e)g(recurren)m(t.)69 b(Clearly)39 b(it)448 4186 y(follo)m(ws)i(from)h(this)e(result)h(that,)46 b(almost)c(surely) -8 b(,)43 b(the)f(Mark)m(o)m(v)i(times)d Fn(\034)3082 4200 y Fm(j)3160 4186 y Fu(and)h Fn(\033)3401 4200 y Fm(j)448 4299 y Fu(de\014ned)29 b(in)h(Eqs.)g(\(97\))i(and)d(\(98\))j (are)f(\014nite.)589 4412 y(An)24 b(imp)s(ortan)m(t)f(ingredien)m(t)f (in)g(the)i(pro)s(of)f(of)h(the)g(results)e(in)h([15)q(])h(is)f(the)h (follo)m(wing)448 4525 y(represen)m(tation)30 b(of)h(the)f(in)m(v)-5 b(arian)m(t)29 b(measure)h Fn(\026)2092 4539 y Fm(")2128 4525 y Fu(:)41 b(Supp)s(ose)28 b Fn(x)2602 4539 y Fm(")2639 4525 y Fu(\(0\))e(=)f Fn(x)g Fk(2)g Fn(@)5 b(U)10 b Fu(.)41 b(Then)448 4638 y Fk(f)p Fn(x)545 4652 y Fm(")582 4638 y Fu(\()p Fn(\034)657 4652 y Fm(j)694 4638 y Fu(\))p Fk(g)35 b Fu(is)e(a)h(Mark)m(o)m(v)h(c)m(hain)f(with)e(a)j(\(compact\)) g(state)h(space)e(giv)m(en)g(b)m(y)g Fn(@)5 b(U)44 b Fu(and)448 4751 y(whic)m(h)32 b(admits)g(an)h(in)m(v)-5 b(arian)m(t)32 b(measure)g Fn(l)1908 4765 y Fm(")1945 4751 y Fu(\()p Fn(dx)p Fu(\).)49 b(The)33 b(follo)m(wing)e(result)h (relates)h(the)448 4863 y(measure)28 b Fn(l)827 4877 y Fm(")891 4863 y Fu(to)h(the)e(in)m(v)-5 b(arian)m(t)27 b(measure)h Fn(\026)1939 4877 y Fm(")1975 4863 y Fu(,)h(see)f(e.g.)41 b([11)q(],)28 b(Chap.)39 b(IV,)28 b(Lemma)g(4.2.)448 4976 y(for)i(a)h(pro)s(of.)1897 5225 y(23)p eop %%Page: 24 24 24 23 bop 448 573 a Fg(Prop)s(osition)37 b(3.4)46 b Fe(L)-5 b(et)32 b(the)i(me)-5 b(asur)g(e)33 b Fn(\027)1897 587 y Fm(")1967 573 y Fe(b)-5 b(e)32 b(de\014ne)-5 b(d)34 b(as)1157 807 y Fn(\027)1202 821 y Fm(")1239 807 y Fu(\()p Fn(D)s Fu(\))41 b(=)1539 692 y Fh(Z)1585 881 y Fm(@)t(U)1700 807 y Fn(l)1727 821 y Fm(")1764 807 y Fu(\()p Fn(dx)p Fu(\))p Fn(E)2005 770 y Fm(")2000 830 y(x)2060 692 y Fh(Z)2143 719 y Fm(\034)2174 728 y Fj(1)2106 881 y Ft(0)2228 807 y Fg(1)2280 821 y Fm(D)2345 807 y Fu(\()p Fn(x)2432 821 y Fm(")2469 807 y Fu(\()p Fn(t)p Fu(\)\))p Fn(dt)15 b(;)503 b Fu(\(102\))448 1044 y Fe(wher)-5 b(e)38 b Fn(D)h Fe(is)e(a)g(Bor)-5 b(el)38 b(set)e(and)i Fg(1)1632 1058 y Fm(D)1733 1044 y Fe(is)e(the)h(char)-5 b(acteristic)39 b(function)e(of)g(the)g(set)f Fn(D)s Fe(.)448 1157 y(Then)d(one)g(has) 1599 1295 y Fn(\026)1654 1309 y Fm(")1691 1295 y Fu(\()p Fn(D)s Fu(\))41 b(=)2003 1233 y Fn(\027)2048 1247 y Fm(")2085 1233 y Fu(\()p Fn(D)s Fu(\))p 2001 1274 235 4 v 2001 1357 a Fn(\027)2046 1371 y Fm(")2083 1357 y Fu(\()p Fn(X)7 b Fu(\))2261 1295 y Fn(:)944 b Fu(\(103\))448 1551 y(Up)27 b(to)g(the)g(normalization,)f(the)h(in)m(v)-5 b(arian)m(t)25 b(measure)i Fn(\026)2388 1565 y Fm(")2451 1551 y Fu(assigns)f(to)h(a)g (set)g Fn(D)i Fu(a)e(mea-)448 1664 y(sure)k(equals)g(to)i(the)e(time)h (sp)s(en)m(t)f(b)m(y)g(the)h(pro)s(cess)f(in)g Fn(D)j Fu(b)s(et)m(w)m(een)e(t)m(w)m(o)h(consecutiv)m(e)448 1777 y(hits)d(on)g Fn(@)5 b(U)10 b Fu(.)589 1890 y(The)30 b(pro)s(of)e(of)i(Theorem)g(3.3)g(is)f(quite)g(long)g(and)g(will)e(b)s (e)i(split)f(in)m(to)i(a)g(sequence)448 2003 y(of)38 b(Lemmas.)61 b(The)37 b(pro)s(of)g(is)f(based)h(on)g(the)h(follo)m (wing)e(ideas:)54 b(As)37 b Fn(")g Fk(!)g Fu(0)h(the)f(in-)448 2115 y(v)-5 b(arian)m(t)39 b(measure)g(is)f(more)h(and)f(more)h(concen) m(trated)h(on)f(a)g(small)e(neigh)m(b)s(orho)s(o)s(d)448 2228 y(of)g(the)f(critical)f(set)i Fk([)1238 2242 y Fm(i)1266 2228 y Fn(K)1343 2242 y Fm(i)1371 2228 y Fu(.)58 b(T)-8 b(o)37 b(estimate)g(the)g(measure)f(of)g(a)h(set)g Fn(D)i Fu(one)d(uses)g(the)448 2341 y(represen)m(tation)e(of)f(the)h(in)m(v)-5 b(arian)m(t)32 b(measure)i(giv)m(en)f(in)f(Prop)s(osition)g(3.4)i (where)f(the)448 2454 y(sets)46 b Fn(U)54 b Fu(and)45 b Fn(V)65 b Fu(are)45 b(neigh)m(b)s(orho)s(o)s(ds)d(of)j(the)g(sets)g Fk(f)p Fn(K)2457 2468 y Fm(i)2486 2454 y Fk(g)p Fu(.)85 b(Let)45 b Fn(\032)k(>)g Fu(0)d(and)e(de-)448 2567 y(note)f Fn(B)5 b Fu(\()p Fn(i;)15 b(\032)p Fu(\))43 b(the)f Fn(\032)p Fu(-neigh)m(b)s(orho)s(o)s(d)e(of)i Fn(K)1987 2581 y Fm(i)2057 2567 y Fu(and)g Fn(B)5 b Fu(\()p Fn(\032)p Fu(\))45 b(=)f Fk([)2658 2581 y Fm(i)2686 2567 y Fn(B)5 b Fu(\()p Fn(i;)15 b(\032)p Fu(\).)76 b(Let)43 b Fn(D)i Fu(b)s(e)448 2680 y(a)f(regular)f(op)s(en)g(set)h(suc)m(h)f(that)h (dist)o(\()p Fk([)1918 2694 y Fm(i)1946 2680 y Fn(K)2023 2694 y Fm(i)2051 2680 y Fn(;)15 b(D)s Fu(\))48 b Fn(>)f Fu(0.)80 b(W)-8 b(e)45 b(c)m(ho)s(ose)f Fn(\032)3038 2647 y Fl(0)3105 2680 y Fu(so)g(small)448 2793 y(that)i(dist)o(\()p Fn(B)5 b Fu(\()p Fn(i;)15 b(\032)1069 2760 y Fl(0)1093 2793 y Fu(\))p Fn(;)g(B)5 b Fu(\()p Fn(j;)15 b(\032)1401 2760 y Fl(0)1426 2793 y Fu(\)\))50 b Fn(>)f Fu(0,)g(for)c Fn(i)k Fk(6)p Fu(=)g Fn(j)i Fu(and)44 b(dist)o(\()p Fn(B)5 b Fu(\()p Fn(i;)15 b(\032)2827 2760 y Fl(0)2851 2793 y Fu(\))p Fn(;)g(D)s Fu(\))50 b Fn(>)f Fu(0,)g(for)448 2906 y Fn(i)30 b Fu(=)f(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(L)p Fu(,)34 b(and)f(w)m(e)g(c)m(ho)s(ose)h Fn(\032)29 b(>)g Fu(0)34 b(suc)m(h)e(that)i(0)c Fn(<)f(\032)g(<)g(\032)2644 2873 y Fl(0)2668 2906 y Fu(.)48 b(W)-8 b(e)34 b(set)f Fn(U)40 b Fu(=)29 b Fn(B)5 b Fu(\()p Fn(\032)p Fu(\))448 3019 y(and)30 b Fn(V)46 b Fu(=)26 b Fn(B)5 b Fu(\()p Fn(\032)977 2986 y Fl(0)1000 3019 y Fu(\).)42 b(W)-8 b(e)32 b(let)e Fn(\033)1442 3033 y Ft(0)1512 3019 y Fu(and)g Fn(\034)1729 3033 y Ft(1)1799 3019 y Fu(b)s(e)g(the)h(Mark)m(o)m(v)h (times)e(de\014ned)g(in)f(Eqs.)h(\(97\))448 3132 y(and)g(\(98\))i(and)e (let)g Fn(\034)1165 3146 y Fm(D)1259 3132 y Fu(b)s(e)g(the)g(Mark)m(o)m (v)j(time)d(de\014ned)f(as)h(follo)m(ws:)1434 3336 y Fn(\034)1474 3350 y Fm(D)1578 3336 y Fu(=)40 b(inf)6 b Fk(f)p Fn(t)41 b Fu(:)f Fn(x)2036 3350 y Fm(")2073 3336 y Fu(\()p Fn(t)p Fu(\))26 b Fk(2)f Fn(D)s Fk(g)15 b Fn(:)779 b Fu(\(104\))589 3540 y(The)35 b(\014rst)e(t)m(w)m(o)j (Lemmas)f(will)d(yield)h(an)h(upp)s(er)f(b)s(ound)f(on)j Fn(\027)2768 3554 y Fm(")2804 3540 y Fu(\()p Fn(D)s Fu(\),)h(the)f (unnor-)448 3653 y(malized)k(measure)h(giv)m(en)g(b)m(y)g(Eq.)g (\(102\).)71 b(The)39 b(\014rst)g(Lemma)h(sho)m(ws)g(that,)j(for)d Fn(")448 3766 y Fu(su\016cien)m(tly)25 b(small,)g(the)i(probabilit)m(y) c(that)k(the)f(di\013usion)d(w)m(anders)i(around)g(without)448 3879 y(hitting)k Fn(B)5 b Fu(\()p Fn(\032)p Fu(\))31 b(or)f Fn(D)j Fu(is)d(negligible.)448 4091 y Fg(Lemma)j(3.5)46 b Fe(F)-7 b(or)34 b(any)f(c)-5 b(omp)g(act)35 b(set)e Fn(K)39 b Fe(one)33 b(has)945 4296 y Fu(lim)912 4355 y Fm(T)10 b Fl(!1)1119 4296 y Fu(lim)15 b(sup)1189 4370 y Fm(")p Fl(!)p Ft(0)1412 4296 y Fn(")g Fu(log)25 b(sup)1603 4373 y Fm(x)p Fl(2)p Fm(K)1769 4296 y Fn(P)1840 4258 y Fm(")1827 4318 y(x)1876 4296 y Fu(\(min)o Fk(f)p Fn(\034)2148 4310 y Fm(D)2233 4296 y Fk(^)20 b Fn(\034)2354 4310 y Ft(1)2393 4296 y Fk(g)25 b Fn(>)g(T)13 b Fu(\))41 b(=)f Fk(\0001)256 b Fu(\(105\))448 4561 y Fe(Pr)-5 b(o)g(of:)40 b Fu(If)25 b Fn(x)g Fk(2)g Fn(D)13 b Fk([)e Fn(B)5 b Fu(\()p Fn(\032)p Fu(\),)28 b Fn(\034)1419 4575 y Fm(D)1493 4561 y Fk(^)11 b Fn(\034)1605 4575 y Ft(1)1669 4561 y Fu(=)25 b(0)h(and)f(there)h(is)f(nothing)g(to)h(pro)m(v)m(e.)40 b(Otherwise)448 4674 y(consider)30 b(the)g(closed)g(sets)664 4878 y Fn(F)722 4892 y Fm(T)818 4878 y Fu(=)41 b Fk(f)p Fn(\036)25 b Fk(2)g(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))43 b(:)d Fn(\036)p Fu(\()p Fn(s)p Fu(\))c Fn(=)-55 b Fk(2)25 b Fn(D)e Fk([)d Fn(B)5 b Fu(\()p Fn(\032)p Fu(\))p Fn(;)46 b Fu(for)30 b(all)f Fn(s)c Fk(2)g Fu([0)p Fn(;)15 b(T)e Fu(])p Fk(g)32 b Fn(:)216 b Fu(\(106\))1897 5225 y(24)p eop %%Page: 25 25 25 24 bop 448 573 a Fu(Clearly)31 b(the)i(ev)m(en)m(t)g Fk(f)p Fn(\034)39 b(>)28 b(T)13 b Fk(g)33 b Fu(is)e(con)m(tained)h(in)f Fk(f)p Fn(x)2244 587 y Fm(")2310 573 y Fk(2)d Fn(F)2457 587 y Fm(T)2512 573 y Fk(g)p Fu(.)47 b(By)33 b(Theorem)f(2.1,)i(w)m(e) 448 686 y(ha)m(v)m(e)e(for)e(all)f Fn(T)38 b(<)25 b Fk(1)p Fu(,)912 857 y(lim)15 b(sup)982 931 y Fm(")p Fl(!)p Ft(0)1205 857 y Fn(")g Fu(log)24 b(sup)1395 934 y Fm(x)p Fl(2)p Fm(K)1561 857 y Fn(P)1632 819 y Fm(")1619 879 y(x)1669 857 y Fu(\()p Fn(x)1756 871 y Fm(")1818 857 y Fk(2)h Fn(F)1962 871 y Fm(T)2018 857 y Fu(\))41 b Fk(\024)f(\000)35 b Fu(inf)2291 916 y Fm(x)p Fl(2)p Fm(K)2492 857 y Fu(inf)2457 917 y Fm(\036)p Fl(2)p Fm(F)2591 928 y Ff(T)2654 857 y Fn(I)2694 871 y Fm(x;T)2808 857 y Fu(\()p Fn(\036)p Fu(\))15 b Fn(:)258 b Fu(\(107\))448 1076 y(In)30 b(order)g(to)h (complete)g(the)f(pro)s(of)g(of)h(the)f(Lemma)h(it)e(is)h(enough)g(to)h (sho)m(w)f(that)1409 1247 y(lim)1376 1307 y Fm(T)10 b Fl(!1)1604 1247 y Fu(inf)1584 1307 y Fm(x)p Fl(2)p Fm(K)1785 1247 y Fu(inf)1750 1307 y Fm(\036)p Fl(2)p Fm(F)1884 1318 y Ff(T)1947 1247 y Fn(I)1987 1261 y Fm(x;T)2101 1247 y Fu(\()p Fn(\036)p Fu(\))42 b(=)e Fk(1)15 b Fn(:)721 b Fu(\(108\))448 1463 y(Let)38 b Fn(\036)672 1430 y Fm(x)753 1463 y Fu(b)s(e)e(the)i(tra)5 b(jectory)38 b(of)f(\(90\))i(starting)e (at)g Fn(x)g Fk(2)f Fn(K)7 b Fu(.)60 b(By)37 b(condition)f Fg(L1)p Fu(,)j Fn(\036)3393 1430 y Fm(x)448 1576 y Fu(hits)32 b Fn(B)5 b Fu(\()p Fn(\032=)p Fu(3\))34 b(in)d(a)j(\014nite)d(time)i Fn(t)1607 1543 y Fm(x)1651 1576 y Fu(.)48 b(By)33 b(the)g(con)m(tin)m (uous)f(dep)s(endence)g(of)h Fn(\036)3135 1543 y Fm(x)3212 1576 y Fu(on)g(its)448 1689 y(initial)c(condition,)i(there)g(is)g(an)g (op)s(en)g(set)h Fn(W)2054 1656 y Fm(x)2129 1689 y Fu(suc)m(h)f(that)h (for)g(all)e Fn(y)g Fk(2)d Fn(W)3063 1656 y Fm(x)3106 1689 y Fu(,)32 b Fn(\036)3217 1656 y Fm(y)3290 1689 y Fu(hits)448 1802 y(the)f(set)g Fn(B)5 b Fu(\(2)p Fn(\032=)p Fu(3\))32 b(b)s(efore)e Fn(t)1411 1769 y Fm(x)1455 1802 y Fu(.)41 b(Since)30 b Fn(K)37 b Fu(is)30 b(compact,)i(there)e(is)g Fn(T)43 b Fu(suc)m(h)30 b(that,)i(for)e(all)448 1915 y Fn(x)c Fk(2)e Fn(K)7 b Fu(,)30 b Fn(\036)804 1882 y Fm(x)878 1915 y Fu(hits)e Fn(B)5 b Fu(\(2)p Fn(\032=)p Fu(3\))31 b(b)s(efore)e Fn(T)13 b Fu(.)40 b(Assume)29 b(no)m(w)h(that)g(the)g(iden)m(tit)m(y)f(\(108\))i(do)s(es)448 2028 y(not)j(hold.)49 b(Then,)34 b(for)f(some)h Fn(M)41 b(<)30 b Fk(1)p Fu(,)35 b(and)e(ev)m(ery)h(in)m(teger)g Fn(n)p Fu(,)g(there)g(is)f Fn( )3112 2042 y Fm(n)3190 2028 y Fk(2)d Fn(F)3339 2042 y Fm(nT)448 2141 y Fu(suc)m(h)g(that)h Fn(I)890 2155 y Fm(nT)988 2141 y Fu(\()p Fn( )1082 2155 y Fm(n)1130 2141 y Fu(\))25 b Fk(\024)g Fn(M)10 b Fu(.)41 b(Consequen)m(tly)-8 b(,)30 b(for)g(some)h Fn( )2457 2156 y Fm(n;k)2588 2141 y Fk(2)25 b Fn(F)2732 2155 y Fm(T)2787 2141 y Fu(,)31 b(w)m(e)g(ha)m(v)m(e)969 2378 y Fn(M)50 b Fk(\025)40 b Fn(I)1258 2392 y Fm(nT)1356 2378 y Fu(\()p Fn( )1450 2392 y Fm(n)1498 2378 y Fu(\))g(=)1727 2272 y Fm(n)1689 2297 y Fh(X)1685 2482 y Fm(k)r Ft(=1)1828 2378 y Fn(I)1868 2392 y Fm(T)1923 2378 y Fu(\()p Fn( )2017 2393 y Fm(n;k)2123 2378 y Fu(\))h Fk(\025)f Fn(n)2434 2301 y Fm(n)2380 2378 y Fu(min)2391 2438 y Fm(k)r Ft(=1)2546 2378 y Fn(I)2586 2392 y Fm(T)2641 2378 y Fu(\()p Fn( )2735 2393 y Fm(n;k)2841 2378 y Fu(\))15 b Fn(:)314 b Fu(\(109\))448 2625 y(Therefore)31 b(there)g(is)f(a)h(sequence)g Fn(\036)1693 2639 y Fm(n)1766 2625 y Fk(2)25 b Fn(F)1910 2639 y Fm(T)1996 2625 y Fu(suc)m(h)31 b(that)g(lim)2525 2639 y Fm(n)p Fl(!1)2729 2625 y Fn(I)2769 2639 y Fm(T)2824 2625 y Fu(\()p Fn(\036)2913 2639 y Fm(n)2960 2625 y Fu(\))26 b(=)g(0.)42 b(Since)448 2738 y(the)k(set)f Fk(f)p Fn(\036)66 b Fu(:)f Fn(I)1071 2752 y Fm(x;T)1185 2738 y Fu(\()p Fn(\036)p Fu(\))51 b Fk(\024)f Fu(1)p Fn(;)15 b(\036)p Fu(\(0\))52 b Fk(2)d Fn(K)7 b Fk(g)45 b Fu(is)f(compact,)51 b Fn(\036)2639 2705 y Fm(n)2731 2738 y Fu(has)45 b(a)g(limit)e(p)s(oin)m(t)448 2851 y Fn(\036)26 b Fk(2)g Fn(F)673 2865 y Fm(T)728 2851 y Fu(.)42 b(Since)30 b Fn(I)1073 2865 y Fm(T)1158 2851 y Fu(is)g(lo)m(w)m(er)h(semicon)m(tin)m(uous,)f(w)m(e)h(ha)m(v)m(e)h Fn(I)2530 2865 y Fm(T)2585 2851 y Fu(\()p Fn(\036)p Fu(\))27 b(=)e(0)31 b(and)g(therefore)448 2963 y Fn(\036)39 b Fu(is)e(tra)5 b(jectory)40 b(of)e(\(90\).)67 b(Since)37 b Fn(\036)i Fk(2)f Fn(F)1931 2977 y Fm(T)1986 2963 y Fu(,)j Fn(\036)d Fu(remains)g(outside)f(of)i Fn(B)5 b Fu(\(2)p Fn(\032=)p Fu(3\))40 b(and)448 3076 y(this)30 b(is)g(a)h(con)m(tradiction)f(with)f(the)i(de\014nition)e(of)h Fn(T)13 b Fu(.)41 b(This)29 b(concludes)h(the)h(pro)s(of)f(of)448 3189 y(Lemma)h(3.5.)p 1004 3197 50 67 v 589 3380 a(Instead)39 b(of)h(the)f(quan)m(tities)g Fn(V)20 b Fu(\()p Fn(K)1811 3394 y Fm(i)1840 3380 y Fn(;)15 b(K)1957 3394 y Fm(j)1994 3380 y Fu(\))39 b(and)g Fn(V)20 b Fu(\()p Fn(K)2439 3394 y Fm(i)2468 3380 y Fn(;)15 b(z)t Fu(\))40 b(de\014ned)e(in)f(Eqs.)i (\(91\))448 3493 y(and)30 b(\(92\),)i(it)e(is)g(useful)e(to)j(in)m(tro) s(duce)f(the)g(follo)m(wing)f(quan)m(tities:)511 3641 y(~)497 3664 y Fn(V)20 b Fu(\()p Fn(K)682 3678 y Fm(i)711 3664 y Fn(;)15 b(K)828 3678 y Fm(j)865 3664 y Fu(\))84 b(=)98 b(inf)1137 3723 y Fm(T)10 b(>)p Ft(0)1294 3664 y Fu(inf)20 b Fk(f)q Fn(I)1505 3678 y Fm(x;T)1619 3664 y Fu(\()p Fn(\036)p Fu(\))p Fn(;)32 b(\036)p Fu(\(0\))26 b Fk(2)f Fn(K)2158 3678 y Fm(i)2186 3664 y Fn(;)15 b(\036)p Fu(\()p Fn(T)e Fu(\))27 b Fk(2)d Fn(K)2605 3678 y Fm(j)2642 3664 y Fn(;)15 b(\036)p Fu(\()p Fn(t)p Fu(\))27 b Fk(62)d([)3012 3679 y Fm(l)q Fl(6)p Ft(=)p Fm(i;j)3169 3664 y Fn(K)3246 3679 y Fm(l)3272 3664 y Fk(g)31 b Fn(;)578 3830 y Fu(~)565 3853 y Fn(V)20 b Fu(\()p Fn(K)750 3867 y Fm(i)779 3853 y Fn(;)15 b(z)t Fu(\))84 b(=)98 b(inf)1137 3912 y Fm(T)10 b(>)p Ft(0)1294 3853 y Fu(inf)20 b Fk(f)q Fn(I)1505 3867 y Fm(x;T)1619 3853 y Fu(\()p Fn(\036)p Fu(\))p Fn(;)32 b(\036)p Fu(\(0\))26 b Fk(2)f Fn(K)2158 3867 y Fm(i)2186 3853 y Fn(;)15 b(\036)p Fu(\()p Fn(T)e Fu(\))27 b(=)e Fn(x;)15 b(\036)p Fu(\()p Fn(t)p Fu(\))26 b Fk(62)f([)2961 3868 y Fm(l)q Fl(6)p Ft(=)p Fm(i)3065 3853 y Fn(K)3142 3868 y Fm(l)3168 3853 y Fk(g)16 b Fn(:)-24 b Fu(\(110\))589 4059 y(The)28 b(follo)m(wing)e(Lemma)h(will)e(yield)h(an)h(upp)s(er)f (b)s(ound)f(on)j(the)g(on)f Fn(\027)2966 4073 y Fm(")3003 4059 y Fu(\()p Fn(D)s Fu(\),)i(where)448 4172 y Fn(\027)493 4186 y Fm(")560 4172 y Fu(is)h(the)g(\(unnormalized\))f(measure)h(giv)m (en)h(b)m(y)f(Eq.)g(\(102\).)448 4352 y Fg(Lemma)j(3.6)46 b Fe(Given)33 b Fn(h)25 b(>)g Fu(0)p Fe(,)33 b(for)g Fu(0)26 b Fn(<)f(\032)g(<)g(\032)2070 4319 y Fl(0)2126 4352 y Fe(su\016ciently)33 b(smal)5 b(l)33 b(one)g(has)530 4523 y Fu(\()p Fn(i)p Fu(\))167 b(lim)15 b(sup)868 4597 y Fm(")p Fl(!)p Ft(0)1092 4523 y Fn(")g Fu(log)118 b(sup)1282 4604 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)1570 4585 y Fa(0)1593 4604 y Ft(\))1636 4523 y Fn(P)1707 4485 y Fm(")1694 4545 y(y)1743 4523 y Fu(\()p Fn(\034)1818 4537 y Fm(D)1908 4523 y Fn(<)25 b(\034)2044 4537 y Ft(1)2083 4523 y Fu(\))41 b Fk(\024)f(\000)p Fu(\()16 b(inf)2376 4582 y Fm(z)s Fl(2)p Fm(D)2547 4500 y Fu(~)2534 4523 y Fn(V)k Fu(\()p Fn(K)2719 4537 y Fm(i)2748 4523 y Fn(;)15 b(z)t Fu(\))21 b Fk(\000)f Fn(h)p Fu(\))15 b Fn(;)122 b Fu(\(111\))499 4746 y(\()p Fn(ii)p Fu(\))167 b(lim)15 b(sup)868 4820 y Fm(")p Fl(!)p Ft(0)1092 4746 y Fn(")g Fu(log)118 b(sup)1282 4827 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)1570 4808 y Fa(0)1593 4827 y Ft(\))1636 4746 y Fn(P)1707 4708 y Fm(")1694 4768 y(y)1743 4746 y Fu(\()p Fn(x)1830 4760 y Fm(")1868 4746 y Fu(\()p Fn(\034)1943 4760 y Ft(1)1982 4746 y Fu(\))26 b Fk(2)f Fn(@)5 b(B)g Fu(\()p Fn(j;)15 b(\032)p Fu(\)\))42 b Fk(\024)e(\000)p Fu(\()2758 4723 y(~)2744 4746 y Fn(V)20 b Fu(\()p Fn(K)2929 4760 y Fm(i)2958 4746 y Fn(;)15 b(K)3075 4760 y Fm(j)3112 4746 y Fu(\))21 b Fk(\000)f Fn(h)p Fu(\))15 b Fn(:)3230 4954 y Fu(\(112\))1897 5225 y(25)p eop %%Page: 26 26 26 25 bop 448 573 a Fe(Pr)-5 b(o)g(of:)51 b Fu(W)-8 b(e)36 b(\014rst)e(pro)m(v)m(e)h(item)g(\(i\).)54 b(If)34 b(inf)1931 587 y Fm(z)s Fl(2)p Fm(D)2107 550 y Fu(~)2093 573 y Fn(V)21 b Fu(\()p Fn(K)2279 587 y Fm(i)2307 573 y Fn(;)15 b(z)t Fu(\))34 b(=)e(+)p Fk(1)i Fu(there)h(is)f(no)h(curv)m(e)448 686 y(connecting)g Fn(K)984 700 y Fm(i)1047 686 y Fu(to)h Fn(z)g Fk(2)c Fn(D)38 b Fu(without)33 b(touc)m(hing)i(the)g(other)f Fn(K)2644 700 y Fm(j)2681 686 y Fu(,)i Fn(j)i Fk(6)p Fu(=)32 b Fn(i)p Fu(..)54 b(Therefore)448 799 y Fn(P)519 766 y Fm(")506 821 y(y)556 799 y Fu(\()p Fn(\034)631 813 y Fm(D)720 799 y Fn(<)25 b(\034)856 813 y Ft(1)896 799 y Fu(\))g(=)g(0)30 b(and)f(there)h(is)f(nothing)g(to)i(pro)m(v)m (e.)41 b(Otherwise,)29 b(for)g Fn(h)d(>)f Fu(0)30 b(w)m(e)g(set)462 903 y(~)448 926 y Fn(V)501 941 y Fm(h)574 926 y Fu(=)d(inf)782 940 y Fm(z)s Fl(2)p Fm(D)958 903 y Fu(~)944 926 y Fn(V)20 b Fu(\()p Fn(K)1129 940 y Fm(i)1158 926 y Fn(;)15 b(z)t Fu(\))22 b Fk(\000)f Fn(h)p Fu(.)45 b(Since)31 b Fn(V)20 b Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))33 b(satis\014es)e(the)h(triangle)f (inequalit)m(y)-8 b(,)31 b(w)m(e)448 1039 y(ha)m(v)m(e,)h(b)m(y)e (condition)f Fg(L2)p Fu(,)i(that,)g(for)f Fn(\032)h Fu(small)e(enough) 616 1218 y(inf)502 1281 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)790 1262 y Fa(0)813 1281 y Ft(\))872 1218 y Fu(inf)856 1277 y Fm(z)s Fl(2)p Fm(D)1027 1195 y Fu(~)1013 1218 y Fn(V)21 b Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))41 b Fk(\025)155 b Fu(inf)1443 1281 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)1731 1262 y Fa(0)1755 1281 y Ft(\))1813 1218 y Fu(inf)1798 1277 y Fm(z)s Fl(2)p Fm(D)1969 1195 y Fu(~)1955 1218 y Fn(V)20 b Fu(\()p Fn(K)2140 1232 y Fm(i)2169 1218 y Fn(;)15 b(z)t Fu(\))21 b Fk(\000)121 b Fu(sup)2402 1299 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)2690 1280 y Fa(0)2714 1299 y Ft(\))2770 1195 y Fu(~)2756 1218 y Fn(V)21 b Fu(\()p Fn(K)2942 1232 y Fm(i)2970 1218 y Fn(;)15 b(y)s Fu(\))41 b Fk(\025)3259 1195 y Fu(~)3245 1218 y Fn(V)3298 1233 y Fm(h)3358 1218 y Fn(:)3230 1400 y Fu(\(113\))448 1513 y(where)599 1669 y(~)585 1692 y Fn(V)20 b Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))42 b(=)55 b(inf)1015 1751 y Fm(T)10 b(>)p Ft(0)1171 1692 y Fu(inf)21 b Fk(f)p Fn(I)1382 1706 y Fm(x;T)1496 1692 y Fu(\()p Fn(\036)p Fu(\))p Fn(;)32 b(\036)p Fu(\(0\))26 b(=)f Fn(y)s(;)15 b(\036)p Fu(\()p Fn(T)e Fu(\))26 b(=)f Fn(z)t(;)15 b(\036)p Fu(\()p Fn(t)p Fu(\))27 b Fk(62)e([)2785 1707 y Fm(l)q Fl(6)p Ft(=)p Fm(i)2889 1692 y Fn(K)2966 1707 y Fm(l)2993 1692 y Fk(g)30 b Fn(:)137 b Fu(\(114\))448 1901 y(By)31 b(Lemma)f(3.5,)i(there)f(is)e Fn(T)38 b(<)25 b Fk(1)30 b Fu(suc)m(h)g(that)1019 2079 y(lim)15 b(sup)1089 2154 y Fm(")p Fl(!)p Ft(0)1312 2079 y Fn(")g Fu(log)119 b(sup)1503 2161 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)1791 2142 y Fa(0)1814 2161 y Ft(\))1857 2079 y Fn(P)1928 2042 y Fm(")1915 2102 y(y)1964 2079 y Fu(\()p Fn(\034)2039 2093 y Fm(D)2124 2079 y Fk(^)19 b Fn(\034)2244 2093 y Ft(1)2309 2079 y Fn(>)25 b(T)13 b Fu(\))40 b Fn(<)g Fk(\000)p Fn(V)2781 2094 y Fm(h)2841 2079 y Fn(:)364 b Fu(\(115\))448 2328 y(Let)34 b Fn(G)686 2342 y Fm(T)773 2328 y Fu(denote)f(the)g (subset)f(of)h Fk(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))34 b(whic)m(h)e(consists)g(of)h(functions)e Fn(\036)p Fu(\()p Fn(t)p Fu(\))i(suc)m(h)448 2441 y(that)i Fn(\036)p Fu(\()p Fn(t)p Fu(\))d Fk(2)p 929 2369 78 4 v 30 w Fn(D)37 b Fu(for)c(some)i Fn(t)c Fk(2)f Fu([0)p Fn(;)15 b(T)e Fu(])35 b(and)e Fn(\036)p Fu(\()p Fn(t)p Fu(\))f Fk(62)f Fn(B)5 b Fu(\()p Fn(\032)p Fu(\))34 b(if)f Fn(t)d Fk(\024)h Fu(inf)6 b Fk(f)p Fn(s)15 b(;)g(\036)p Fu(\()p Fn(s)p Fu(\))42 b Fn(=)-55 b Fk(2)31 b Fn(D)s Fk(g)p Fu(.)448 2554 y(The)f(set)h Fn(G)849 2568 y Fm(T)934 2554 y Fu(is)e(closed)i(as) f(is)g(seen)g(b)m(y)g(considering)f(its)h(complemen)m(t.)589 2666 y(W)-8 b(e)32 b(ha)m(v)m(e)1022 2845 y(inf)908 2909 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)1196 2890 y Fa(0)1219 2909 y Ft(\))1303 2845 y Fu(inf)1262 2905 y Fm(\036)p Fl(2)p Fm(G)1406 2916 y Ff(T)1470 2845 y Fn(I)1510 2859 y Fm(y)r(;T)1621 2845 y Fu(\()p Fn(\036)p Fu(\))42 b Fk(\025)154 b Fu(inf)1898 2909 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)2186 2890 y Fa(0)2209 2909 y Ft(\))2268 2845 y Fu(inf)2252 2921 y Fm(z)s Fl(2)p 2335 2866 60 4 v Fm(D)2423 2822 y Fu(~)2410 2845 y Fn(V)20 b Fu(\()p Fn(y)s(;)15 b(z)t Fu(\))41 b Fk(\025)g Fn(V)2893 2860 y Fm(h)2953 2845 y Fn(;)252 b Fu(\(116\))448 3071 y(and)30 b(th)m(us)g(b)m(y)g(Theorem)g(2.1,)i(w)m(e)f(ha)m(v)m(e)553 3249 y(lim)15 b(sup)623 3324 y Fm(")p Fl(!)p Ft(0)847 3249 y Fn(")g Fu(log)118 b(sup)1037 3331 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)1325 3312 y Fa(0)1348 3331 y Ft(\))1391 3249 y Fn(P)1462 3212 y Fm(")1449 3272 y(y)1499 3249 y Fu(\()p Fn(x)1586 3263 y Fm(")1648 3249 y Fk(2)25 b Fn(G)1806 3263 y Fm(T)1860 3249 y Fu(\))41 b Fk(\024)f(\000)129 b Fu(inf)2133 3313 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)2421 3294 y Fa(0)2445 3313 y Ft(\))2528 3249 y Fu(inf)2487 3309 y Fm(\036)p Fl(2)p Fm(G)2631 3320 y Ff(T)2695 3249 y Fn(I)2735 3263 y Fm(y)r(;T)2847 3249 y Fu(\()p Fn(\036)p Fu(\))41 b Fk(\024)f(\000)p Fn(V)3247 3264 y Fm(h)3307 3249 y Fn(:)3230 3432 y Fu(\(117\))448 3545 y(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(inequalit)m(y)988 3724 y Fn(P)1059 3686 y Fm(")1046 3746 y(y)1096 3724 y Fu(\()p Fn(\034)1171 3738 y Fm(D)1260 3724 y Fn(<)25 b(\034)1396 3738 y Ft(1)1435 3724 y Fu(\))h Fk(\024)f Fn(P)1663 3686 y Fm(")1650 3746 y(y)1700 3724 y Fu(\()p Fn(\034)1775 3738 y Fm(D)1859 3724 y Fk(^)20 b Fn(\034)1980 3738 y Ft(1)2044 3724 y Fn(>)25 b(T)13 b Fu(\))20 b(+)g Fn(P)2423 3686 y Fm(")2410 3746 y(y)2460 3724 y Fu(\()p Fn(x)2547 3738 y Fm(")2609 3724 y Fk(2)25 b Fn(G)2767 3738 y Fm(T)2822 3724 y Fu(\))15 b Fn(;)333 b Fu(\(118\))448 3902 y(and)30 b(com)m(bining)f(the)i(estimates)g(\(115\))h(and)e (\(117\))i(yields)1112 4081 y(lim)15 b(sup)1182 4155 y Fm(")p Fl(!)p Ft(0)1406 4081 y Fn(")g Fu(log)118 b(sup)1596 4162 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)1884 4143 y Fa(0)1908 4162 y Ft(\))1950 4081 y Fn(P)2021 4043 y Fm(")2008 4103 y(y)2058 4081 y Fu(\()p Fn(\034)2133 4095 y Fm(D)2217 4081 y Fk(^)20 b Fn(\034)2338 4095 y Ft(1)2377 4081 y Fu(\))41 b Fk(\024)f(\000)p Fn(V)2688 4096 y Fm(h)2748 4081 y Fn(:)457 b Fu(\(119\))448 4329 y(This)29 b(completes)i(the)f(pro)s(of)g(of)g(item)g(\(i\))h(of)f (Lemma)h(3.6.)589 4442 y(The)f(pro)s(of)g(of)g(part)g(\(ii\))f(of)i (the)f(Lemma)g(is)g(v)m(ery)g(similar)e(to)j(the)f(\014rst)g(part)g (and)448 4555 y(follo)m(ws)37 b(closely)h(the)g(corresp)s(onding)e (estimates)j(in)e([8],)k(Chapter)c(6,)k(Lemma)d(2.1.)448 4668 y(The)30 b(details)g(are)g(left)g(to)i(the)e(reader.)p 1866 4676 50 67 v 589 4863 a(The)k(follo)m(wing)e(Lemma)h(will)e(yield) h(a)i(lo)m(w)m(er)g(b)s(ound)d(on)j Fn(\027)2670 4877 y Fm(")2706 4863 y Fu(\()p Fn(D)s Fu(\).)51 b(It)34 b(mak)m(es)g(full) 448 4976 y(use)c(of)h(the)f(information)f(con)m(tained)i(in)e(Lemmas)h (2.3)i(and)d(2.8.)1897 5225 y(26)p eop %%Page: 27 27 27 26 bop 448 573 a Fg(Lemma)33 b(3.7)46 b Fe(Given)33 b Fn(h)25 b(>)g Fu(0)p Fe(,)33 b(for)g Fu(0)26 b Fn(<)f(\032)1902 540 y Fl(0)1950 573 y Fn(<)g(\032)33 b Fe(su\016ciently)g(smal)5 b(l)33 b(one)g(has)553 763 y Fu(\()p Fn(i)p Fu(\))167 b(lim)15 b(inf)878 820 y Fm(")p Fl(!)p Ft(0)1088 763 y Fn(")g Fu(log)122 b(inf)1278 827 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)p Ft(\))1613 763 y Fn(P)1684 726 y Fm(")1671 786 y(x)1720 763 y Fu(\()p Fn(\034)1795 777 y Fm(D)1885 763 y Fn(<)25 b(\034)2021 777 y Ft(1)2060 763 y Fu(\))41 b Fk(\025)f(\000)p Fu(\()16 b(inf)2353 822 y Fm(z)s Fl(2)p Fm(D)2524 740 y Fu(~)2511 763 y Fn(V)k Fu(\()p Fn(K)2696 777 y Fm(i)2724 763 y Fn(;)15 b(z)t Fu(\))22 b(+)e Fn(h)p Fu(\))15 b Fn(:)145 b Fu(\(120\))522 969 y(\()p Fn(ii)p Fu(\))167 b(lim)15 b(inf)878 1025 y Fm(")p Fl(!)p Ft(0)1088 969 y Fn(")g Fu(log)122 b(inf)1278 1032 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)p Ft(\))1613 969 y Fn(P)1684 931 y Fm(")1671 991 y(x)1720 969 y Fu(\()p Fn(x)1807 983 y Fm(")1844 969 y Fu(\()p Fn(\034)1919 983 y Ft(1)1959 969 y Fu(\))26 b Fk(2)e Fn(@)5 b(B)g Fu(\()p Fn(j;)15 b(\032)p Fu(\)\))43 b Fk(\025)d(\000)p Fu(\()2735 946 y(~)2721 969 y Fn(V)20 b Fu(\()p Fn(K)2906 983 y Fm(i)2935 969 y Fn(;)15 b(K)3052 983 y Fm(j)3089 969 y Fu(\))21 b(+)e Fn(h)p Fu(\))c Fn(:)3230 1159 y Fu(\(121\))448 1357 y Fe(Pr)-5 b(o)g(of:)47 b Fu(W)-8 b(e)35 b(start)e(with)f(the)h(pro)s(of)g(of)g(item)f(\(i\).)49 b(If)33 b(inf)2412 1371 y Fm(z)s Fl(2)p Fm(D)2588 1334 y Fu(~)2574 1357 y Fn(V)20 b Fu(\()p Fn(K)2759 1371 y Fm(i)2788 1357 y Fn(;)15 b(z)t Fu(\))31 b(=)e(+)p Fk(1)k Fu(there)448 1470 y(is)i(nothing)f(to)j(pro)m(v)m(e.)57 b(Otherwise)34 b(let)h Fn(h)f(>)g Fu(0)i(b)s(e)f(giv)m(en.)56 b(By)36 b(condition)e Fg(L2)p Fu(,)j(\(see)448 1583 y(Corollary)29 b(2.8\),)j(there)f(are)f Fn(\032)g Fu(and)g Fn(\032)1742 1550 y Fl(0)1791 1583 y Fn(>)25 b Fu(0)30 b(with)f Fn(\032)c(<)g(\032) 2384 1550 y Fl(0)2408 1583 y Fn(=)p Fu(3)31 b(and)e Fn(T)2758 1597 y Ft(0)2823 1583 y Fn(<)c Fk(1)30 b Fu(suc)m(h)g(that,)448 1696 y(for)23 b(all)f Fn(x)j Fk(2)g Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\),)26 b(there)e(is)e(a)h(path)g Fn( )1873 1663 y Fm(x)1943 1696 y Fk(2)i(C)5 b Fu(\([0)p Fn(;)15 b(T)2280 1710 y Ft(0)2321 1696 y Fu(]\))24 b(whic)m(h)d (satis\014es)i Fn(I)3025 1710 y Fm(x;T)3126 1719 y Fj(0)3164 1696 y Fu(\()p Fn( )3261 1663 y Fm(x)3306 1696 y Fu(\))i Fk(\024)448 1809 y Fn(h=)p Fu(3)d(with)d Fn( )871 1776 y Fm(x)916 1809 y Fu(\(0\))26 b(=)f Fn(x)c Fu(and)f Fn( )1455 1776 y Fm(x)1499 1809 y Fu(\()p Fn(T)1587 1823 y Ft(0)1627 1809 y Fu(\))26 b(=)f Fn(x)1836 1823 y Ft(0)1900 1809 y Fk(2)g Fn(K)2063 1823 y Fm(i)2112 1809 y Fu(and)20 b Fn( )2341 1776 y Fm(x)2385 1809 y Fu(\()p Fn(t)p Fu(\))26 b Fk(2)f Fn(B)5 b Fu(\(2)p Fn(\032)2801 1776 y Fl(0)2825 1809 y Fn(=)p Fu(3\),)24 b(0)h Fk(\024)g Fn(t)g Fk(\024)g Fn(T)3372 1823 y Ft(0)3412 1809 y Fu(.)589 1922 y(By)30 b(condition)e Fg(L3)p Fu(,)i(there)f(are)h Fn(z)f Fk(2)c Fn(D)s Fu(,)k Fn(T)2023 1936 y Ft(1)2088 1922 y Fn(<)c Fk(1)k Fu(and)g Fn(\036)2534 1936 y Ft(1)2599 1922 y Fk(2)c(C)5 b Fu(\([0)p Fn(;)15 b(T)2936 1936 y Ft(1)2977 1922 y Fu(]\))30 b(suc)m(h)e(that)448 2035 y Fn(I)488 2049 y Fm(x)528 2058 y Fj(0)563 2049 y Fm(;T)624 2058 y Fj(1)662 2035 y Fu(\()p Fn(\036)751 2049 y Ft(1)791 2035 y Fu(\))e Fk(\024)f Fu(inf)1058 2049 y Fm(z)s Fl(2)p Fm(D)1234 2012 y Fu(~)1220 2035 y Fn(V)20 b Fu(\()p Fn(K)1405 2049 y Fm(i)1434 2035 y Fn(;)15 b(z)t Fu(\))f(+)g Fn(h=)p Fu(3)29 b(and)e Fn(\036)2053 2049 y Ft(1)2093 2035 y Fu(\(0\))f(=)f Fn(x)2382 2049 y Ft(0)2447 2035 y Fk(2)g Fn(K)2610 2049 y Fm(i)2665 2035 y Fu(and)i Fn(\036)2893 2049 y Ft(1)2933 2035 y Fu(\()p Fn(T)3021 2049 y Ft(1)3061 2035 y Fu(\))e(=)g Fn(z)32 b Fu(and)448 2148 y Fn(\036)502 2162 y Ft(1)567 2148 y Fu(do)s(es)25 b(not)g(touc)m(h)h Fn(K)1245 2162 y Fm(j)1282 2148 y Fu(,)g(with)e Fn(j)31 b Fk(6)p Fu(=)25 b Fn(i)p Fu(.)39 b(W)-8 b(e)27 b(ma)m(y)f(and)e(will)f (assume)i(that)g Fn(\032)h Fu(and)e Fn(\032)3267 2115 y Fl(0)3316 2148 y Fu(are)448 2261 y(c)m(hosen)37 b(suc)m(h)e(that)h(2) p Fn(\032)1249 2228 y Fl(0)1307 2261 y Fk(\024)e Fu(dist)o(\()p Fn(\036)1648 2275 y Ft(1)1688 2261 y Fu(\()p Fn(t)p Fu(\))p Fn(;)15 b Fk([)1892 2276 y Fm(j)t Fl(6)p Ft(=)p Fm(i)2008 2261 y Fn(K)2085 2275 y Fm(j)2122 2261 y Fu(.)56 b(W)-8 b(e)37 b(note)g(\001)d(=)f(dist)o(\()p Fn(z)t(;)15 b(@)5 b(D)s Fu(\).)59 b(Let)448 2373 y Fn(x)500 2387 y Ft(1)571 2373 y Fu(b)s(e)32 b(the)g(p)s(oin)m(t)f(of)h(last)f(in)m(tersection)h (of)g Fn(\036)2021 2387 y Ft(1)2092 2373 y Fu(with)f Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\))33 b(and)e(let)h Fn(t)2993 2387 y Ft(1)3064 2373 y Fu(suc)m(h)f(that)448 2486 y Fn(\036)502 2500 y Ft(1)542 2486 y Fu(\()p Fn(t)610 2500 y Ft(1)650 2486 y Fu(\))d(=)g Fn(x)864 2500 y Ft(1)903 2486 y Fu(.)46 b(W)-8 b(e)33 b(note)g Fn(\036)1390 2500 y Ft(2)1458 2486 y Fk(2)28 b(C)5 b Fu(\([0)p Fn(;)15 b(T)1798 2500 y Ft(2)1839 2486 y Fu(]\),)33 b(with)e Fn(T)2219 2500 y Ft(2)2287 2486 y Fu(=)d Fn(T)2439 2500 y Ft(1)2499 2486 y Fk(\000)22 b Fn(t)2625 2500 y Ft(1)2664 2486 y Fu(,)33 b(the)f(path)g(obtained)448 2599 y(from)g Fn(\036)719 2613 y Ft(1)791 2599 y Fu(b)m(y)h(deleting)e(up)h(to)h (time)f Fn(t)1754 2613 y Ft(1)1826 2599 y Fu(and)g(translating)f(in)g (time.)47 b(Notice)34 b(that)f(the)448 2712 y(path)d Fn(\036)714 2726 y Ft(2)784 2712 y Fu(ma)m(y)h(hit)e(sev)m(eral)i (times)e Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)1943 2679 y Fl(0)1968 2712 y Fu(\),)30 b(but)g(hits)f Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\))31 b(only)f(one)g(time)g(\(at)448 2825 y(time)g(0\).)42 b(Denote)32 b(as)1376 3015 y Fn(\033)c Fu(=)d(inf)6 b Fk(f)p Fn(t)25 b Fu(:)h Fn(\036)1871 3029 y Ft(2)1910 3015 y Fu(\()p Fn(t)p Fu(\))g Fk(2)f Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)2405 2978 y Fl(0)2429 3015 y Fu(\))p Fk(g)721 b Fu(\(122\))448 3206 y(the)23 b(\014rst)e(time)h Fn(\036)1029 3220 y Ft(2)1068 3206 y Fu(\()p Fn(t)p Fu(\))h(hits)e Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)1643 3173 y Fl(0)1667 3206 y Fu(\).)38 b(W)-8 b(e)23 b(c)m(ho)s(ose)g(\001)2269 3173 y Fl(0)2314 3206 y Fu(so)g(small)d(that)j(if)e Fn( )28 b Fk(2)d(C)5 b Fu(\([0)p Fn(;)15 b(T)3335 3220 y Ft(2)3377 3206 y Fu(]\))448 3319 y(b)s(elongs)24 b(to)i(the)f(\001)1102 3286 y Fl(0)1126 3319 y Fu(-neigh)m(b)s(orho)s(o)s(d)e(of)i Fn(\036)1874 3333 y Ft(2)1913 3319 y Fu(,)i(then)e Fn( )s Fu(\()p Fn(t)p Fu(\))h(do)s(es)e(not)i(in)m(tersect)f Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\))p Fk(g)448 3432 y Fu(and)30 b Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)905 3399 y Fl(0)929 3432 y Fu(\))p Fk(g)31 b Fu(for)f(0)c Fn(<)f(t)g(<)g(\033)33 b Fu(and)d(and)g(do)s(es)f(not)i(in)m(tersect)g Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\))p Fk(g)32 b Fu(for)e Fn(t)25 b(>)g(\033)s Fu(.)589 3545 y(By)g(condition)f Fg(L2)p Fu(,)i(there)f(are)g Fn(T)1712 3559 y Ft(3)1776 3545 y Fn(<)g Fk(1)g Fu(and)f Fn(\036)2213 3559 y Ft(3)2278 3545 y Fk(2)h(C)5 b Fu(\([0)p Fn(;)15 b(T)2615 3559 y Ft(3)2656 3545 y Fu(]\))25 b(suc)m(h)f(that)h Fn(\036)3185 3559 y Ft(3)3225 3545 y Fu(\(0\))h(=)448 3658 y Fn(x)500 3672 y Ft(0)540 3658 y Fu(,)46 b Fn(\036)665 3672 y Ft(3)705 3658 y Fu(\()p Fn(T)793 3672 y Ft(3)833 3658 y Fu(\))h(=)f Fn(x)1084 3672 y Ft(1)1123 3658 y Fu(,)h Fn(\036)1249 3672 y Ft(3)1288 3658 y Fu(\()p Fn(t)p Fu(\))g Fk(2)f Fn(B)5 b Fu(\(2)p Fn(\032)1746 3625 y Fl(0)1770 3658 y Fn(=)p Fu(3\),)48 b(0)f Fk(\024)f Fn(t)g Fk(\024)g Fn(T)2426 3672 y Ft(3)2466 3658 y Fu(,)g(and)d Fn(I)2767 3672 y Fm(x)2807 3681 y Fj(0)2841 3672 y Fm(;T)2902 3681 y Fj(3)2941 3658 y Fu(\()p Fn(\036)3030 3672 y Ft(3)3070 3658 y Fu(\))j Fk(\024)h Fn(h=)p Fu(3.)448 3770 y(Concatenating)39 b Fn( )1124 3737 y Fm(x)1169 3770 y Fu(,)h Fn(\036)1288 3784 y Ft(3)1366 3770 y Fu(and)e Fn(\036)1605 3784 y Ft(2)1645 3770 y Fu(,)i(w)m(e)f(obtain)f(a)h(path)f Fn(\036)2501 3737 y Fm(x)2583 3770 y Fk(2)h(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))40 b(with)d Fn(T)51 b Fu(=)448 3883 y Fn(T)501 3897 y Ft(0)561 3883 y Fu(+)20 b Fn(T)705 3897 y Ft(3)764 3883 y Fu(+)g Fn(T)908 3897 y Ft(2)977 3883 y Fu(and)30 b Fn(I)1194 3897 y Fm(x;T)1308 3883 y Fu(\()p Fn(\036)1397 3850 y Fm(x)1442 3883 y Fu(\))25 b Fk(\024)g Fu(inf)1709 3897 y Fm(z)s Fl(2)p Fm(D)1885 3860 y Fu(~)1871 3883 y Fn(V)20 b Fu(\()p Fn(K)2056 3897 y Fm(i)2085 3883 y Fn(;)15 b(z)t Fu(\))21 b(+)e Fn(h)p Fu(.)41 b(By)31 b(construction)f (the)g(path)448 3996 y Fn(\036)502 3963 y Fm(x)574 3996 y Fu(a)m(v)m(oids)f Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\))p Fk(g)30 b Fu(after)f(the)f(time)g Fn(T)1863 4010 y Ft(0)1918 3996 y Fu(+)16 b Fn(T)2058 4010 y Ft(3)2113 3996 y Fu(+)f Fn(\033)32 b Fu(where)27 b Fn(\033)32 b Fu(de\014ned)27 b(in)g(Eq.)h(\(122\).)589 4109 y(W)-8 b(e)32 b(consider)d(the)i(op)s(en)f(set)622 4344 y Fn(U)684 4358 y Fm(T)780 4344 y Fu(=)983 4263 y Fh([)891 4451 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1182 4225 y Fh(\032)1244 4344 y Fn( )f Fk(2)24 b(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))43 b(:)e Fk(k)p Fn( )24 b Fk(\000)c Fn(\036)2123 4358 y Fm(x)2167 4344 y Fk(k)25 b(\024)g Fu(min)o Fk(f)2540 4282 y Fn(\032)p 2540 4323 48 4 v 2541 4406 a Fu(3)2597 4344 y Fn(;)2648 4282 y Fu(\001)p 2648 4323 76 4 v 2663 4406 a(2)2733 4344 y Fn(;)2784 4282 y Fu(\001)2860 4249 y Fl(0)p 2784 4323 100 4 v 2810 4406 a Fu(2)2893 4344 y Fk(g)2938 4225 y Fh(\033)3031 4344 y Fn(:)174 b Fu(\(123\))448 4630 y(By)27 b(construction)f(the)h (ev)m(en)m(t)h Fk(f)p Fn(x)1590 4644 y Fm(")1627 4630 y Fu(\()p Fn(t)p Fu(\))e Fk(2)f Fn(U)1904 4644 y Fm(T)1959 4630 y Fk(g)i Fu(is)e(con)m(tained)i(in)e(the)i(ev)m(en)m(t)h Fk(f)p Fn(\034)3102 4644 y Fm(D)3191 4630 y Fk(\024)d Fn(\034)3327 4644 y Ft(1)3366 4630 y Fk(g)p Fu(.)448 4743 y(By)31 b(Theorem)f(2.1)i(w)m(e)e(ha)m(v)m(e)572 4933 y(lim)15 b(inf)629 4990 y Fm(")p Fl(!)p Ft(0)840 4933 y Fn(")g Fu(log)99 b(inf)1030 4997 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1320 4933 y Fn(P)1391 4896 y Fm(")1378 4956 y(x)1428 4933 y Fu(\()p Fn(\034)1503 4947 y Fm(D)1592 4933 y Fn(<)25 b(\034)1728 4947 y Ft(1)1767 4933 y Fu(\))99 b Fk(\025)f Fu(lim)15 b(inf)2127 4990 y Fm(")p Fl(!)p Ft(0)2337 4933 y Fn(")g Fu(log)100 b(inf)2527 4997 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))2818 4933 y Fn(P)2889 4896 y Fm(")2876 4956 y(x)2925 4933 y Fu(\()p Fn(x)3012 4947 y Fm(")3075 4933 y Fk(2)25 b Fn(U)3223 4947 y Fm(T)3278 4933 y Fu(\))1897 5225 y(27)p eop %%Page: 28 28 28 27 bop 1901 573 a Fk(\025)98 b(\000)84 b Fu(sup)2156 654 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))2487 573 y Fu(inf)2446 633 y Fm( )r Fl(2)p Fm(U)2589 644 y Ff(T)2653 573 y Fn(I)2693 587 y Fm(x;T)2808 573 y Fu(\()p Fn( )s Fu(\))1901 780 y Fk(\025)98 b(\000)84 b Fu(sup)2156 862 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))2446 780 y Fn(I)2486 794 y Fm(x;T)2601 780 y Fu(\()p Fn(\036)2690 743 y Fm(x)2734 780 y Fu(\))1901 1004 y Fk(\025)98 b(\000)p Fu(\()16 b(inf)2176 1063 y Fm(z)s Fl(2)p Fm(D)2347 981 y Fu(~)2334 1004 y Fn(V)k Fu(\()p Fn(K)2519 1018 y Fm(i)2548 1004 y Fn(;)15 b(z)t Fu(\))21 b(+)f Fn(h)p Fu(\))15 b Fn(:)322 b Fu(\(124\))448 1235 y(This)29 b(concludes)g(the)i(pro)s(of)f (of)g(item)g(\(i\).)589 1348 y(The)35 b(pro)s(of)e(of)i(\(ii\))f(follo) m(ws)g(v)m(ery)h(closely)f(the)h(corresp)s(onding)d(estimate)k(in)d([8) q(],)448 1461 y(Chapter)27 b(6,)h(Lemma)f(2.1.,)i(whic)m(h)d(considers) f(the)i(case)h(where)f(the)g(generator)h(of)f(the)448 1574 y(di\013usion)j(is)h(elliptic:)43 b(for)32 b(an)m(y)h Fn(h)c(>)f Fu(0)33 b(one)f(construct)h(paths)f Fn(\036)2695 1541 y Fm(xy)2805 1574 y Fk(2)c(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))35 b(from)448 1687 y Fn(x)26 b Fk(2)e Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\))26 b(to)f Fn(y)j Fk(2)d Fn(@)5 b(B)g Fu(\()p Fn(j;)15 b(\032)p Fu(\))26 b(suc)m(h)e(that)g Fn(I)1992 1701 y Fm(x;T)2107 1687 y Fu(\()p Fn(\036)2196 1654 y Fm(xy)2277 1687 y Fu(\))i Fk(\024)2448 1664 y Fu(~)2434 1687 y Fn(V)20 b Fu(\()p Fn(K)2619 1701 y Fm(i)2648 1687 y Fn(;)15 b(K)2765 1701 y Fm(j)2802 1687 y Fu(\))8 b(+)g Fn(h=)p Fu(2)26 b(and)d(suc)m(h)448 1799 y(that)35 b(if)e Fn(x)788 1813 y Fm(")824 1799 y Fu(\()p Fn(t)p Fu(\))h(is)f(in)g(a)h(small)e(neigh)m(b)s(orho)s(o)s(d)g (of)i Fn(\036)2222 1766 y Fm(xy)2303 1799 y Fu(,)h(then)e Fn(x)2625 1813 y Fm(")2662 1799 y Fu(\()p Fn(\034)2737 1813 y Ft(1)2777 1799 y Fu(\))e Fk(2)g Fn(@)5 b(B)g Fu(\()p Fn(j;)15 b(\032)p Fu(\).)52 b(As)448 1912 y(in)29 b(part)h(\(i\))g(of)g (the)g(Lemma,)g(the)h(k)m(ey)f(elemen)m(t)h(to)g(construct)f(the)g (paths)g Fn(\036)3109 1879 y Fm(xy)3220 1912 y Fu(is)f(the)448 2025 y(condition)34 b Fg(L2)h Fu(of)g(small-time)e(con)m(trollabilit)m (y)g(around)h(the)h(sets)g Fn(K)2880 2039 y Fm(i)2909 2025 y Fu(.)54 b(The)34 b(details)448 2138 y(are)d(left)f(to)h(the)g (reader.)589 2251 y(This)e(concludes)h(the)g(pro)s(of)g(of)g(lemma)g (3.7.)p 2229 2259 50 67 v 589 2458 a(The)e(follo)m(wing)f(t)m(w)m(o)i (Lemmas)f(giv)m(e)h(upp)s(er)d(and)i(lo)m(w)m(er)g(b)s(ounds)e(on)i (the)h(normal-)448 2571 y(ization)h(constan)m(t)i Fn(\027)1158 2585 y Fm(")1195 2571 y Fu(\()p Fn(X)7 b Fu(\),)31 b(where)f Fn(\027)1711 2585 y Fm(")1778 2571 y Fu(is)g(de\014ned)f(in)g(Eq.)h (\(102\).)448 2775 y Fg(Lemma)j(3.8)46 b Fe(F)-7 b(or)34 b(any)f Fn(h)26 b(>)f Fu(0)p Fe(,)32 b(we)h(have)1439 2971 y Fu(lim)15 b(inf)1496 3028 y Fm(")p Fl(!)p Ft(0)1706 2971 y Fn(")g Fu(log)i Fn(\027)1941 2985 y Fm(")1978 2971 y Fu(\()p Fn(X)7 b Fu(\))42 b Fk(\025)e(\000)p Fn(h)15 b(:)784 b Fu(\(125\))448 3206 y Fe(Pr)-5 b(o)g(of:)40 b Fu(W)-8 b(e)26 b(c)m(ho)s(ose)h(an)e(arbitrary)g Fn(h)g(>)g Fu(0.)40 b(F)-8 b(or)26 b(an)m(y)g Fn(\032)2320 3173 y Fl(0)2368 3206 y Fn(>)f Fu(0)h(w)m(e)g(ha)m(v)m(e)h(the)e(inequalit)m (y:)1011 3402 y Fn(\027)1056 3416 y Fm(")1093 3402 y Fu(\()p Fn(X)7 b Fu(\))99 b Fk(\025)f Fn(\027)1558 3416 y Fm(")1595 3402 y Fu(\()p Fn(B)5 b Fu(\()p Fn(\032)1786 3364 y Fl(0)1809 3402 y Fu(\)\))1344 3575 y(=)1513 3460 y Fh(Z)1559 3649 y Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1767 3575 y Fn(l)1794 3589 y Fm(")1831 3575 y Fu(\()p Fn(dx)p Fu(\))p Fn(E)2072 3537 y Fm(")2067 3597 y(x)2127 3460 y Fh(Z)2210 3486 y Fm(\034)2241 3495 y Fj(1)2173 3649 y Ft(0)2296 3575 y Fg(1)2348 3593 y Fm(B)s Ft(\()p Fm(\032)2467 3574 y Fa(0)2490 3593 y Ft(\))2522 3575 y Fu(\()p Fn(x)2609 3589 y Fm(")2646 3575 y Fu(\()p Fn(t)p Fu(\)\))p Fn(dt)1344 3816 y Fk(\025)1513 3701 y Fh(Z)1559 3889 y Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1767 3816 y Fn(l)1794 3830 y Fm(")1831 3816 y Fu(\()p Fn(dx)p Fu(\))p Fn(E)2072 3778 y Fm(")2067 3838 y(x)2127 3701 y Fh(Z)2210 3727 y Fm(\033)2250 3736 y Fj(0)2173 3889 y Ft(0)2305 3816 y Fg(1)2357 3834 y Fm(B)s Ft(\()p Fm(\032)2476 3815 y Fa(0)2499 3834 y Ft(\))2531 3816 y Fu(\()p Fn(x)2618 3830 y Fm(")2655 3816 y Fu(\()p Fn(t)p Fu(\)\))p Fn(dt)1344 4054 y Fu(=)1513 3939 y Fh(Z)1559 4128 y Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1767 4054 y Fn(l)1794 4068 y Fm(")1831 4054 y Fu(\()p Fn(dx)p Fu(\))p Fn(E)2072 4017 y Fm(")2067 4077 y(x)2112 4054 y Fu(\()p Fn(\033)2199 4068 y Ft(0)2239 4054 y Fu(\))15 b Fn(:)448 4299 y Fu(W)-8 b(e)34 b(use)e(a)g(construction)g(similar)e (as)j(that)f(used)g(in)f(Lemma)h(3.7.)48 b(Using)31 b(condition)448 4412 y Fg(L2)p Fu(,)39 b(there)d(are)h Fn(\032)g Fu(and)f Fn(\032)1337 4379 y Fl(0)1395 4412 y Fn(>)g Fu(0)g(with)g Fn(\032)f(<)g(\032)2032 4379 y Fl(0)2056 4412 y Fn(=)p Fu(3)i(suc)m(h)f(that)h(suc)m(h)f(that)h(for)g(all)e Fn(x)h Fk(2)448 4525 y Fn(@)5 b(B)g Fu(\()p Fn(\032)p Fu(\),)29 b(there)e(are)h Fn(T)1177 4539 y Ft(1)1242 4525 y Fn(<)d Fk(1)i Fu(and)g Fn( )1692 4492 y Fm(x)1761 4525 y Fk(2)e(C)5 b Fu(\([0)p Fn(;)15 b(T)2098 4539 y Ft(1)2139 4525 y Fu(]\))28 b(suc)m(h)f(that)h Fn( )2685 4492 y Fm(x)2729 4525 y Fu(\(0\))f(=)e Fn(x)p Fu(,)j Fn( )3134 4492 y Fm(x)3178 4525 y Fu(\()p Fn(T)3266 4539 y Ft(1)3306 4525 y Fu(\))d(=)448 4638 y Fn(x)500 4652 y Ft(0)565 4638 y Fk(2)g([)712 4652 y Fm(i)740 4638 y Fn(K)817 4652 y Fm(i)845 4638 y Fu(,)31 b Fn( )963 4605 y Fm(x)1008 4638 y Fu(\()p Fn(t)p Fu(\))26 b Fk(2)f Fn(B)5 b Fu(\(2)p Fn(\032=)p Fu(3\),)32 b(0)26 b Fk(\024)f Fn(t)h Fk(\024)f Fn(T)1981 4652 y Ft(1)2051 4638 y Fu(and)30 b Fn(I)2268 4652 y Fm(x;T)2369 4661 y Fj(1)2407 4638 y Fu(\()p Fn( )2504 4605 y Fm(x)2548 4638 y Fu(\))c Fk(\024)g Fn(h=)p Fu(4.)42 b(F)-8 b(urthermore,)448 4751 y(using)28 b(Corollary)g(2.4,)j(for)e Fn(\032)1442 4718 y Fl(0)1495 4751 y Fu(small)f(enough,)h(there)h(are)f Fn(z)h Fk(2)25 b Fn(@)5 b(B)g Fu(\()p Fn(\032)2818 4718 y Fl(0)2841 4751 y Fu(\),)30 b Fn(T)2984 4765 y Ft(2)3049 4751 y Fn(<)25 b Fk(1)p Fu(,)30 b(and)448 4863 y Fn( )h Fk(2)c(C)5 b Fu(\([0)p Fn(;)15 b(T)877 4877 y Ft(2)918 4863 y Fu(]\))33 b(suc)m(h)e(that)h Fn( )s Fu(\(0\))d(=)e Fn(x)1771 4877 y Ft(0)1811 4863 y Fu(,)32 b Fn( )s Fu(\()p Fn(T)2018 4877 y Ft(2)2058 4863 y Fu(\))c(=)f Fn(z)t Fu(,)32 b Fn( )s Fu(\()p Fn(t)p Fu(\))d Fk(2)e Fn(B)5 b Fu(\()p Fn(\032)2760 4830 y Fl(0)2783 4863 y Fu(\))32 b(for)f(0)d Fk(\024)f Fn(t)g Fk(\024)g Fn(T)3372 4877 y Ft(2)3412 4863 y Fu(,)448 4976 y(and)41 b Fn(I)676 4990 y Fm(x)716 4999 y Fj(0)750 4990 y Fm(;T)811 4999 y Fj(2)849 4976 y Fu(\()p Fn( )s Fu(\))k Fk(\024)e Fn(h=)p Fu(4.)74 b(W)-8 b(e)42 b(denote)f Fn(\036)1905 4943 y Fm(x)1993 4976 y Fk(2)h(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\),)46 b(with)40 b Fn(T)56 b Fu(=)42 b Fn(T)2984 4990 y Ft(1)3051 4976 y Fu(+)27 b Fn(T)3202 4990 y Ft(2)3242 4976 y Fu(,)44 b(the)1897 5225 y(28)p eop %%Page: 29 29 29 28 bop 448 573 a Fu(path)35 b(obtained)f(b)m(y)h(concatenating)h Fn( )1821 540 y Fm(x)1900 573 y Fu(and)e Fn( )s Fu(.)55 b(It)35 b(satis\014es)f Fn(I)2705 587 y Fm(x;T)2819 573 y Fu(\()p Fn( )s Fu(\))g Fk(\024)f Fn(h=)p Fu(2.)55 b(W)-8 b(e)448 686 y(consider)30 b(the)g(op)s(en)g(set)999 925 y Fn(V)1052 939 y Fm(T)1148 925 y Fu(=)1351 845 y Fh([)1259 1033 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1550 806 y Fh(\032)1612 925 y Fn( )f Fk(2)24 b(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))43 b(:)d Fk(k)p Fn( )25 b Fk(\000)20 b Fn(\036)2491 939 y Fm(x)2535 925 y Fk(k)25 b Fn(<)2711 864 y(\032)p 2711 904 48 4 v 2712 988 a Fu(3)2768 806 y Fh(\033)2861 925 y Fn(:)344 b Fu(\(126\))448 1220 y(Applying)28 b(Theorem)i(2.1,)i(one)f(obtains)1158 1462 y(lim)15 b(inf)1215 1519 y Fm(")p Fl(!)p Ft(0)1425 1462 y Fn(")g Fu(log)100 b(inf)1615 1526 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1906 1462 y Fn(P)1977 1425 y Fm(")1964 1485 y(x)2014 1462 y Fu(\()p Fn(x)2101 1476 y Fm(")2163 1462 y Fk(2)25 b Fn(V)2302 1476 y Fm(T)2357 1462 y Fu(\))41 b Fk(\025)f(\000)2625 1401 y Fn(h)p 2625 1441 53 4 v 2628 1525 a Fu(2)2702 1462 y Fn(:)503 b Fu(\(127\))448 1720 y(There)27 b(is)f Fn(T)862 1687 y Fl(\003)926 1720 y Fn(>)f Fu(0,)j(suc)m(h)f(that)g(for) g(an)m(y)g Fn(\036)f Fk(2)e Fn(V)2037 1734 y Fm(T)2119 1720 y Fu(the)j(time)g(sp)s(en)m(t)f(in)g Fn(B)5 b Fu(\()p Fn(\032)2974 1687 y Fl(0)2997 1720 y Fu(\))27 b(is)f(at)i(least)448 1833 y Fn(T)514 1800 y Fl(\003)554 1833 y Fu(.)40 b(Therefore,)31 b(for)f Fn(")g Fu(small)f(enough)h(w)m(e)h(obtain)f(the)h(b)s(ound)952 1960 y Fh(Z)999 2149 y Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1206 2075 y Fn(l)1233 2089 y Fm(")1270 2075 y Fu(\()p Fn(dx)p Fu(\))p Fn(E)1511 2038 y Fm(")1506 2098 y(x)1551 2075 y Fu(\()p Fn(\033)1638 2089 y Ft(0)1678 2075 y Fu(\))41 b Fk(\025)25 b Fn(T)1916 2038 y Fl(\003)1970 2075 y Fu(exp)15 b(\()p Fk(\000)2258 2014 y Fn(h)p 2240 2054 88 4 v 2240 2138 a Fu(2)p Fn(")2338 2075 y Fu(\))41 b Fk(\025)f Fu(exp)15 b(\()p Fk(\000)2795 2014 y Fn(h)p 2795 2054 53 4 v 2800 2138 a(")2857 2075 y Fu(\))h Fn(;)297 b Fu(\(128\))448 2328 y(and)30 b(this)f(completes)i(the)g(pro)s(of)e(of)i(Lemma)f(3.8.)p 2236 2336 50 67 v 589 2541 a(T)-8 b(o)43 b(get)g(an)f(upp)s(er)f(b)s (ound)f(on)i(the)g(normalization)f(constan)m(t)i Fn(\027)2918 2555 y Fm(")2955 2541 y Fu(\()p Fn(X)7 b Fu(\))43 b(w)m(e)g(will)448 2654 y(need)28 b(an)f(upp)s(er)f(b)s(ound)g(on)h(the)h(escap)s(e)h (time)e(out)h(of)g(the)g(ball)e Fn(B)5 b Fu(\()p Fn(\032)2826 2621 y Fl(0)2849 2654 y Fu(\))28 b(around)f Fk([)3279 2668 y Fm(i)3306 2654 y Fn(K)3383 2668 y Fm(i)3412 2654 y Fu(,)448 2767 y(starting)j(from)g Fn(x)c Fk(2)e Fn(@)5 b(B)g Fu(\()p Fn(\032)p Fu(\).)448 2979 y Fg(Lemma)33 b(3.9)46 b Fe(Given)33 b Fn(h)25 b(>)g Fu(0)p Fe(,)33 b(for)g Fu(0)26 b Fn(<)f(\032)g(<)g(\032)2070 2946 y Fl(0)2126 2979 y Fe(su\016ciently)33 b(smal)5 b(l,)1312 3183 y Fu(lim)15 b(sup)1382 3258 y Fm(")p Fl(!)p Ft(0)1606 3183 y Fn(")g Fu(log)86 b(sup)1796 3265 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))2086 3183 y Fn(E)2158 3146 y Fm(")2153 3206 y(x)2197 3183 y Fu(\()p Fn(\033)2284 3197 y Ft(0)2324 3183 y Fu(\))26 b Fk(\024)f Fn(h)15 b(:)657 b Fu(\(129\))448 3467 y Fe(Pr)-5 b(o)g(of:)50 b Fu(Fix)34 b Fn(h)e(>)f Fu(0)j(arbitrary)-8 b(.)52 b(As)34 b(in)f(Lemma)h(3.8,)j(w)m(e)e(see)f(that,)i(for)e(0)e Fn(<)g(\032)f(<)h(\032)3414 3434 y Fl(0)448 3580 y Fu(su\016cien)m(tly) 26 b(small)f(and)h(for)g(all)g Fn(x)f Fk(2)g Fn(@)5 b(B)g Fu(\()p Fn(\032)p Fu(\),)28 b(there)f(are)g Fn(T)2455 3594 y Ft(1)2494 3580 y Fn(;)15 b(T)2587 3594 y Ft(2)2653 3580 y Fk(\024)25 b(1)p Fu(,)i Fn(z)40 b(=)-55 b Fk(2)25 b Fn(B)5 b Fu(\()p Fn(\032)3206 3547 y Fl(0)3229 3580 y Fu(\))27 b(and)448 3693 y Fn(\036)502 3660 y Fm(x)572 3693 y Fk(2)d(C)5 b Fu(\([0)p Fn(;)15 b(T)908 3707 y Ft(1)954 3693 y Fu(+)t Fn(T)1082 3707 y Ft(2)1121 3693 y Fu(]\))23 b(suc)m(h)f(that)h Fn(\036)1644 3660 y Fm(x)1688 3693 y Fu(\(0\))k(=)e Fn(x)p Fu(,)f Fn(\036)2081 3660 y Fm(x)2125 3693 y Fu(\()p Fn(T)2213 3707 y Ft(1)2253 3693 y Fu(\))h Fk(2)g([)2460 3707 y Fm(i)2488 3693 y Fn(K)2565 3707 y Fm(i)2593 3693 y Fu(,)g(and)c Fn(\036)2865 3660 y Fm(x)2909 3693 y Fu(\()p Fn(T)2997 3707 y Ft(1)3042 3693 y Fu(+)t Fn(T)3170 3707 y Ft(2)3209 3693 y Fu(\))26 b(=)f Fn(z)t Fu(,)448 3806 y(and)35 b Fn(I)670 3820 y Fm(x;T)771 3829 y Fj(1)805 3820 y Ft(+)p Fm(T)901 3829 y Fj(2)939 3806 y Fu(\()p Fn(\036)1028 3773 y Fm(x)1073 3806 y Fu(\))e Fk(\024)g Fn(h=)p Fu(2.)56 b(W)-8 b(e)36 b(set)g Fn(T)1829 3820 y Ft(0)1902 3806 y Fu(=)c Fn(T)2058 3820 y Ft(1)2121 3806 y Fu(+)23 b Fn(T)2268 3820 y Ft(2)2343 3806 y Fu(and)34 b(\001)f(=)g(dist)o(\()p Fn(z)t(;)15 b(B)5 b Fu(\()p Fn(\032)3161 3773 y Fl(0)3185 3806 y Fu(\)\))36 b(and)448 3919 y(consider)30 b(the)g(op)s(en)g(set)943 4160 y Fn(W)1029 4174 y Fm(T)1070 4183 y Fj(0)1149 4160 y Fu(=)1352 4079 y Fh([)1260 4267 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1551 4041 y Fh(\032)1613 4160 y Fn( )f Fk(2)24 b(C)5 b Fu(\([0)p Fn(;)15 b(T)2037 4174 y Ft(0)2079 4160 y Fu(]\))41 b(:)f Fk(k)p Fn( )24 b Fk(\000)c Fn(\036)2518 4174 y Fm(x)2562 4160 y Fk(k)26 b Fn(<)2739 4099 y Fu(\001)p 2739 4139 76 4 v 2754 4222 a(2)2825 4041 y Fh(\033)2917 4160 y Fn(;)288 b Fu(\(130\))448 4462 y(so)38 b(that)h(if)e Fn( )k Fk(2)c Fn(W)1147 4476 y Fm(T)1188 4485 y Fj(0)1264 4462 y Fu(it)h(escap)s(es)g(from)f Fn(B)5 b Fu(\()p Fn(\032)2070 4429 y Fl(0)2093 4462 y Fu(\))39 b(in)d(a)i(time)g(less)f(than)g Fn(T)3025 4476 y Ft(0)3065 4462 y Fu(,)j(i.e.,)g(the)448 4575 y(ev)m(en)m(t)p Fk(f)p Fn(x)753 4589 y Fm(")819 4575 y Fk(2)26 b Fn(W)992 4589 y Fm(T)1033 4598 y Fj(0)1071 4575 y Fk(g)32 b Fu(is)e(con)m (tained)h(in)f(the)h(ev)m(en)m(t)i Fk(f)p Fn(\033)2254 4589 y Ft(0)2320 4575 y Fn(<)27 b(T)2471 4589 y Ft(0)2510 4575 y Fk(g)32 b Fu(.)43 b(Using)30 b(Theorem)h(2.1,)448 4688 y(w)m(e)g(see)g(that)g(for)f(su\016cien)m(tly)f(small)g Fn(")p Fu(,)i(one)g(has)f(the)g(b)s(ound)1135 4930 y Fn(q)43 b Fk(\021)122 b Fu(inf)1330 4993 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1621 4930 y Fn(P)1692 4892 y Fm(")1679 4952 y(x)1728 4930 y Fu(\()p Fn(\033)1815 4944 y Ft(0)1880 4930 y Fn(<)25 b(T)2029 4944 y Ft(0)2069 4930 y Fu(\))41 b Fk(\025)f Fu(exp)2410 4811 y Fh(\022)2471 4930 y Fk(\000)2569 4868 y Fn(h)p 2552 4909 88 4 v 2552 4992 a Fu(2)p Fn(")2649 4811 y Fh(\023)2726 4930 y Fn(:)479 b Fu(\(131\))1897 5225 y(29)p eop %%Page: 30 30 30 29 bop 448 573 a Fu(Consider)29 b(the)h(ev)m(en)m(ts)i Fn(\033)1312 587 y Ft(0)1377 573 y Fn(>)25 b(k)s(T)1576 587 y Ft(0)1616 573 y Fu(,)30 b Fn(k)e Fu(=)d(1)p Fn(;)15 b Fu(2)p Fn(;)g(:)g(:)g(:)s Fu(.)41 b(Using)29 b(the)i(Mark)m(o)m(v)h (prop)s(ert)m(y)e(one)448 686 y(obtains)g(the)h(b)s(ound)538 852 y Fn(P)609 815 y Fm(")596 875 y(x)646 852 y Fu(\()p Fn(\033)733 866 y Ft(0)798 852 y Fn(>)25 b Fu(\()p Fn(k)f Fu(+)19 b(1\))p Fn(T)1223 866 y Ft(0)1264 852 y Fu(\))98 b Fk(\024)g Fu([)q(1)20 b Fk(\000)g Fn(P)1819 815 y Fm(")1806 875 y(x)1856 852 y Fu(\()p Fn(k)s(T)1994 866 y Ft(0)2059 852 y Fn(<)25 b(\033)2207 866 y Ft(0)2272 852 y Fk(\024)g Fu(\()p Fn(k)e Fu(+)d(1\))p Fn(T)2697 866 y Ft(0)2738 852 y Fu(\)])15 b Fn(P)2884 815 y Fm(")2871 875 y(x)2921 852 y Fu(\()p Fn(\033)3008 866 y Ft(0)3073 852 y Fn(>)25 b(k)s(T)3272 866 y Ft(0)3312 852 y Fu(\))1397 990 y Fk(\024)98 b Fu(\(1)21 b Fk(\000)f Fn(q)s Fu(\))p Fn(P)1908 953 y Fm(")1895 1013 y(x)1945 990 y Fu(\()p Fn(\033)2032 1004 y Ft(0)2097 990 y Fn(>)25 b(k)s(T)2296 1004 y Ft(0)2336 990 y Fu(\))p Fn(:)834 b Fu(\(132\))448 1157 y(Iterating)31 b(o)m(v)m(er)h Fn(k)h Fu(yields)1347 1323 y(sup)1278 1404 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1568 1323 y Fn(P)1639 1286 y Fm(")1626 1346 y(x)1676 1323 y Fu(\()p Fn(\033)1763 1337 y Ft(0)1828 1323 y Fn(>)25 b(k)s(T)2027 1337 y Ft(0)2067 1323 y Fu(\))40 b Fk(\024)h Fu(\(1)21 b Fk(\000)e Fn(q)s Fu(\))2524 1286 y Fm(k)2582 1323 y Fn(:)623 b Fu(\(133\))448 1554 y(Therefore)924 1764 y(sup)855 1845 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1146 1764 y Fn(E)1218 1726 y Fm(")1213 1786 y(x)1257 1764 y Fu(\()p Fn(\033)1344 1778 y Ft(0)1383 1764 y Fu(\))99 b Fk(\024)f Fn(T)1739 1778 y Ft(0)1794 1620 y Fh(")1842 1764 y Fu(1)21 b(+)2028 1658 y Fl(1)2003 1683 y Fh(X)1999 1867 y Fm(k)r Ft(=1)2212 1764 y Fu(sup)2142 1845 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))2433 1764 y Fn(P)2504 1726 y Fm(")2491 1786 y(x)2541 1764 y Fu(\()p Fn(\033)2628 1778 y Ft(0)2693 1764 y Fn(>)k(k)s(T)2892 1778 y Ft(0)2931 1764 y Fu(\))2966 1620 y Fh(#)1517 2051 y Fk(\024)98 b Fn(T)1739 2065 y Ft(0)1823 1946 y Fl(1)1798 1971 y Fh(X)1794 2155 y Fm(k)r Ft(=0)1922 2051 y Fu(\(1)21 b Fk(\000)f Fn(q)s Fu(\))2193 2014 y Fm(k)2276 2051 y Fu(=)2397 1990 y Fn(T)2450 2004 y Ft(0)p 2397 2030 93 4 v 2422 2114 a Fn(q)2515 2051 y(:)690 b Fu(\(134\))448 2307 y(Since)30 b Fn(q)e Fk(\025)d Fu(exp)15 b(\()p Fk(\000)1134 2271 y Fm(h)p 1121 2286 68 4 v 1121 2339 a Ft(2)p Fm(")1199 2307 y Fu(\))30 b(one)h(obtains,)f(for)g(su\016cien)m(tly)f(small)g Fn(")1138 2525 y Fu(sup)1069 2606 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1360 2525 y Fn(E)1432 2488 y Fm(")1427 2548 y(x)1471 2525 y Fu(\()p Fn(\033)1558 2539 y Ft(0)1597 2525 y Fu(\))41 b Fk(\024)f Fn(T)1837 2539 y Ft(0)1892 2525 y Fu(exp)2046 2406 y Fh(\022)2135 2464 y Fn(h)p 2117 2504 88 4 v 2117 2587 a Fu(2)p Fn(")2215 2406 y Fh(\023)2316 2525 y Fk(\024)g Fu(exp)2581 2406 y Fh(\022)2653 2464 y Fn(h)p 2653 2504 53 4 v 2658 2587 a(")2715 2406 y Fh(\023)2791 2525 y Fn(:)414 b Fu(\(135\))448 2756 y(This)29 b(concludes)g(the)i(pro)s(of)f(of)g(Lemma)h(3.9.)p 2119 2764 50 67 v 589 2944 a(With)h(this)f(Lemma)i(w)m(e)f(ha)m(v)m(e)i (pro)m(v)m(ed)e(all)f(large)h(deviations)g(estimates)g(needed)448 3057 y(in)37 b(the)g(pro)s(of)g(of)h(Theorem)f(3.3.)64 b(W)-8 b(e)38 b(will)d(need)j(upp)s(er)d(and)i(lo)m(w)m(er)h(estimates) g(on)448 3170 y Fn(l)475 3184 y Fm(")512 3170 y Fu(\()p Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\)\))39 b(where)e Fn(l)1233 3184 y Fm(")1307 3170 y Fu(is)f(the)h(in)m(v)-5 b(arian)m(t)37 b(measure)g(of)g(the)h(Mark)m(o)m(v)g(c)m(hain)f Fn(x)3228 3184 y Fm(")3265 3170 y Fu(\()p Fn(\034)3340 3184 y Fm(j)3377 3170 y Fu(\).)448 3282 y(These)26 b(estimates)h(are)f (pro)m(v)m(ed)g(in)f([8)q(],)i(Chapter)f(6,)h(Section)f(3)g(and)g(4)g (and)f(are)i(purely)448 3395 y(com)m(binatorial)33 b(and)g(rely)g(on)h (the)g(represen)m(tation)f(of)h(the)g(in)m(v)-5 b(arian)m(t)33 b(measure)h(of)g(a)448 3508 y(Mark)m(o)m(v)39 b(c)m(hain)d(with)g(a)h (\014nite)f(state)i(space)g(via)e(graphs)h(on)f(the)i(state)g(space.)61 b(By)448 3621 y(Lemma)35 b(3.6,)j(\(ii\))c(and)g(3.7,)k(\(ii\))c(w)m(e) h(ha)m(v)m(e)i(the)e(follo)m(wing)e(estimates)j(on)f(the)g(prob-)448 3734 y(abilit)m(y)h(transition)f Fn(q)s Fu(\()p Fn(x;)15 b(y)s Fu(\),)40 b Fn(x;)15 b(y)39 b Fk(2)d Fn(@)5 b(B)g Fu(\()p Fn(\032)p Fu(\))37 b(of)g(the)h(Mark)m(o)m(v)g(c)m(hain)f Fn(x)2942 3748 y Fm(")2978 3734 y Fu(\()p Fn(\034)3053 3748 y Fm(j)3090 3734 y Fu(\):)54 b(Giv)m(en)448 3847 y Fn(h)26 b(>)f Fu(0,)31 b(for)f(0)c Fn(<)f(\032)g(<)g(\032)1244 3814 y Fl(0)1298 3847 y Fu(su\016cien)m(tly)k(small)578 4047 y(exp)16 b Fk(\000)814 3985 y Fu(1)p 814 4026 46 4 v 816 4109 a Fn(")869 4047 y Fu(\()918 4024 y(~)904 4047 y Fn(V)k Fu(\()p Fn(K)1089 4061 y Fm(i)1118 4047 y Fn(;)15 b(K)1235 4061 y Fm(j)1272 4047 y Fu(\))21 b(+)f Fn(h)p Fu(\))41 b Fk(\024)f Fn(q)s Fu(\()p Fn(x;)15 b(@)5 b(B)g Fu(\()p Fn(j;)15 b(\032)p Fu(\)\))43 b Fk(\024)d Fu(exp)15 b Fk(\000)2574 3985 y Fu(1)p 2574 4026 V 2576 4109 a Fn(")2629 4047 y Fu(\()2678 4024 y(~)2664 4047 y Fn(V)20 b Fu(\()p Fn(K)2849 4061 y Fm(i)2878 4047 y Fn(;)15 b(K)2995 4061 y Fm(j)3032 4047 y Fu(\))21 b Fk(\000)f Fn(h)p Fu(\))c Fn(;)3230 4186 y Fu(\(136\))448 4298 y(for)35 b(all)f Fn(x)f Fk(2)g Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\))36 b(and)f(su\016cien)m(tly)f(small)f Fn(")p Fu(.)56 b(It)35 b(is)f(sho)m(wn)h(in)e([8)q(],)k(Chapter)d(6,)448 4411 y(Lemmas)e(3.1)i(and)d(3.2)i(that)g(the)f(b)s(ound)e(\(136\))k (implies)c(a)i(b)s(ound)e(on)i Fn(l)2988 4425 y Fm(")3025 4411 y Fu(\()p Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\)\).)448 4524 y(One)30 b(obtains)908 4709 y(exp)1062 4589 y Fh(\022)1123 4709 y Fk(\000)1204 4647 y Fu(1)p 1204 4687 V 1206 4771 a Fn(")1259 4709 y Fu(\()1321 4686 y(~)1294 4709 y Fn(W)13 b Fu(\()p Fn(K)1505 4723 y Fm(i)1534 4709 y Fu(\))20 b Fk(\000)g Fu(min)1740 4766 y Fm(j)1873 4686 y Fu(~)1847 4709 y Fn(W)12 b Fu(\()p Fn(K)2057 4723 y Fm(j)2094 4709 y Fu(\))21 b(+)f Fn(h)p Fu(\))2328 4589 y Fh(\023)2430 4709 y Fk(\024)40 b Fn(l)2568 4723 y Fm(")2605 4709 y Fu(\()p Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\)\))42 b Fk(\024)1395 4943 y(\024)f Fu(exp)1661 4824 y Fh(\022)1722 4943 y Fk(\000)1803 4882 y Fu(1)p 1803 4922 V 1805 5005 a Fn(")1858 4943 y Fu(\()1920 4920 y(~)1893 4943 y Fn(W)13 b Fu(\()p Fn(K)2104 4957 y Fm(i)2132 4943 y Fu(\))21 b Fk(\000)f Fu(min)2338 5001 y Fm(j)2472 4920 y Fu(~)2446 4943 y Fn(W)12 b Fu(\()p Fn(K)2656 4957 y Fm(j)2693 4943 y Fu(\))21 b Fk(\000)f Fn(h)p Fu(\))2927 4824 y Fh(\023)3230 4943 y Fu(\(137\))1897 5225 y(30)p eop %%Page: 31 31 31 30 bop 448 573 a Fu(for)30 b(su\016cien)m(tly)g(small)e Fn(")p Fu(,)j(where)1222 754 y(~)1195 777 y Fn(W)12 b Fu(\()p Fn(K)1405 791 y Fm(i)1434 777 y Fu(\))41 b(=)81 b(min)1621 841 y Fm(g)r Fl(2)p Fm(G)p Fl(f)p Fm(i)p Fl(g)1966 696 y Fh(X)1869 884 y Ft(\()p Fm(m)p Fl(!)p Fm(n)p Ft(\))p Fl(2)p Fm(g)2212 754 y Fu(~)2198 777 y Fn(V)20 b Fu(\()p Fn(K)2383 791 y Fm(m)2450 777 y Fn(;)15 b(K)2567 791 y Fm(n)2615 777 y Fu(\))g Fn(:)540 b Fu(\(138\))448 1093 y(Also)34 b(in)f([8)q(],)j(Chapter)e(6,)h(Lemma)g(4.1,)1919 1070 y(~)1893 1093 y Fn(W)12 b Fu(\()p Fn(K)2103 1107 y Fm(i)2132 1093 y Fu(\))34 b(is)g(sho)m(wn)f(to)j(b)s(e)d(in)g(fact)i (equal)f(to)448 1206 y Fn(W)13 b Fu(\()p Fn(K)659 1220 y Fm(i)688 1206 y Fu(\))30 b(de\014ned)f(in)g(Eq.)i(\(93\):)1184 1387 y(~)1157 1410 y Fn(W)13 b Fu(\()p Fn(K)1368 1424 y Fm(i)1397 1410 y Fu(\))98 b(=)139 b(min)1699 1473 y Fm(g)r Fl(2)p Fm(G)p Fl(f)p Fm(i)p Fl(g)2044 1329 y Fh(X)1947 1517 y Ft(\()p Fm(m)p Fl(!)p Fm(n)p Ft(\))p Fl(2)p Fm(g)2276 1410 y Fn(V)20 b Fu(\()p Fn(K)2461 1424 y Fm(m)2528 1410 y Fn(;)15 b(K)2645 1424 y Fm(n)2693 1410 y Fu(\))1530 1643 y(=)98 b Fn(W)13 b Fu(\()p Fn(K)1910 1657 y Fm(i)1938 1643 y Fu(\))i Fn(:)1217 b Fu(\(139\))448 1848 y(F)-8 b(urthermore)33 b(it)g(is)f(sho)m(wn)h(Lemma)g(4.2)h(there)f(that)h (the)f(function)f Fn(W)13 b Fu(\()p Fn(x)p Fu(\),)34 b(de\014ned)448 1961 y(b)m(y)d(Eq.)f(\(94\),)i(satis\014es)e(the)h (iden)m(tit)m(y)943 2165 y Fn(W)13 b Fu(\()p Fn(x)p Fu(\))99 b(=)f(min)1495 2223 y Fm(i)1583 2165 y Fu(\()p Fn(W)13 b Fu(\()p Fn(K)1829 2179 y Fm(i)1858 2165 y Fu(\))20 b(+)g Fn(V)g Fu(\()p Fn(K)2189 2179 y Fm(i)2218 2165 y Fn(;)15 b(x)p Fu(\)\))21 b Fk(\000)f Fu(min)2552 2223 y Fm(j)2659 2165 y Fn(W)13 b Fu(\()p Fn(K)2870 2179 y Fm(j)2907 2165 y Fu(\))1263 2361 y(=)98 b(min)1495 2418 y Fm(i)1583 2361 y Fu(\()1645 2338 y(~)1618 2361 y Fn(W)13 b Fu(\()p Fn(K)1829 2375 y Fm(i)1858 2361 y Fu(\))20 b(+)2018 2338 y(~)2004 2361 y Fn(V)g Fu(\()p Fn(K)2189 2375 y Fm(i)2218 2361 y Fn(;)15 b(x)p Fu(\)\))21 b Fk(\000)f Fu(min)2552 2418 y Fm(j)2686 2338 y Fu(~)2659 2361 y Fn(W)13 b Fu(\()p Fn(K)2870 2375 y Fm(j)2907 2361 y Fu(\))288 b(\(140\))589 2602 y(W)-8 b(e)32 b(can)f(turn)e(to)i(the)g(pro)s(of)e (of)i(Theorem)f(3.3.)448 2715 y Fe(Pr)-5 b(o)g(of)34 b(of)f(The)-5 b(or)g(em)35 b(3.3:)589 2828 y Fu(In)j(order)h(to)g(pro)m (v)m(e)g(Eq.)g(\(95\),)k(it)38 b(is)g(enough)g(to)i(sho)m(w)e(that,)k (for)c(an)m(y)h Fn(h)h(>)e Fu(0,)448 2941 y(there)31 b(is)e Fn(")814 2955 y Ft(0)879 2941 y Fn(>)c Fu(0)31 b(suc)m(h)f(that,)h(for)g Fn(")25 b(<)g(")1823 2955 y Ft(0)1893 2941 y Fu(w)m(e)31 b(ha)m(v)m(e)h(the)e(inequalities:)1152 3181 y Fn(\026)1207 3195 y Fm(")1243 3181 y Fu(\()p Fn(D)s Fu(\))99 b Fk(\025)f Fu(exp)1813 3062 y Fh(\022)1874 3181 y Fk(\000)1955 3119 y Fu(1)p 1955 3160 46 4 v 1957 3243 a Fn(")2010 3181 y Fu(\()16 b(inf)2045 3240 y Fm(z)s Fl(2)p Fm(D)2203 3181 y Fn(W)d Fu(\()p Fn(z)t Fu(\))21 b(+)e Fn(h)p Fu(\))2616 3062 y Fh(\023)2693 3181 y Fn(;)512 b Fu(\(141\))1152 3413 y Fn(\026)1207 3427 y Fm(")1243 3413 y Fu(\()p Fn(D)s Fu(\))99 b Fk(\024)f Fu(exp)1813 3294 y Fh(\022)1874 3413 y Fk(\000)1955 3352 y Fu(1)p 1955 3392 V 1957 3476 a Fn(")2010 3413 y Fu(\()16 b(inf)2045 3472 y Fm(z)s Fl(2)p Fm(D)2203 3413 y Fn(W)d Fu(\()p Fn(z)t Fu(\))21 b Fk(\000)e Fn(h)p Fu(\))2616 3294 y Fh(\023)2693 3413 y Fn(:)512 b Fu(\(142\))589 3660 y(W)-8 b(e)40 b(let)e Fn(\032)940 3627 y Fl(0)1002 3660 y Fn(>)g Fu(0)h(suc)m(h)f(that)h Fn(\032)1660 3627 y Fl(0)1722 3660 y Fn(<)f Fu(dist)o(\()p Fn(x)2065 3674 y Ft(min)2187 3660 y Fn(;)15 b(D)s Fu(\).)66 b(Recall)38 b(that)h Fn(\034)2958 3674 y Fm(D)3060 3660 y Fu(=)f(inf)6 b Fk(f)p Fn(t)54 b Fu(:)448 3773 y Fn(x)500 3787 y Fm(")537 3773 y Fu(\()p Fn(t)p Fu(\))43 b Fk(2)g Fn(D)s Fk(g)e Fu(is)f(the)h(\014rst)f(hitting) f(time)i(of)g(the)g(set)g Fn(D)s Fu(.)72 b(w)m(e)41 b(ha)m(v)m(e)i(the) e(follo)m(wing)448 3885 y(b)s(ound)29 b(on)h(the)g Fn(\027)1056 3899 y Fm(")1093 3885 y Fu(\()p Fn(D)s Fu(\))665 4125 y Fn(\027)710 4139 y Fm(")746 4125 y Fu(\()p Fn(D)s Fu(\))99 b Fk(\024)1162 4044 y Fh(X)1210 4227 y Fm(i)1297 4125 y Fn(l)1324 4139 y Fm(")1361 4125 y Fu(\()p Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\)\))108 b(sup)1761 4206 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)p Ft(\))2097 4125 y Fn(E)2169 4088 y Fm(")2164 4148 y(x)2223 4010 y Fh(Z)2306 4036 y Fm(\034)2337 4045 y Fj(1)2269 4199 y Ft(0)2391 4125 y Fg(1)2443 4139 y Fm(D)2508 4125 y Fu(\()p Fn(x)2595 4139 y Fm(")2632 4125 y Fu(\()p Fn(t)p Fu(\)\))p Fn(dt)993 4336 y Fk(\024)98 b Fn(L)15 b Fu(max)1311 4394 y Fm(i)1423 4336 y Fn(l)1450 4350 y Fm(")1487 4336 y Fu(\()p Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\)\))108 b(sup)1887 4418 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)p Ft(\))2223 4336 y Fn(P)2294 4299 y Fm(")2281 4359 y(x)2331 4336 y Fu(\()p Fn(\034)2406 4350 y Fm(D)2495 4336 y Fk(\024)25 b Fn(\034)2631 4350 y Ft(1)2671 4336 y Fu(\))39 b(sup)2721 4414 y Fm(y)r Fl(2)p Fm(@)t(D)2921 4336 y Fn(E)2993 4299 y Fm(")2988 4359 y(y)3031 4336 y Fu(\()p Fn(\034)3106 4350 y Ft(1)3145 4336 y Fu(\))15 b Fn(:)10 b Fu(\(143\))448 4605 y(By)31 b Fg(L1)p Fu(,)g(there)f(exist)g(a)h(constan)m(t)h Fn(C)37 b Fu(indep)s(enden)m(t)28 b(of)i Fn(")h Fu(suc)m(h)f(that)1606 4809 y(sup)1581 4887 y Fm(y)r Fl(2)p Fm(@)t(D)1782 4809 y Fn(E)1854 4772 y Fm(")1849 4832 y(y)1891 4809 y Fu(\()p Fn(\034)1966 4823 y Ft(1)2005 4809 y Fu(\))41 b Fk(\024)f Fn(C)22 b(;)926 b Fu(\(144\))1897 5225 y(31)p eop %%Page: 32 32 32 31 bop 448 573 a Fu(for)30 b Fn(")c Fk(\024)f Fn(")793 587 y Ft(0)833 573 y Fu(.)40 b(F)-8 b(rom)31 b(Lemma)f(3.6,)i(\(i\),)e (giv)m(en)g Fn(h)c(>)f Fu(0,)31 b(for)f(su\016cien)m(tly)f(small)f(0)e Fn(<)f(\032)g(<)448 686 y(\032)495 653 y Fl(0)519 686 y Fu(,)30 b(w)m(e)h(ha)m(v)m(e)h(the)e(b)s(ound)990 925 y Fn(P)1061 888 y Fm(")1048 948 y(x)1098 925 y Fu(\()p Fn(\034)1173 939 y Fm(D)1262 925 y Fn(<)25 b(\034)1398 939 y Ft(1)1437 925 y Fu(\))41 b Fk(\024)f Fu(exp)1778 806 y Fh(\022)1839 925 y Fk(\000)1920 864 y Fu(1)p 1920 904 46 4 v 1922 988 a Fn(")1975 925 y Fu(\()17 b(inf)2010 985 y Fm(z)s Fl(2)p Fm(D)2182 902 y Fu(~)2168 925 y Fn(V)j Fu(\()p Fn(K)2353 939 y Fm(i)2382 925 y Fn(;)15 b(z)t Fu(\))21 b Fk(\000)f Fn(h=)p Fu(4\))2792 806 y Fh(\023)2870 925 y Fn(;)335 b Fu(\(145\))448 1170 y(for)35 b(su\016cien)m(tly)e (small)g Fn(")p Fu(.)53 b(F)-8 b(rom)35 b(Eq.)f(\(137\),)k(giv)m(en)c Fn(h)f(>)e Fu(0,)36 b(for)e(su\016cien)m(tly)g(small)448 1283 y(0)26 b Fn(<)f(\032)g(<)g(\032)830 1250 y Fl(0)853 1283 y Fu(,)31 b(w)m(e)g(ha)m(v)m(e)g(the)g(b)s(ound)752 1523 y Fn(l)779 1537 y Fm(")816 1523 y Fu(\()p Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\)\))42 b Fk(\024)e Fu(exp)1508 1404 y Fh(\022)1569 1523 y Fk(\000)1650 1461 y Fu(1)p 1650 1502 V 1652 1585 a Fn(")1705 1523 y Fu(\()1767 1500 y(~)1740 1523 y Fn(W)13 b Fu(\()p Fn(K)1951 1537 y Fm(i)1980 1523 y Fu(\))20 b Fk(\000)g Fu(min)2186 1580 y Fm(j)2319 1500 y Fu(~)2293 1523 y Fn(W)12 b Fu(\()p Fn(K)2503 1537 y Fm(j)2541 1523 y Fu(\))20 b Fk(\000)g Fn(h=)p Fu(4\))2864 1404 y Fh(\023)3230 1523 y Fu(\(146\))448 1769 y(F)-8 b(rom)28 b(the)f(estimates)h(\(143\)-\(146\),)k(and)26 b(the)i(iden)m(tit)m(y)e(\(140\))j(w)m(e)e(obtain)g(the)g(b)s(ound)1172 2014 y Fn(\027)1217 2028 y Fm(")1254 2014 y Fu(\()p Fn(D)s Fu(\))40 b Fk(\024)h Fu(exp)1708 1895 y Fh(\022)1769 2014 y Fk(\000)1850 1953 y Fu(1)p 1850 1993 V 1852 2077 a Fn(")1905 2014 y Fu(\(min)1944 2073 y Fm(z)s Fl(2)p Fm(D)2107 2014 y Fn(W)13 b Fu(\()p Fn(z)t Fu(\))20 b(+)g Fn(h=)p Fu(2\))2610 1895 y Fh(\023)2688 2014 y Fn(;)517 b Fu(\(147\))448 2254 y(for)30 b(su\016cien)m(tly)f(small)g Fn(")p Fu(.)41 b(F)-8 b(rom)31 b(Lemma)g(3.8,)g(giv)m(en)g Fn(h)25 b(>)g Fu(0,)31 b(for)g(su\016cien)m(tly)e(small)448 2367 y(0)d Fn(<)f(\032)g(<)g(\032)830 2334 y Fl(0)853 2367 y Fu(,)31 b(w)m(e)g(ha)m(v)m(e)g(the)g(b)s(ound)1502 2609 y Fn(\027)1547 2623 y Fm(")1584 2609 y Fu(\()p Fn(X)7 b Fu(\))41 b Fk(\025)f Fu(exp)2042 2490 y Fh(\022)2103 2609 y Fk(\000)2202 2547 y Fn(h)p 2184 2588 88 4 v 2184 2671 a Fu(2)p Fn(")2282 2490 y Fh(\023)2358 2609 y Fn(;)847 b Fu(\(148\))448 2853 y(for)41 b(su\016cien)m(tly)e(small)g Fn(")p Fu(.)71 b(Com)m(bining)38 b(estimates)j(\(147\))i(and)c (\(148\),)46 b(w)m(e)41 b(obtain)448 2966 y(that)1217 3097 y Fn(\026)1272 3111 y Fm(")1308 3097 y Fu(\()p Fn(D)s Fu(\))g Fk(\024)f Fu(exp)1762 2978 y Fh(\022)1824 3097 y Fk(\000)1905 3035 y Fu(1)p 1905 3076 46 4 v 1907 3159 a Fn(")1960 3097 y Fu(\()16 b(inf)1995 3156 y Fm(z)s Fl(2)p Fm(D)2153 3097 y Fn(W)d Fu(\()p Fn(z)t Fu(\))20 b Fk(\000)g Fn(h)p Fu(\))2566 2978 y Fh(\023)2643 3097 y Fn(;)562 b Fu(\(149\))448 3304 y(for)30 b(su\016cien)m(tly)g(small)e Fn(")j Fu(and)f(this)f(giv)m(es)i(the)f(b)s(ound)f(\(142\).)589 3417 y(In)22 b(order)f(to)i(pro)m(v)m(e)g(\(141\),)j(w)m(e)c(consider)f (the)i(set)f Fn(D)2348 3432 y Fm(\016)2412 3417 y Fu(=)j Fk(f)p Fn(x)g Fk(2)g Fn(D)43 b Fu(:)e(dist)o(\()p Fn(x;)15 b(@)5 b(D)s Fu(\))26 b Fk(\025)448 3530 y Fn(\016)s Fk(g)p Fu(.)42 b(F)-8 b(or)32 b Fn(\016)i Fu(su\016cien)m(tly)29 b(small,)g Fn(D)1634 3545 y Fm(\016)1698 3530 y Fk(6)p Fu(=)c Fk(;)p Fu(.)41 b(By)31 b Fg(L3)p Fu(,)2232 3507 y(~)2218 3530 y Fn(V)21 b Fu(\()p Fn(K)2404 3544 y Fm(i)2432 3530 y Fn(;)15 b(z)t Fu(\))32 b(is)d(upp)s(er)g(semicon)m(tin)m(u-)448 3643 y(ous)h(in)g Fn(z)k Fu(so)d(that)1116 3620 y(~)1102 3643 y Fn(V)20 b Fu(\()p Fn(K)1287 3657 y Fm(i)1316 3643 y Fn(;)15 b(z)1402 3610 y Fl(0)1426 3643 y Fu(\))25 b Fk(\024)1596 3620 y Fu(~)1582 3643 y Fn(V)20 b Fu(\()p Fn(K)1767 3657 y Fm(i)1796 3643 y Fn(;)15 b(z)t Fu(\))21 b(+)f Fn(h=)p Fu(4,)32 b(for)e Fn(z)2413 3610 y Fl(0)2457 3643 y Fk(\000)20 b Fn(z)29 b Fk(\024)c Fn(\016)s Fu(.)42 b(Therefore)1257 3847 y(inf)1225 3906 y Fm(z)s Fl(2)p Fm(D)1366 3918 y Ff(\016)1429 3824 y Fu(~)1415 3847 y Fn(V)20 b Fu(\()p Fn(K)1600 3861 y Fm(i)1629 3847 y Fn(;)15 b(z)t Fu(\))26 b Fk(\024)41 b Fu(inf)1872 3906 y Fm(z)s Fl(2)p Fm(D)2044 3824 y Fu(~)2030 3847 y Fn(V)20 b Fu(\()p Fn(K)2215 3861 y Fm(i)2244 3847 y Fn(;)15 b(z)t Fu(\))21 b(+)f Fn(h=)p Fu(4)15 b Fn(:)571 b Fu(\(150\))448 4089 y(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(b)s(ound)480 4306 y Fn(\027)525 4320 y Fm(")562 4306 y Fu(\()p Fn(D)s Fu(\))41 b Fk(\025)f Fu(max)934 4364 y Fm(i)1046 4306 y Fn(l)1073 4320 y Fm(")1110 4306 y Fu(\()p Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\)\))121 b(inf)1510 4370 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)p Ft(\))1846 4306 y Fn(P)1917 4269 y Fm(")1904 4329 y(x)1954 4306 y Fu(\()p Fn(\034)2029 4320 y Fm(D)2087 4332 y Ff(\016)2151 4306 y Fn(<)24 b(\034)2286 4320 y Ft(1)2326 4306 y Fu(\))70 b(inf)2376 4366 y Fm(x)p Fl(2)p Fm(@)t(D)2562 4378 y Ff(\016)2611 4306 y Fn(E)2683 4269 y Fm(")2678 4329 y(x)2737 4191 y Fh(Z)2820 4217 y Fm(\034)2851 4226 y Fj(1)2783 4380 y Ft(0)2906 4306 y Fg(1)2958 4320 y Fm(D)3022 4306 y Fu(\()p Fn(x)3109 4320 y Fm(")3146 4306 y Fu(\()p Fn(t)p Fu(\)\))p Fn(dt)15 b(:)3230 4471 y Fu(\(151\))448 4584 y(There)30 b(is)g Fn(")845 4598 y Ft(0)910 4584 y Fn(>)25 b Fu(0)30 b(and)g(a)h(constan)m(t)p 1701 4512 72 4 v 32 w Fn(C)g(>)25 b Fu(0)31 b(suc)m(h)f(that)h(w)m(e)g(ha)m(v)m(e)h(the) e(b)s(ound)1273 4815 y(inf)1239 4874 y Fm(x)p Fl(2)p Fm(D)1384 4886 y Ff(\016)1433 4815 y Fn(E)1505 4777 y Fm(")1500 4837 y(x)1559 4700 y Fh(Z)1642 4726 y Fm(\034)1673 4735 y Fj(1)1605 4888 y Ft(0)1727 4815 y Fg(1)1779 4829 y Fm(D)1843 4815 y Fu(\()p Fn(x)1930 4829 y Fm(")1967 4815 y Fu(\()p Fn(t)p Fu(\)\))p Fn(dt)42 b Fk(\025)p 2338 4743 V 40 w Fn(C)47 b(>)40 b Fu(0)15 b Fn(;)584 b Fu(\(152\))1897 5225 y(32)p eop %%Page: 33 33 33 32 bop 448 573 a Fu(uniformly)37 b(in)h Fn(")j Fk(\024)g Fn(")1224 587 y Ft(0)1263 573 y Fu(.)69 b(F)-8 b(rom)40 b(Eq.)f(\(137\),)45 b(giv)m(en)39 b Fn(h)i(>)f Fu(0,)j(for)c (su\016cien)m(tly)g(small)448 686 y(0)26 b Fn(<)f(\032)g(<)g(\032)830 653 y Fl(0)853 686 y Fu(,)31 b(w)m(e)g(ha)m(v)m(e)g(the)g(b)s(ound)731 925 y Fn(l)758 939 y Fm(")795 925 y Fu(\()p Fn(@)5 b(B)g Fu(\()p Fn(i;)15 b(\032)p Fu(\)\))43 b Fk(\025)d Fu(exp)1488 806 y Fh(\022)1549 925 y Fk(\000)1630 864 y Fu(1)p 1630 904 46 4 v 1632 988 a Fn(")1685 925 y Fu(\()1747 902 y(~)1720 925 y Fn(W)13 b Fu(\()p Fn(K)1931 939 y Fm(i)1960 925 y Fu(\))20 b Fk(\000)g Fu(min)2166 983 y Fm(j)2299 902 y Fu(~)2273 925 y Fn(W)12 b Fu(\()p Fn(K)2483 939 y Fm(j)2520 925 y Fu(\))21 b(+)f Fn(h=)p Fu(4\))2844 806 y Fh(\023)2922 925 y Fn(;)283 b Fu(\(153\))448 1172 y(for)36 b(su\016cien)m(tly)f(small)f Fn(")p Fu(.)58 b(F)-8 b(urthermore,)37 b(b)m(y)f(Lemma)g(3.7)h(and)f(inequalit)m(y)e (\(150\),)448 1285 y(giv)m(en)d Fn(h)25 b(>)g Fu(0,)31 b(for)f(0)c Fn(<)f(\032)g(<)g(\032)1481 1252 y Fl(0)1535 1285 y Fu(su\016cien)m(tly)k(small,)g(w)m(e)i(ha)m(v)m(e)807 1525 y(inf)703 1588 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(i;\032)p Ft(\))1037 1525 y Fn(P)1108 1487 y Fm(")1095 1547 y(x)1145 1525 y Fu(\()p Fn(\034)1220 1539 y Fm(D)1278 1551 y Ff(\016)1342 1525 y Fk(\024)25 b Fn(\034)1478 1539 y Ft(1)1517 1525 y Fu(\))40 b Fk(\025)h Fu(exp)1858 1406 y Fh(\022)1919 1525 y Fk(\000)2000 1463 y Fu(1)p 2000 1504 V 2002 1587 a Fn(")2055 1525 y Fu(\()16 b(inf)2090 1584 y Fm(z)s Fl(2)p Fm(D)2262 1502 y Fu(~)2248 1525 y Fn(V)k Fu(\()p Fn(K)2433 1539 y Fm(i)2462 1525 y Fn(;)15 b(z)t Fu(\))21 b(+)f Fn(h=)p Fu(4\))2872 1406 y Fh(\023)2950 1525 y Fn(;)255 b Fu(\(154\))448 1781 y(for)28 b(su\016cien)m(tly)e (small)g Fn(")p Fu(.)40 b(Com)m(bining)26 b(estimates)i (\(151\){\(154\))k(and)27 b(iden)m(tit)m(y)g(\(140\))448 1894 y(w)m(e)k(\014nd)1176 2025 y Fn(\027)1221 2039 y Fm(")1258 2025 y Fu(\()p Fn(D)s Fu(\))41 b Fk(\025)f Fu(exp)1712 1906 y Fh(\022)1773 2025 y Fk(\000)1854 1963 y Fu(1)p 1854 2004 V 1856 2087 a Fn(")1909 2025 y Fu(\()17 b(inf)1944 2084 y Fm(z)s Fl(2)p Fm(D)2102 2025 y Fn(W)c Fu(\()p Fn(z)t Fu(\))21 b(+)f Fn(h=)p Fu(2\))2606 1906 y Fh(\023)2684 2025 y Fn(:)521 b Fu(\(155\))448 2232 y(In)34 b(order)g(to)i(giv)m(e)f(an)g(upp)s(er)d(b)s(ound)h(on)i(the)f (normalization)g(constan)m(t)i Fn(\027)3083 2246 y Fm(")3119 2232 y Fu(\()p Fn(X)7 b Fu(\),)38 b(w)m(e)448 2345 y(use)30 b(Eq.)h(\(102\).)43 b(Using)29 b(the)i(Mark)m(o)m(v)h(prop)s(ert)m(y)-8 b(,)30 b(w)m(e)h(obtain)888 2582 y Fn(\027)933 2596 y Fm(")970 2582 y Fu(\()p Fn(X)7 b Fu(\))99 b(=)1390 2467 y Fh(Z)1436 2656 y Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1644 2582 y Fn(l)1671 2596 y Fm(")1708 2582 y Fu(\()p Fn(dx)p Fu(\))p Fn(E)1949 2545 y Fm(")1944 2605 y(x)1989 2582 y Fu(\()p Fn(\034)2064 2596 y Ft(1)2103 2582 y Fu(\))1221 2821 y(=)1390 2706 y Fh(Z)1436 2894 y Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1644 2821 y Fn(l)1671 2835 y Fm(")1708 2821 y Fu(\()p Fn(dx)p Fu(\))1892 2727 y Fh(\020)1943 2821 y Fn(E)2015 2783 y Fm(")2010 2843 y(x)2054 2821 y Fu(\()p Fn(\033)2141 2835 y Ft(0)2180 2821 y Fu(\))21 b(+)f Fn(E)2399 2783 y Fm(")2394 2843 y(x)2438 2821 y Fu(\()p Fn(E)2545 2783 y Fm(")2540 2846 y(x)2580 2854 y Ff(")2613 2846 y Ft(\()p Fm(\033)2680 2855 y Fj(0)2716 2846 y Ft(\))2747 2821 y Fu(\()p Fn(\034)2822 2835 y Ft(1)2862 2821 y Fu(\)\))2932 2727 y Fh(\021)1221 3012 y Fk(\024)167 b Fu(sup)1390 3094 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1681 3012 y Fn(E)1753 2975 y Fm(")1748 3035 y(x)1792 3012 y Fu(\()p Fn(\033)1879 3026 y Ft(0)1918 3012 y Fu(\))21 b(+)99 b(sup)2065 3094 y Fm(y)r Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)2309 3075 y Fa(0)2333 3094 y Ft(\))2375 3012 y Fn(E)2447 2975 y Fm(")2442 3035 y(y)2484 3012 y Fu(\()p Fn(\034)2559 3026 y Ft(1)2599 3012 y Fu(\))15 b Fn(:)556 b Fu(\(156\))448 3288 y(By)36 b(Lemma)f(3.9,)i(giv)m(en)e Fn(h)f(>)f Fu(0,)k(for)d(su\016cien)m(tly)g (small)g(0)f Fn(<)g(\032)g(<)g(\032)2900 3255 y Fl(0)2958 3288 y Fu(w)m(e)j(ha)m(v)m(e)g(the)448 3401 y(estimate)1442 3534 y(sup)1373 3615 y Fm(x)p Fl(2)p Fm(@)t(B)s Ft(\()p Fm(\032)p Ft(\))1663 3534 y Fn(E)1735 3496 y Fm(")1730 3556 y(x)1774 3534 y Fu(\()p Fn(\033)1861 3548 y Ft(0)1901 3534 y Fu(\))41 b Fk(\024)f Fu(exp)2242 3415 y Fh(\022)2331 3472 y Fn(h)p 2313 3513 88 4 v 2313 3596 a Fu(2)p Fn(")2411 3415 y Fh(\023)2487 3534 y Fn(;)718 b Fu(\(157\))448 3766 y(for)36 b(su\016cien)m(tly)e(small)g Fn(")p Fu(.)56 b(By)36 b Fg(L1)p Fu(,)h(the)e(second)h(term)f(on)h(the)f(righ)m(t)g (hand)f(side)h(of)448 3879 y(\(156\))g(is)c(b)s(ounded)g(b)m(y)h(a)h (constan)m(t,)h(uniformly)c(in)h(0)e Fk(\024)g Fn(")g Fk(\024)f Fn(")2666 3893 y Ft(0)2706 3879 y Fu(.)47 b(Therefore)32 b(for)h(w)m(e)448 3991 y(obtain)d(the)h(estimate)1537 4124 y Fn(\027)1582 4138 y Fm(")1619 4124 y Fu(\()p Fn(X)7 b Fu(\))42 b Fk(\024)e Fu(exp)2078 4005 y Fh(\022)2167 4063 y Fn(h)p 2149 4103 V 2149 4187 a Fu(2)p Fn(")2247 4005 y Fh(\023)2323 4124 y Fn(;)882 b Fu(\(158\))448 4332 y(for)30 b(su\016cien)m(tly)e(small)g Fn(")p Fu(.)41 b(Com)m(bining)27 b(estimates)j(\(155\))i(and)d(\(158\))j(w)m(e)e (obtain)f(the)448 4445 y(b)s(ound)1217 4575 y Fn(\026)1272 4589 y Fm(")1308 4575 y Fu(\()p Fn(D)s Fu(\))41 b Fk(\025)f Fu(exp)1762 4456 y Fh(\022)1824 4575 y Fk(\000)1905 4514 y Fu(1)p 1905 4554 46 4 v 1907 4638 a Fn(")1960 4575 y Fu(\()16 b(inf)1995 4635 y Fm(z)s Fl(2)p Fm(D)2153 4575 y Fn(W)d Fu(\()p Fn(z)t Fu(\))20 b(+)g Fn(h)p Fu(\))2566 4456 y Fh(\023)2643 4575 y Fn(;)562 b Fu(\(159\))448 4783 y(and)30 b(this)f(is)h(the)g(b)s(ound)f(\(141\).)42 b(This)29 b(concludes)h(the)g(pro)s(of)g(of)g(Theorem)g(3.3.)p 3334 4791 50 67 v 1897 5225 a(33)p eop %%Page: 34 34 34 33 bop 448 573 a Fo(4)135 b(Prop)t(erties)46 b(of)f(the)g(rate)h (function)448 776 y Fu(In)27 b(this)e(section)j(w)m(e)f(pro)m(v)m(e)h (assertions)f(\(16\))h(and)e(\(17\))j(of)e(Theorem)g(1.4)h(and)e(Prop)s (o-)448 905 y(sition)42 b(1.7.)81 b(Recall)44 b(that)g(for)f(Eq.)g (\(5\),)48 b(the)c(rate)g(function,)i Fn(I)2761 857 y Ft(\()p Fm(\021)r Ft(\))2754 932 y Fm(x;T)2868 905 y Fu(\()p Fn(\036)p Fu(\),)i(tak)m(es)d(the)448 1017 y(follo)m(wing)29 b(form:)40 b(F)-8 b(or)32 b Fn(\036)p Fu(\()p Fn(t)p Fu(\))25 b(=)g(\()p Fn(p)p Fu(\()p Fn(t)p Fu(\))p Fn(;)15 b(q)s Fu(\()p Fn(t)p Fu(\))p Fn(;)g(r)s Fu(\()p Fn(t)p Fu(\)\),)905 1274 y Fn(I)952 1226 y Ft(\()p Fm(\021)r Ft(\))945 1301 y Fm(x;T)1059 1274 y Fu(\()p Fn(\036)p Fu(\))41 b(=)1418 1212 y(1)p 1345 1253 191 4 v 1345 1336 a(4)p Fn(\015)5 b(\025)1495 1310 y Ft(2)1560 1159 y Fh(Z)1643 1185 y Fm(T)1607 1348 y Ft(0)1698 1274 y Fu(\()14 b(_)-39 b Fn(r)24 b Fu(+)19 b Fn(\015)5 b(\025)1993 1236 y Ft(2)2033 1274 y Fk(r)2109 1288 y Fm(r)2147 1274 y Fn(G)p Fu(\))p Fn(D)2332 1236 y Fl(\000)p Ft(1)2426 1274 y Fu(\()14 b(_)-39 b Fn(r)23 b Fu(+)d Fn(\015)5 b(\025)2721 1236 y Ft(2)2760 1274 y Fk(r)2836 1288 y Fm(r)2874 1274 y Fn(G)p Fu(\))249 b(\(160\))448 1517 y(if)1479 1630 y(_)-42 b Fn(q)28 b Fu(=)d Fk(r)1703 1644 y Fm(p)1742 1630 y Fn(G)15 b(;)124 b Fu(_)-43 b Fn(p)25 b Fu(=)g Fk(\000r)2274 1644 y Fm(q)2312 1630 y Fn(G)15 b(;)806 b Fu(\(161\))448 1796 y(and)36 b(is)g(+)p Fk(1)f Fu(otherwise.)59 b(Recall)36 b(that)h(for)f(a)h(path)f Fn(\036)f Fk(2)g(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))38 b(with)e Fn(\036)p Fu(\(0\))g(=)f Fn(x)448 1909 y Fu(and)d Fn(\036)p Fu(\()p Fn(T)13 b Fu(\))30 b(=)e Fn(y)36 b Fu(w)m(e)d(denote)1470 1885 y(~)1458 1909 y Fn(\036)f Fu(the)h(time)f(rev)m(ersed)h(path)f(whic)m (h)f(satisfy)3043 1885 y(~)3031 1909 y Fn(\036)q Fu(\(0\))f(=)e Fn(J)9 b(y)448 2022 y Fu(and)637 1998 y(~)625 2022 y Fn(\036)p Fu(\()p Fn(T)k Fu(\))26 b(=)f Fn(J)9 b(x)p Fu(.)448 2151 y Fe(Pr)-5 b(o)g(of)34 b(of)f(Pr)-5 b(op)g(osition)36 b(1.7:)41 b Fu(W)-8 b(e)31 b(rewrite)f(the)g(rate)h(function)f Fn(I)2684 2103 y Ft(\()p Fm(\021)r Ft(\))2677 2178 y Fm(x;T)2791 2151 y Fu(\()p Fn(\036)p Fu(\))h(as)714 2389 y Fn(I)761 2341 y Ft(\()p Fm(\021)r Ft(\))754 2416 y Fm(x;T)869 2389 y Fu(\()p Fn(\036)p Fu(\))26 b(=)730 2596 y(=)923 2535 y(1)p 851 2575 V 851 2659 a(4)p Fn(\015)5 b(\025)1001 2632 y Ft(2)1066 2481 y Fh(Z)1149 2508 y Fm(T)1112 2670 y Ft(0)1204 2596 y Fu(\()14 b(_)-39 b Fn(r)23 b Fu(+)d Fn(\015)5 b(\025)1499 2559 y Ft(2)1538 2596 y Fk(r)1614 2610 y Fm(r)1652 2596 y Fn(G)p Fu(\))p Fn(D)1837 2559 y Fl(\000)p Ft(1)1931 2596 y Fu(\()14 b(_)-39 b Fn(r)23 b Fu(+)d Fn(\015)5 b(\025)2226 2559 y Ft(2)2266 2596 y Fk(r)2342 2610 y Fm(r)2379 2596 y Fn(G)p Fu(\))p Fn(dt)714 2844 y Fu(=)893 2782 y(1)p 820 2822 V 820 2906 a(4)p Fn(\015)g(\025)970 2880 y Ft(2)1036 2729 y Fh(Z)1119 2755 y Fm(T)1082 2917 y Ft(0)1174 2844 y Fu(\()14 b(_)-39 b Fn(r)23 b Fk(\000)d Fn(\015)5 b(\025)1469 2806 y Ft(2)1508 2844 y Fk(r)1584 2858 y Fm(r)1622 2844 y Fn(G)p Fu(\))p Fn(D)1807 2806 y Fl(\000)p Ft(1)1901 2844 y Fu(\()14 b(_)-39 b Fn(r)23 b Fk(\000)d Fn(\015)5 b(\025)2196 2806 y Ft(2)2235 2844 y Fk(r)2311 2858 y Fm(r)2349 2844 y Fn(G)p Fu(\))p Fn(dt)20 b Fu(+)2647 2729 y Fh(Z)2730 2755 y Fm(T)2693 2917 y Ft(0)2785 2844 y Fu(\()p Fk(r)2896 2858 y Fm(r)2934 2844 y Fn(G)p Fu(\))p Fn(D)3119 2806 y Fl(\000)p Ft(1)3228 2844 y Fu(_)-40 b Fn(r)s(dt)714 3025 y Fk(\021)25 b Fn(K)887 3039 y Ft(1)927 3025 y Fu(\()p Fn(\036)p Fu(\))c(+)f Fn(K)1240 3039 y Ft(2)1280 3025 y Fu(\()p Fn(\036)p Fu(\))1826 b(\(162\))448 3229 y(The)42 b(term)f Fn(K)952 3243 y Ft(1)992 3229 y Fu(\()p Fn(\036)p Fu(\))i(has)e(the)h(follo)m(wing)e(in)m (terpretation:)63 b(It)42 b(is)f(the)h(rate)g(function)448 3342 y(corresp)s(onding)29 b(to)i(the)f(the)h(set)g(of)f(sto)s(c)m (hastic)i(di\013eren)m(tial)c(equations)1091 3546 y Fn(dq)101 b Fu(=)d Fk(r)1525 3560 y Fm(p)1564 3546 y Fn(Gdt)15 b(;)1089 3684 y(dp)98 b Fu(=)g Fk(\000r)1596 3698 y Fm(q)1633 3684 y Fn(Gdt)15 b(;)1091 3831 y(dr)101 b Fu(=)d(+)p Fn(\015)5 b(\025)1625 3794 y Ft(2)1664 3831 y Fk(r)1740 3845 y Fm(r)1778 3831 y Fn(Gdt)20 b Fu(+)g Fn(")2083 3794 y Ft(1)p Fm(=)p Ft(2)2193 3831 y Fu(\(2)p Fn(\015)5 b(\025)2378 3794 y Ft(2)2418 3831 y Fn(D)s Fu(\))2531 3794 y Ft(1)p Fm(=)p Ft(2)2641 3831 y Fn(dw)18 b(:)434 b Fu(\(163\))448 4035 y(In)38 b(particular)f(there)i(is)f Fn(u)i Fk(2)e Fn(L)1592 4003 y Ft(2)1632 4035 y Fu(\([0)p Fn(;)15 b(T)e Fu(]\))40 b(suc)m(h)f(that)g(for)g Fn(\036)p Fu(\()p Fn(t)p Fu(\))h(=)f(\()p Fn(p)p Fu(\()p Fn(t)p Fu(\))p Fn(;)15 b(q)s Fu(\()p Fn(t)p Fu(\))p Fn(;)g(r)s Fu(\()p Fn(t)p Fu(\)\),)448 4148 y(with)29 b Fn(\036)p Fu(\(0\))e(=)e Fn(x)30 b Fu(and)g Fn(\036)p Fu(\()p Fn(T)13 b Fu(\))26 b(=)f Fn(y)33 b Fu(w)m(e)e(ha)m(v)m(e)1279 4353 y(_)-42 b Fn(q)101 b Fu(=)d Fk(r)1649 4367 y Fm(p)1688 4353 y Fn(G)15 b(;)1278 4490 y Fu(_)-43 b Fn(p)98 b Fu(=)g Fk(\000r)1720 4504 y Fm(q)1757 4490 y Fn(G)15 b(;)1276 4638 y Fu(_)-39 b Fn(r)101 b Fu(=)d(+)p Fn(\015)5 b(\025)1749 4600 y Ft(2)1788 4638 y Fk(r)1864 4652 y Fm(r)1902 4638 y Fn(G)20 b Fu(+)f(\(2)p Fn(\015)5 b(\025)2269 4600 y Ft(2)2310 4638 y Fn(D)s Fu(\))2423 4600 y Ft(1)p Fm(=)p Ft(2)2533 4638 y Fn(u)15 b(:)605 b Fu(\(164\))448 4842 y(Consider)38 b(no)m(w)i(the)g(transformation)f(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\))41 b Fk(!)g Fn(J)9 b Fu(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\))40 b(and)f Fn(t)i Fk(!)g(\000)p Fn(t)p Fu(.)68 b(This)448 4955 y(transformation)36 b(maps)g(the)g (solution)f(of)h(Eq.)g(\(164\))i(in)m(to)2556 4931 y(~)2544 4955 y Fn(\036)p Fu(\()p Fn(t)p Fu(\))d(=)g(\()8 b(~)-53 b Fn(p)p Fu(\()p Fn(t)p Fu(\))p Fn(;)23 b Fu(~)-53 b Fn(q)t Fu(\()p Fn(t)p Fu(\))p Fn(;)20 b Fu(~)-50 b Fn(r)s Fu(\()p Fn(t)p Fu(\)\))1897 5225 y(34)p eop %%Page: 35 35 35 34 bop 448 573 a Fu(whic)m(h)29 b(is)h(the)g(solution)f(to)1219 756 y(_)1217 777 y(~)-52 b Fn(q)101 b Fu(=)d Fk(r)1597 791 y Fm(p)1636 777 y Fn(G)15 b(;)1219 893 y Fu(_)1216 915 y(~)-53 b Fn(p)98 b Fu(=)g Fk(\000r)1668 929 y Fm(q)1706 915 y Fn(G)15 b(;)1220 1040 y Fu(_)1215 1062 y(~)-50 b Fn(r)101 b Fu(=)d Fk(\000)p Fn(\015)5 b(\025)1697 1024 y Ft(2)1736 1062 y Fk(r)1812 1076 y Fm(r)1850 1062 y Fn(G)20 b Fu(+)g(\(2)p Fn(\015)5 b(\025)2218 1024 y Ft(2)2258 1062 y Fn(D)s Fu(\))2371 1024 y Ft(1)p Fm(=)p Ft(2)2487 1062 y Fu(~)-51 b Fn(u)p Fu(\()p Fn(t)p Fu(\))15 b Fn(;)554 b Fu(\(165\))448 1266 y(with)667 1242 y(~)655 1266 y Fn(\036)q Fu(\(0\))26 b(=)f Fn(J)9 b(y)s Fu(,)1121 1242 y(~)1109 1266 y Fn(\036)q Fu(\()p Fn(T)k Fu(\))25 b(=)g Fn(J)9 b(x)p Fu(.)41 b(This)29 b(implies)e(the)k(equalit)m(y)826 1523 y Fn(K)903 1537 y Ft(1)942 1523 y Fu(\()p Fn(\036)p Fu(\))99 b(=)1416 1461 y(1)p 1344 1502 191 4 v 1344 1585 a(4)p Fn(\015)5 b(\025)1494 1559 y Ft(2)1559 1408 y Fh(Z)1642 1434 y Fm(T)1605 1596 y Ft(0)1697 1523 y Fu(\()14 b(_)-39 b Fn(r)23 b Fk(\000)d Fn(\015)5 b(\025)1992 1485 y Ft(2)2032 1523 y Fk(r)2108 1537 y Fm(r)2146 1523 y Fn(G)p Fu(\))p Fn(D)2331 1485 y Fl(\000)p Ft(1)2425 1523 y Fu(\()14 b(_)-39 b Fn(r)23 b Fk(\000)d Fn(\015)5 b(\025)2720 1485 y Ft(2)2759 1523 y Fk(r)2835 1537 y Fm(r)2873 1523 y Fn(G)p Fu(\))1165 1770 y(=)1416 1708 y(1)p 1344 1749 V 1344 1832 a(4)p Fn(\025)1442 1806 y Ft(2)1482 1832 y Fn(\015)1559 1655 y Fh(Z)1642 1681 y Fm(T)1605 1844 y Ft(0)1697 1770 y Fu(\()1741 1748 y(_)1736 1770 y(~)-49 b Fn(r)23 b Fu(+)d Fn(\015)5 b(\025)1992 1732 y Ft(2)2032 1770 y Fk(r)2108 1784 y Fm(r)2146 1770 y Fn(G)p Fu(\))p Fn(D)2331 1732 y Fl(\000)p Ft(1)2425 1770 y Fu(\()2469 1748 y(_)2464 1770 y(~)-49 b Fn(r)23 b Fu(+)d Fn(\015)5 b(\025)2720 1732 y Ft(2)2759 1770 y Fk(r)2835 1784 y Fm(r)2873 1770 y Fn(G)p Fu(\))p Fn(dt)1165 1981 y Fu(=)98 b Fn(I)1381 1933 y Ft(\()p Fm(\021)r Ft(\))1374 2008 y Fm(J)6 b(y)r(;T)1531 1981 y Fu(\()1578 1957 y(~)1566 1981 y Fn(\036)p Fu(\))15 b Fn(:)1535 b Fu(\(166\))448 2185 y(This)27 b(means)g(that)i Fn(K)1201 2199 y Ft(1)1241 2185 y Fu(\()p Fn(\036)p Fu(\))g(is)e(nothing)g(but)g(the)h(w)m(eigh)m (t)h(of)f(the)g(time)g(rev)m(ersed)g(path,)448 2298 y(i.e.)41 b(the)30 b(path)h(starting)f(at)h(time)f(0)h(from)f Fn(J)9 b(y)33 b Fu(and)d(leading)f(to)i Fn(J)9 b(x)30 b Fu(at)h(time)f Fn(T)13 b Fu(.)589 2411 y(W)-8 b(e)46 b(no)m(w)f(consider)e(the)i (second)f(term)h Fn(K)2119 2425 y Ft(2)2158 2411 y Fu(\()p Fn(\036)p Fu(\))h(in)d(Eq.)h(\(162\).)85 b(W)-8 b(e)46 b(consider)448 2524 y(separately)37 b(the)h(equilibrium)32 b(case)38 b(\(i.e.,)h Fn(\021)g Fu(=)d(0\))i(and)e(the)h (non-equilibrium)32 b(case)448 2637 y(\(i.e.)41 b Fn(\021)29 b Fk(6)p Fu(=)c(0\).)41 b(F)-8 b(or)31 b Fn(\021)e Fu(=)c(0)31 b(the)f(matrix)g Fn(D)j Fu(is)d(the)g(iden)m(tit)m(y)g(and)g(w)m(e)h (\014nd)1432 2893 y Fn(K)1509 2907 y Ft(2)1549 2893 y Fu(\()p Fn(\036)p Fu(\))41 b(=)1825 2778 y Fh(Z)1908 2805 y Fm(T)1871 2967 y Ft(0)1978 2893 y Fk(r)2054 2907 y Fm(r)2092 2893 y Fn(G)14 b Fu(_)-39 b Fn(r)r(dt)943 b Fu(\(167\))448 3130 y(Using)44 b(the)h(constrain)m(ts)61 b(_)-42 b Fn(q)52 b Fu(=)d Fk(r)1655 3144 y Fm(p)1694 3130 y Fn(G)44 b Fu(and)62 b(_)-43 b Fn(p)49 b Fu(=)f Fk(\000r)2362 3144 y Fm(q)2400 3130 y Fn(G)c Fu(w)m(e)h(obtain)f(the)h (iden)m(tit)m(y)448 3243 y Fk(r)524 3257 y Fm(p)564 3243 y Fn(G)17 b Fu(_)-42 b Fn(p)19 b Fu(+)h Fk(r)868 3257 y Fm(q)906 3243 y Fn(G)c Fu(_)-41 b Fn(q)27 b Fu(=)e(0)31 b(and)f(therefore)h(w)m(e)g(get)962 3388 y Fh(Z)1045 3414 y Fm(T)1008 3577 y Ft(0)1115 3503 y Fk(r)1191 3517 y Fm(r)1229 3503 y Fn(G)14 b Fu(_)-39 b Fn(r)r(dt)98 b Fu(=)1691 3388 y Fh(Z)1774 3414 y Fm(T)1737 3577 y Ft(0)1845 3503 y Fu(\()p Fk(r)1956 3517 y Fm(r)1994 3503 y Fn(G)14 b Fu(_)-39 b Fn(r)22 b Fu(+)e Fk(r)2296 3517 y Fm(p)2335 3503 y Fn(G)d Fu(_)-42 b Fn(p)20 b Fu(+)f Fk(r)2639 3517 y Fm(q)2677 3503 y Fn(G)e Fu(_)-42 b Fn(q)s Fu(\))15 b Fn(dt)1522 3744 y Fu(=)1691 3629 y Fh(Z)1774 3655 y Fm(T)1737 3817 y Ft(0)1871 3682 y Fn(d)p 1854 3723 81 4 v 1854 3806 a(dt)1945 3744 y(Gdt)40 b Fu(=)g Fn(G)p Fu(\()p Fn(y)s Fu(\))20 b Fk(\000)g Fn(G)p Fu(\()p Fn(x)p Fu(\))15 b Fn(;)447 b Fu(\(168\))448 3985 y(and)30 b(this)f(pro)m(v)m(es)i(Eq.)g(\(22\))h(in)d(the)h(case)i Fn(\021)c Fu(=)d(0.)589 4098 y(T)-8 b(o)31 b(pro)m(v)m(e)g(Eq.)g (\(22\))h(in)d(the)h(case)i Fn(\021)d Fk(6)p Fu(=)24 b(0)31 b(observ)m(e)g(that)g(w)m(e)g(ha)m(v)m(e)h(the)e(iden)m(tit)m(y) -8 b(,)521 4284 y Fn(d)p 504 4324 V 504 4408 a(dt)604 4284 y Fu(1)p 604 4324 46 4 v 604 4408 a(2)660 4345 y(\()p Fn(\025)748 4308 y Fl(\000)p Ft(1)843 4345 y Fn(r)884 4359 y Fm(i)932 4345 y Fk(\000)20 b Fn(\025q)1117 4359 y Fm(i)1145 4345 y Fu(\))1180 4308 y Ft(2)1260 4345 y Fu(=)40 b(\()p Fn(\025)1459 4308 y Fl(\000)p Ft(1)1554 4345 y Fn(r)1595 4359 y Fm(i)1643 4345 y Fk(\000)20 b Fn(\025q)1828 4359 y Fm(i)1856 4345 y Fu(\)\()p Fn(\025)1979 4308 y Fl(\000)p Ft(1)2088 4345 y Fu(_)-39 b Fn(r)2115 4359 y Fm(i)2163 4345 y Fk(\000)20 b Fn(\025)d Fu(_)-42 b Fn(q)2348 4359 y Fm(i)2376 4345 y Fu(\))41 b(=)f Fn(\025)p Fk(r)2692 4359 y Fm(r)2724 4369 y Ff(i)2754 4345 y Fn(G)p Fu(\()p Fn(\025)2914 4308 y Fl(\000)p Ft(1)3022 4345 y Fu(_)-39 b Fn(r)3049 4359 y Fm(i)3097 4345 y Fk(\000)20 b Fn(\025p)3287 4359 y Fm(i)3315 4345 y Fu(\))15 b Fn(;)3230 4484 y Fu(\(169\))448 4597 y(for)30 b Fn(i)c Fu(=)f(1)p Fn(;)15 b(n)p Fu(,)31 b(and)f(therefore)1116 4839 y Fk(r)1192 4853 y Fm(r)1224 4863 y Ff(i)1255 4839 y Fn(G)14 b Fu(_)-39 b Fn(r)1368 4853 y Fm(i)1420 4839 y Fu(=)25 b Fn(\025)1569 4801 y Ft(2)1609 4839 y Fk(r)1685 4853 y Fm(r)1717 4863 y Ff(i)1747 4839 y Fn(Gp)1865 4853 y Fm(i)1913 4839 y Fu(+)2030 4777 y Fn(d)p 2013 4818 81 4 v 2013 4901 a(dt)2114 4777 y Fu(1)p 2114 4818 46 4 v 2114 4901 a(2)2169 4839 y(\()p Fn(\025)2257 4801 y Fl(\000)p Ft(1)2352 4839 y Fn(r)2393 4853 y Fm(i)2441 4839 y Fk(\000)20 b Fn(\025q)2626 4853 y Fm(i)2654 4839 y Fu(\))2689 4801 y Ft(2)2744 4839 y Fn(:)461 b Fu(\(170\))1897 5225 y(35)p eop %%Page: 36 36 36 35 bop 448 573 a Fu(Hence,)32 b(using)d(the)h(de\014nition)e (\(21\),)k(w)m(e)f(obtain)552 829 y Fn(K)629 843 y Ft(2)668 829 y Fu(\()p Fn(\036)p Fu(\))41 b(=)945 714 y Fh(Z)1028 741 y Fm(T)991 903 y Ft(0)1098 829 y Fk(r)1174 843 y Fm(r)1212 829 y Fn(D)1290 792 y Fl(\000)p Ft(1)1384 829 y Fn(G)14 b Fu(_)-39 b Fn(r)r(dt)40 b Fu(=)g Fn(R)q Fu(\()p Fn(\036)p Fu(\()p Fn(T)13 b Fu(\)\))22 b Fk(\000)d Fn(R)q Fu(\()p Fn(\036)p Fu(\(0\)\))j Fk(\000)2594 714 y Fh(Z)2677 741 y Fm(T)2640 903 y Ft(0)2748 829 y Fu(\002\()p Fn(\036)p Fu(\()p Fn(t)p Fu(\)\))p Fn(dt)104 b Fu(\(171\))448 1066 y(This)29 b(completes)i(the)f(pro)s(of)g(of)g(Prop)s(osition)f(1.7.)p 2294 1074 50 67 v 589 1278 a(With)h(generalized)g(detailed)g(balance)g (w)m(e)h(sho)m(w)f(the)h(follo)m(wing)448 1466 y Fg(Prop)s(osition)37 b(4.1)46 b Fe(If)32 b Fn(\021)d Fu(=)c(0)33 b Fe(then)g Fn(W)1837 1433 y Ft(\(0\))1931 1466 y Fu(\()p Fn(x)p Fu(\))25 b(=)g Fn(G)p Fu(\()p Fn(x)p Fu(\))c Fk(\000)f Fu(min)2631 1480 y Fm(x)2690 1466 y Fn(G)p Fu(\()p Fn(x)p Fu(\))p Fe(.)448 1654 y(Pr)-5 b(o)g(of:)42 b Fu(The)30 b(function)f Fn(W)1370 1621 y Ft(\(0\))1464 1654 y Fu(\()p Fn(x)p Fu(\))i(is)e(giv)m(en)i(b)m(y)681 1873 y Fn(W)780 1836 y Ft(\(0\))874 1873 y Fu(\()p Fn(x)p Fu(\))41 b(=)f(min)1211 1931 y Fm(i)1314 1779 y Fh(\020)1364 1873 y Fn(W)1463 1836 y Ft(\(0\))1557 1873 y Fu(\()p Fn(K)1669 1887 y Fm(i)1698 1873 y Fu(\))20 b(+)g Fn(V)1917 1836 y Ft(\(0\))2012 1873 y Fu(\()p Fn(K)2124 1887 y Fm(i)2152 1873 y Fn(;)15 b(x)p Fu(\))2279 1779 y Fh(\021)2350 1873 y Fk(\000)20 b Fu(min)2500 1931 y Fm(j)2607 1873 y Fn(W)2706 1836 y Ft(\(0)2773 1873 y Fu(\()p Fn(K)2885 1887 y Fm(j)2922 1873 y Fu(\))15 b Fn(:)233 b Fu(\(172\))448 2135 y(where)20 b(the)h(minim)m(um)d(is)h(tak)m(en)j(o)m(v)m(er)g(all)d(compact)j(sets) f Fn(K)2464 2149 y Fm(i)2492 2135 y Fu(.)37 b(In)20 b(Eq.)g(\(172\),)25 b Fn(W)3167 2102 y Ft(\(0\))3261 2135 y Fu(\()p Fn(K)3373 2149 y Fm(i)3402 2135 y Fu(\))448 2248 y(is)30 b(giv)m(en)g(b)m(y)1118 2361 y Fn(W)1217 2323 y Ft(\(0\))1311 2361 y Fu(\()p Fn(K)1423 2375 y Fm(i)1452 2361 y Fu(\))41 b(=)80 b(min)1639 2424 y Fm(g)r Fl(2)p Fm(G)p Fl(f)p Fm(i)p Fl(g)1983 2280 y Fh(X)1887 2468 y Ft(\()p Fm(m)p Fl(!)p Fm(n)p Ft(\))p Fl(2)p Fm(g)2215 2361 y Fn(V)2289 2323 y Ft(\(0\))2383 2361 y Fn(K)2460 2375 y Fm(m)2527 2361 y Fn(;)15 b(K)2644 2375 y Fm(n)2691 2361 y Fu(\))g Fn(:)464 b Fu(\(173\))448 2623 y(The)35 b(sets)h Fn(K)900 2637 y Fm(j)972 2623 y Fu(are)g(the)g(critical)e(sets)i(of)g(the)g(generalized)f (Hamiltonian)f Fn(G)p Fu(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\),)448 2736 y(therefore)36 b Fn(G)e Fu(is)f(constan)m(t)j(on)f Fn(K)1615 2750 y Fm(j)1686 2736 y Fu(and)g(w)m(e)g(set)g Fn(G)p Fu(\()p Fn(x)p Fu(\))e(=)f Fn(G)2554 2750 y Fm(j)2625 2736 y Fu(for)i(all)g Fn(x)e Fk(2)g Fn(K)3153 2750 y Fm(j)3190 2736 y Fu(.)54 b(F)-8 b(ur-)448 2849 y(thermore)27 b(if)f(\()p Fn(p;)15 b(q)s(;)g(r)s Fu(\))26 b Fk(2)f Fn(K)1387 2863 y Fm(j)1423 2849 y Fu(,)j(then)f Fn(p)e Fu(=)f(0)k(and)e(therefore)h(the)g(sets)g Fn(K)2874 2863 y Fm(j)2938 2849 y Fu(are)g(in)m(v)-5 b(arian)m(t)448 2962 y(under)29 b(time)h(rev)m(ersal:)40 b Fn(J)9 b(K)1418 2976 y Fm(j)1480 2962 y Fu(=)25 b Fn(K)1653 2976 y Fm(j)1690 2962 y Fu(.)41 b(Using)29 b(the)h(generalized)g(detailed)f(balance,)i (w)m(e)448 3075 y(see)h(that)g(for)f(an)m(y)h(path)f Fn(\036)c Fk(2)g(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))33 b(with)d Fn(\036)p Fu(\(0\))e(=)f Fn(x)g Fk(2)g Fn(K)2593 3089 y Fm(m)2691 3075 y Fu(and)j Fn(\036)p Fu(\()p Fn(t)p Fu(\))e(=)f Fn(y)i Fk(2)e Fn(K)3390 3089 y Fm(n)448 3188 y Fu(w)m(e)k(ha)m(v)m(e)686 3392 y Fn(I)733 3344 y Ft(\(0\))726 3419 y Fm(x;T)841 3392 y Fu(\()p Fn(\036)p Fu(\))41 b(=)f Fn(I)1164 3344 y Ft(\(0\))1157 3419 y Fm(J)6 b(y)r(;T)1314 3392 y Fu(\()1361 3368 y(~)1349 3392 y Fn(\036)p Fu(\))21 b(+)f Fn(G)p Fu(\()p Fn(y)s Fu(\))g Fk(\000)g Fn(G)p Fu(\()p Fn(x)p Fu(\))40 b(=)g Fn(I)2243 3344 y Ft(\(0\))2236 3419 y Fm(y)r(;T)2348 3392 y Fu(\()2395 3368 y(~)2383 3392 y Fn(\036)q Fu(\))20 b(+)g Fn(G)2656 3406 y Fm(n)2723 3392 y Fk(\000)g Fn(G)2886 3406 y Fm(m)2967 3392 y Fn(:)238 b Fu(\(174\))448 3596 y(T)-8 b(aking)30 b(the)h(in\014m)m(um)d(o)m(v)m (er)k(all)d(paths)h Fn(\036)g Fu(and)g(all)f(time)h Fn(T)13 b Fu(,)31 b(w)m(e)g(obtain)e(the)i(iden)m(tit)m(y)1061 3800 y Fn(V)1134 3763 y Ft(\(0\))1228 3800 y Fu(\()p Fn(K)1340 3814 y Fm(m)1407 3800 y Fn(;)15 b(K)1524 3814 y Fm(n)1572 3800 y Fu(\))41 b(=)f Fn(V)1832 3763 y Ft(\(0\))1926 3800 y Fu(\()p Fn(K)2038 3814 y Fm(n)2086 3800 y Fn(;)15 b(K)2203 3814 y Fm(m)2270 3800 y Fu(\))21 b(+)f Fn(G)2489 3814 y Fm(m)2575 3800 y Fk(\000)g Fn(G)2738 3814 y Fm(n)2799 3800 y Fn(:)406 b Fu(\(175\))448 4005 y(In)35 b(Eq.)g(\(173\))j(the)d (minim)m(um)e(is)h(tak)m(en)j(o)m(v)m(er)f(all)f Fk(f)p Fn(i)p Fk(g)p Fu(-graphs)h(\(see)g(the)g(paragraph)448 4118 y(ab)s(o)m(v)m(e)f(Theorem)e(1.4)i(in)d(the)i(in)m(tro)s (duction\).)49 b(Giv)m(en)33 b(an)h Fk(f)p Fn(i)p Fk(g)p Fu(-graph)g(and)f(a)h Fn(j)39 b Fu(with)448 4231 y Fn(j)e Fk(6)p Fu(=)30 b Fn(i)p Fu(,)35 b(there)e(is)g(a)h(sequence)g(of)g (arro)m(ws)f(leading)g(from)g Fn(j)39 b Fu(to)34 b Fn(i)p Fu(.)51 b(Consider)31 b(no)m(w)j(the)448 4343 y(graph)f(obtained)f(b)m (y)h(rev)m(ersing)f(all)g(the)h(arro)m(ws)f(leading)g(from)g Fn(j)39 b Fu(to)33 b Fn(i)p Fu(;)i(in)c(this)h(w)m(a)m(y)448 4456 y(w)m(e)g(obtain)e(a)i Fk(f)p Fn(j)5 b Fk(g)p Fu(-graph.)44 b(Using)30 b(the)h(iden)m(tit)m(y)g(\(175\))i(the)e(w)m(eigh)m(t)g(of)h (this)e(graph)g(is)448 4569 y(equal)f(to)h(the)g(w)m(eigh)m(t)g(of)f (the)h(original)d(graph)i(plus)f Fn(G)2346 4583 y Fm(j)2400 4569 y Fk(\000)18 b Fn(G)2561 4583 y Fm(i)2588 4569 y Fu(.)41 b(T)-8 b(aking)29 b(the)g(in\014m)m(um)448 4682 y(o)m(v)m(er)j(all)d(graphs)h(w)m(e)h(obtain)f(the)g(iden)m(tit)m(y) 1259 4886 y Fn(W)1358 4849 y Ft(\(0\))1452 4886 y Fu(\()p Fn(K)1564 4900 y Fm(i)1592 4886 y Fu(\))41 b(=)f Fn(W)1878 4849 y Ft(\(0\))1972 4886 y Fu(\()p Fn(K)2084 4900 y Fm(j)2121 4886 y Fu(\))21 b(+)f Fn(G)2340 4900 y Fm(j)2396 4886 y Fk(\000)g Fn(G)2559 4900 y Fm(i)2601 4886 y Fn(;)604 b Fu(\(176\))1897 5225 y(36)p eop %%Page: 37 37 37 36 bop 448 573 a Fu(and)30 b(therefore)h(w)m(e)g(ha)m(v)m(e)1453 746 y Fn(W)1552 709 y Ft(\(0\))1646 746 y Fu(\()p Fn(K)1758 760 y Fm(i)1787 746 y Fu(\))40 b(=)h Fn(G)2046 760 y Fm(i)2093 746 y Fu(+)20 b(const)c Fn(;)798 b Fu(\(177\))448 920 y(and)30 b(so)1062 1033 y Fn(W)1161 995 y Ft(\(0\))1254 1033 y Fu(\()p Fn(x)p Fu(\))42 b(=)e(min)1592 1090 y Fm(i)1680 1033 y Fu(\()p Fn(G)1787 1047 y Fm(i)1835 1033 y Fu(+)20 b Fn(V)1999 995 y Ft(\(0\))2094 1033 y Fu(\()p Fn(K)2206 1047 y Fm(i)2235 1033 y Fn(;)15 b(x)p Fu(\)\))21 b Fk(\000)f Fu(min)2568 1090 y Fm(j)2675 1033 y Fn(G)2747 1047 y Fm(j)2799 1033 y Fn(:)406 b Fu(\(178\))448 1227 y(The)30 b(second)h(term)f(in)f(Eq.)i(\(178\))h(is)d(equal)h(to)h(min) 2249 1241 y Fm(x)2308 1227 y Fn(G)p Fu(\()p Fn(x)p Fu(\),)g(since)e Fn(G)p Fu(\()p Fn(x)p Fu(\))i(is)e(b)s(ounded)448 1340 y(b)s(elo)m(w.)589 1453 y(W)-8 b(e)28 b(no)m(w)e(deriv)m(e)f(upp)s(er)f (and)i(lo)m(w)m(er)g(b)s(ounds)e(on)i(the)g(\014rst)g(term)g(in)f(Eq.)h (\(178\).)41 b(A)448 1565 y(lo)m(w)m(er)31 b(b)s(ound)d(follo)m(ws)h (easily)g(from)h(Prop)s(osition)e(1.7:)42 b(F)-8 b(or)31 b(an)m(y)f(path)g Fn(\036)25 b Fk(2)g(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))448 1678 y(with)29 b Fn(\036)p Fu(\(0\))e(=)e Fn(z)k Fk(2)c Fn(K)1181 1692 y Fm(i)1240 1678 y Fu(and)30 b Fn(\036)p Fu(\()p Fn(T)13 b Fu(\))26 b(=)f Fn(x)30 b Fu(w)m(e)h(obtain)f(the)g(inequalit)m(y)1011 1868 y Fn(I)1058 1820 y Ft(\(0\))1051 1895 y Fm(z)s(;T)1161 1868 y Fu(\()p Fn(\036)p Fu(\))42 b(=)e Fn(I)1485 1820 y Ft(\(0\))1478 1895 y Fm(J)6 b(x;T)1637 1868 y Fu(\()1684 1844 y(~)1672 1868 y Fn(\036)p Fu(\))21 b(+)f Fn(G)p Fu(\()p Fn(x)p Fu(\))g Fk(\000)g Fn(G)2250 1882 y Fm(i)2318 1868 y Fk(\025)40 b Fn(G)p Fu(\()p Fn(x)p Fu(\))20 b Fk(\000)g Fn(G)2806 1882 y Fm(i)2849 1868 y Fn(;)356 b Fu(\(179\))448 2041 y(since)28 b(the)g(rate)h(function)e(is)g (nonnegativ)m(e.)41 b(T)-8 b(aking)28 b(in\014m)m(um)e(o)m(v)m(er)j (all)f(paths)f Fn(\036)i Fu(and)448 2154 y(time)h Fn(T)43 b Fu(w)m(e)31 b(obtain)1356 2267 y Fn(W)1455 2229 y Ft(\(0\))1549 2267 y Fu(\()p Fn(x)p Fu(\))41 b Fk(\025)f Fn(G)p Fu(\()p Fn(x)p Fu(\))20 b Fk(\000)g Fu(min)2184 2317 y Fm(x)2295 2267 y Fn(G)p Fu(\()p Fn(x)p Fu(\))15 b Fn(:)701 b Fu(\(180\))448 2457 y(T)-8 b(o)40 b(pro)m(v)m(e)f(the)g(lo)m(w)m(er)g(b)s(ound)e(w)m (e)i(consider)f(the)h(tra)5 b(jectory)2659 2433 y(~)2647 2457 y Fn(\036)39 b Fu(starting)f(at)i Fn(J)9 b(x)38 b Fu(at)448 2570 y(time)30 b(0)h(whic)m(h)e(is)h(the)g(solution)f(of)i (the)f(equation)1323 2743 y(_)-41 b Fn(q)1348 2757 y Fm(i)1458 2743 y Fu(=)83 b Fk(r)1688 2757 y Fm(p)1724 2767 y Ff(i)1754 2743 y Fn(G)181 b(i)26 b Fu(=)f(1)p Fn(;)15 b Fk(\001)g(\001)g(\001)i Fn(;)e(n)g(;)1319 2881 y Fu(_)-43 b Fn(p)1347 2895 y Fm(i)1458 2881 y Fu(=)83 b Fk(\000r)1759 2895 y Fm(q)1791 2905 y Ff(i)1821 2881 y Fn(G)181 b(i)25 b Fu(=)g(1)p Fn(;)15 b Fk(\001)g(\001)g(\001)i Fn(;)e(n)g(;)662 b Fu(\(181\))1320 3019 y(_)-39 b Fn(r)1347 3033 y Fm(i)1458 3019 y Fu(=)83 b Fk(\000)p Fn(\015)5 b(\025)1788 2981 y Ft(2)1827 3019 y Fk(r)1903 3033 y Fm(r)1935 3043 y Ff(i)1965 3019 y Fn(G)182 b(i)25 b Fu(=)g(1)p Fn(;)15 b(n)g(:)589 3192 y Fu(By)43 b(Lemma)f(2.2,)k(there)d(is)e(some) h Fn(K)1930 3206 y Fm(j)2009 3192 y Fu(suc)m(h)g(that)g(lim)2561 3206 y Fm(t)p Fl(!1)2759 3168 y Fu(~)2747 3192 y Fn(\036)p Fu(\()p Fn(t)p Fu(\))j Fk(2)f Fn(K)3131 3206 y Fm(j)3168 3192 y Fu(.)76 b(F)-8 b(ur-)448 3305 y(thermore,)36 b(since)1107 3281 y(~)1095 3305 y Fn(\036)f Fu(is)f(a)h(solution)e(of)i(Eq.)f (\(181\),)k(the)d(rate)h(function)d(of)i(this)e(path)448 3434 y(v)-5 b(anishes)30 b Fn(I)853 3386 y Ft(\(0\))846 3461 y Fm(J)6 b(x;T)1005 3434 y Fu(\()1052 3410 y(~)1040 3434 y Fn(\036)q Fu(\))27 b(=)f(0,)32 b(for)f(an)m(y)g Fn(T)39 b(>)26 b Fu(0.)44 b(Note)32 b(that)g(an)f(in\014nite)e(amoun)m (t)i(of)g(time)448 3547 y(is)c(needed)h(to)g(reac)m(h)h Fn(K)1261 3561 y Fm(j)1326 3547 y Fu(in)d(general.)40 b(No)m(w)29 b(consider)e(the)h(time)f(rev)m(ersed)h(path)g Fn(\036)p Fu(\()p Fn(t)p Fu(\).)448 3660 y(It)k(starts)g(at)g Fn(t)26 b Fu(=)h Fk(\000)p Fn(T)44 b Fu(with)30 b Fn(T)39 b Fk(\024)27 b(1)k Fu(at)h Fn(K)1949 3674 y Fm(i)2009 3660 y Fu(and)f(reac)m(hes)h Fn(x)f Fu(at)h(time)f(0.)45 b(F)-8 b(or)32 b(suc)m(h)f(a)448 3773 y(path)f(w)m(e)h(ha)m(v)m(e)762 3957 y(lim)729 4016 y Fm(T)10 b Fl(!1)936 3957 y Fn(I)983 3909 y Ft(\(0\))976 3984 y Fm(z)s(;T)1087 3957 y Fu(\()p Fn(\036)p Fu(\))41 b(=)73 b(lim)1363 4016 y Fm(T)10 b Fl(!1)1570 3957 y Fn(I)1617 3909 y Ft(\(0\))1610 3984 y Fm(J)c(x)1712 3957 y Fu(\()1759 3933 y(~)1747 3957 y Fn(\036)p Fu(\))21 b(+)f Fn(G)p Fu(\()p Fn(x)p Fu(\))g Fk(\000)g Fn(G)2325 3971 y Fm(i)2393 3957 y Fu(=)40 b Fn(G)p Fu(\()p Fn(x)p Fu(\))20 b Fk(\000)g Fn(G)2881 3971 y Fm(i)2924 3957 y Fn(;)281 b Fu(\(182\))448 4156 y(and)30 b(therefore)1426 4269 y Fn(V)1500 4232 y Ft(\(0\))1594 4269 y Fu(\()p Fn(K)1706 4283 y Fm(i)1735 4269 y Fn(;)15 b(x)p Fu(\))41 b Fk(\024)f Fn(G)p Fu(\()p Fn(x)p Fu(\))20 b Fk(\000)g Fn(G)2391 4283 y Fm(i)2434 4269 y Fn(:)771 b Fu(\(183\))448 4421 y(W)-8 b(e)32 b(\014nally)c(obtain)638 4594 y Fn(W)737 4557 y Ft(\(0\))830 4594 y Fu(\()p Fn(x)p Fu(\))41 b Fk(\024)g Fn(G)1177 4608 y Fm(i)1224 4594 y Fu(+)20 b Fn(V)1389 4557 y Ft(\(0\))1483 4594 y Fu(\()p Fn(K)1595 4608 y Fm(i)1624 4594 y Fn(;)15 b(x)p Fu(\))21 b Fk(\000)e Fu(min)1918 4644 y Fm(x)2029 4594 y Fn(G)p Fu(\()p Fn(x)p Fu(\))41 b Fk(\024)f Fn(G)p Fu(\()p Fn(x)p Fu(\))20 b Fk(\000)g Fu(min)2736 4644 y Fm(x)2847 4594 y Fn(G)p Fu(\()p Fn(x)p Fu(\))189 b(\(184\))448 4784 y(and)30 b(this)f(concludes)h(the)h(pro)s(of)e(of)i(Prop)s(osition)d (4.1.)p 2431 4792 50 67 v 589 4976 a(W)-8 b(e)32 b(ha)m(v)m(e)g(the)e (follo)m(wing)f(b)s(ound)f(on)j(the)f(rate)h(function:)1897 5225 y(37)p eop %%Page: 38 38 38 37 bop 448 573 a Fg(Lemma)33 b(4.2)46 b Fe(If)33 b Fn(\021)28 b Fk(\025)d Fu(0)33 b Fe(then)g(for)g(any)h Fn(\036)25 b Fk(2)g(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))p Fe(,)982 793 y Fu(\(1)21 b(+)f Fn(\021)s Fu(\))1257 755 y Fl(\000)p Ft(1)1352 793 y Fn(I)1399 745 y Ft(\(0\))1392 820 y Fm(x;T)1507 793 y Fu(\()p Fn(\036)p Fu(\))41 b Fk(\024)f Fn(I)1830 745 y Ft(\()p Fm(\021)r Ft(\))1823 820 y Fm(x;T)1937 793 y Fu(\()p Fn(\036)p Fu(\))i Fk(\024)e Fu(\(1)21 b Fk(\000)f Fn(\021)s Fu(\))2489 755 y Fl(\000)p Ft(1)2584 793 y Fn(I)2631 745 y Ft(\(0\))2624 820 y Fm(x;T)2738 793 y Fu(\()p Fn(\036)p Fu(\))15 b Fn(;)328 b Fu(\(185\))448 997 y Fe(and)34 b(a)f(similar)h(b)-5 b(ound)33 b(for)g Fn(\021)c Fk(\024)c Fu(0)p Fe(.)448 1210 y(Pr)-5 b(o)g(of:)42 b Fu(The)28 b(pro)s(of)h(follo)m(ws)f(from)h(the)h(fact)g(that)g(the)g (subset)e(of)i Fk(C)5 b Fu(\([0)p Fn(;)15 b(T)e Fu(]\))31 b(on)e(whic)m(h)448 1338 y Fn(I)495 1291 y Ft(\()p Fm(\021)r Ft(\))488 1366 y Fm(x;T)603 1338 y Fu(\()p Fn(\036)p Fu(\))36 b Fn(<)g Fk(1)h Fu(is)e(indep)s(enden)m(t)g(of)i Fn(\021)s Fu(.)60 b(This)35 b(is)h(seen)h(from)f(the)h(de\014nition)d (of)j(rate)448 1451 y(function)29 b(\(24\).)43 b(Insp)s(ection)29 b(of)h(Eq.)g(\(160\))j(implies)27 b(the)k(b)s(ound)d(\(185\).)p 3046 1459 50 67 v 589 1664 a(F)-8 b(rom)31 b(this)f(w)m(e)g(obtain)g (immediately)448 1876 y Fg(Corollary)35 b(4.3)46 b Fe(If)33 b Fn(\021)c Fk(\025)c Fu(0)32 b Fe(then)505 2081 y Fu(\(1)21 b(+)f Fn(\021)s Fu(\))780 2043 y Fl(\000)p Ft(1)875 2081 y Fu(\()p Fn(G)p Fu(\()p Fn(x)p Fu(\))h Fk(\000)f Fu(min)1272 2130 y Fm(x)1383 2081 y Fn(G)p Fu(\()p Fn(x)p Fu(\)\))40 b Fk(\024)h Fn(W)1863 2043 y Ft(\()p Fm(\021)r Ft(\))1959 2081 y Fu(\()p Fn(x)p Fu(\))g Fk(\024)f Fu(\(1)21 b Fk(\000)f Fn(\021)s Fu(\))2508 2043 y Fl(\000)p Ft(1)2603 2081 y Fu(\()p Fn(G)p Fu(\()p Fn(x)p Fu(\))h Fk(\000)f Fu(min)3000 2130 y Fm(x)3110 2081 y Fn(G)p Fu(\()p Fn(x)p Fu(\)\))15 b Fn(:)3230 2215 y Fu(\(186\))448 2328 y Fe(and)34 b(a)f(similar)h(b)-5 b(ound)33 b(for)g Fn(\021)c Fk(\024)c Fu(0)p Fe(.)589 2541 y Fu(This)k(concludes)h(the)g(pro)s(of)g(of)g(Theorem)g(1.4.)448 2795 y Fg(Ac)m(kno)m(wledgemen)m(ts)98 b Fu(W)-8 b(e)33 b(w)m(ould)e(lik)m(e)g(to)h(thank)f(J.-P)-8 b(.)33 b(Ec)m(kmann,)f(M.)g (Hairer,)448 2908 y(J.)37 b(Leb)s(o)m(witz,)g(C.-A.)g(Pillet,)g(and)e (H.)i(Sp)s(ohn)d(for)i(useful)f(discussions.)55 b(This)35 b(w)m(ork)448 3021 y(w)m(as)24 b(partially)e(supp)s(orted)f(b)m(y)j (Swiss)d(National)j(Science)f(F)-8 b(oundation)23 b(\(L.R.-B.\))j(and) 448 3134 y(NSF)31 b(gran)m(t)g(DMS)f(980139)j(\(L.E.T\).)448 3420 y Fo(References)494 3623 y Fu([1])46 b(R.)c(Azencott,)81 b(Grandes)42 b(deviations)f(et)h(applications.)74 b(In)42 b Fe(Ec)-5 b(ole)43 b(d')n(\023)-44 b(et)n(\023)g(e)44 b(de)635 3736 y(pr)-5 b(ob)g(abilit)n(\023)-44 b(es)43 b(de)d(Saint-Flour)i(VIII-1978)p Fu(,)g(Lectures)d(notes)g(in)f (mathematics)635 3849 y(778,)32 b(2{176,)h(Springer)28 b(\(1980\).)494 4036 y([2])46 b(G.)28 b(Ben)g(Arous)e(and)h(R.)h(L)m (\023)-43 b(eandre,)37 b(D)m(\023)-43 b(ecroissance)29 b(exp)s(onen)m(tielle)d(du)g(no)m(y)m(au)i(de)635 4149 y(la)36 b(c)m(haleur)f(sur)g(la)g(diagonale.)h(I)g(and)f(I)s(I.)56 b(Probab.)36 b(Theory)f(Related)h(Fields,)635 4262 y Fg(90)p Fu(,)31 b(175{202,)j(and)c(377{402)j(\(1991\).)494 4450 y([3])46 b(A.)37 b(Dem)m(b)s(o)h(and)e(O.)h(Zeitouni,)60 b Fe(L)-5 b(ar)g(ge)39 b(deviations)h(te)-5 b(chniques)39 b(and)g(applic)-5 b(a-)635 4563 y(tions)p Fu(.)42 b(Applications)28 b(of)i(mathematics,)i(V)-8 b(ol)30 b(38,)i(Springer)c(\(1998\).)494 4750 y([4])46 b(D.)33 b(J.)e(Ev)-5 b(ans,)32 b(E.)g(G.)g(D.)h(Cohen,)f (and)f(G.)h(P)-8 b(.)32 b(Morris,)44 b(Probabilit)m(y)30 b(of)i(second)635 4863 y(la)m(w)23 b(violations)g(in)f(shearing)g (steady)i(states.)31 b(Ph)m(ys.)23 b(Rev.)h(Lett.)g Fg(71)p Fu(,)i(2401{2404)635 4976 y(\(1993\))1897 5225 y(38)p eop %%Page: 39 39 39 38 bop 494 573 a Fu([5])46 b(M.)38 b(Hairer)f(and)f(J.-P)-8 b(.)38 b(Ec)m(kmann,)63 b(Non-equilibrium)33 b(statistical)k(mec)m (hanics)635 686 y(of)42 b(strongly)e(anharmonic)h(c)m(hains)f(of)i (oscillators.)72 b(Preprin)m(t,)43 b(Univ)m(ersit)m(y)e(of)635 799 y(Genev)-5 b(a)32 b(\(1999\).)494 986 y([6])46 b(J.-P)-8 b(.)39 b(Ec)m(kmann,)h(C.-A.)f(Pillet)d(and)h(L.)h(Rey-Bellet,)66 b(Non-equilibrium)34 b(sta-)635 1099 y(tistical)h(mec)m(hanics)g(of)g (anharmonic)g(c)m(hains)f(coupled)h(to)h(t)m(w)m(o)g(heat)g(baths)f(at) 635 1212 y(di\013eren)m(t)30 b(temp)s(eratures.)40 b(Comm)m(un.)30 b(Math.)h(Ph)m(ys.,)g Fg(201)p Fu(,)g(657{697)j(\(1999\).)494 1400 y([7])46 b(J.-P)-8 b(.)32 b(Ec)m(kmann,)g(C.-A.)h(Pillet)c(and)i (L.)h(Rey-Bellet,)45 b(En)m(trop)m(y)32 b(pro)s(duction)d(in)635 1513 y(non-linear,)k(thermally)f(driv)m(en)g(Hamiltonian)g(systems.)50 b(J.)33 b(Stat.)i(Ph)m(ys.,)f Fg(95)p Fu(,)635 1626 y(305{331)g (\(1999\).)494 1813 y([8])46 b(M.I.)37 b(F)-8 b(reidlin)33 b(and)j(A.D.)h(W)-8 b(en)m(tzell,)60 b Fe(R)-5 b(andom)40 b(p)-5 b(erturb)g(ations)41 b(of)d(dynamic)-5 b(al)635 1926 y(systems)p Fu(.)42 b(Grundlehren)28 b(der)i(mathematisc)m(hen)h (Wissensc)m(haft)f(260,)i(Springer)635 2039 y(\(1984\).)494 2227 y([9])46 b(G.)31 b(Galla)m(v)m(otti)g(and)e(E.)h(G.)h(D.)g(Cohen,) 39 b(Dynamical)30 b(ensem)m(bles)f(in)g(stationary)635 2340 y(states.)42 b(J.)30 b(Stat.)i(Ph)m(ys.)e Fg(80)p Fu(,)h(931{970)i(\(1995\).)448 2527 y([10])47 b(R.)27 b(Graham,)36 b(W)-8 b(eak)28 b(noise)f(limit)d(and)i(nonequilibrium)c (p)s(oten)m(tials)k(of)h(dissipa-)635 2640 y(tiv)m(e)k(dynamical)e (systems.)42 b(In)30 b Fe(Instabilities)k(and)f(none)-5 b(quilibrium)34 b(structur)-5 b(es)p Fu(,)635 2753 y(271{290,)34 b(Math.)d(Appl.,)e(33,)j(Reidel)d(\(1987\).)448 2941 y([11])47 b(R.Z.)32 b(Has'minskii,)43 b Fe(Sto)-5 b(chastic)36 b(stability)f(of)g(di\013er)-5 b(ential)35 b(e)-5 b(quations)p Fu(.)46 b(Sijtho\013)635 3054 y(and)30 b(No)s(ordho\013)g(\(1980\).)448 3241 y([12])47 b(L)28 b(H\177)-45 b(ormander,)38 b Fe(The)31 b(A)n(nalysis)g(of)g(line)-5 b(ar)32 b(p)-5 b(artial)34 b(di\013er)-5 b(ential)32 b(op)-5 b(er)g(ators)p Fu(.)41 b(V)-8 b(ol)635 3354 y(I)s(I)s(I,)29 b(Springer)f(\(1985\).)448 3542 y([13])47 b(C.)25 b(Jarzynski,)33 b(Hamiltonian)24 b(deriv)-5 b(ation)24 b(of)h(a)h(detailed)f(\015uctuation)f(theorem.) 635 3655 y(T)-8 b(o)31 b(app)s(ear)f(in)f(J.)h(Stat.)h(Ph)m(ys.)448 3842 y([14])47 b(V.)37 b(Jak)-5 b(\024)-41 b(si)m(\023)e(c)39 b(and)d(C.-A.)h(Pillet,)60 b(Ergo)s(dic)36 b(prop)s(erties)f(of)i (classical)f(dissipativ)m(e)635 3955 y(systems.)31 b(I.)40 b(Acta)32 b(Math.)f Fg(181)g Fu(245{282)j(\(1998\).)448 4143 y([15])47 b(W.)29 b(Kliemann,)35 b(Recurrence)28 b(and)f(in)m(v)-5 b(arian)m(t)28 b(measures)f(for)h(degenerate)i (di\013u-)635 4256 y(sions.)40 b(Ann.)29 b(of)i(Prob.,)f Fg(15)p Fu(,)h(690{702)j(\(1987\).)448 4443 y([16])47 b(A.)75 b(Komec)m(h,)40 b(H.)74 b(Sp)s(ohn,)37 b(and)g(M.)75 b(Kunze,)62 b(Long-time)37 b(asymptotics)g(for)635 4556 y(a)31 b(classical)f(particle)g(in)m(teracting)h(with)f(a)h(scalar)g(w) m(a)m(v)m(e)h(\014eld.)41 b(Comm.)30 b(P)m(artial)635 4669 y(Di\013eren)m(tial)g(Equations)f Fg(22)i Fu(307{335)j(\(1997\).) 448 4857 y([17])47 b(J.)33 b(Kurc)m(han,)47 b(Fluctuation)31 b(theorem)i(for)g(sto)s(c)m(hastic)g(dynamics.)46 b(J.)32 b(Ph)m(ys.)h(A)635 4970 y Fg(31)p Fu(,)e(3719{3729)k(\(1998\).)1897 5225 y(39)p eop %%Page: 40 40 40 39 bop 448 573 a Fu([18])47 b(J.)36 b(L.)h(Leb)s(o)m(witz)f(and)g (H.)g(Sp)s(ohn,)58 b(A)37 b(Galla)m(v)m(otti-Cohen-t)m(yp)s(e)g (symmetry)f(in)635 686 y(the)e(large)f(deviation)g(functional)f(for)h (sto)s(c)m(hastic)i(dynamics.)48 b(J.)34 b(Stat.)g(Ph)m(ys.,)635 799 y Fg(95)p Fu(,)d(333{365)j(\(1999\).)448 986 y([19])47 b(E.)32 b(B.)h(Lee)g(and)e(L.)h(Markus,)46 b Fe(F)-7 b(oundations)37 b(of)d(optimal)j(c)-5 b(ontr)g(ol)36 b(the)-5 b(ory)p Fu(.)48 b(The)635 1099 y(SIAM)30 b(ser.)h(in)e(Appl.)g (Math.,)i(Wiley)f(\(1967\).)448 1287 y([20])47 b(C.)30 b(Maes,)40 b(The)29 b(\015uctuation)g(theorem)h(as)g(a)g(Gibbs)e(prop)s (ert)m(y)-8 b(.)39 b(J.)29 b(Stat.)i(Ph)m(ys.,)635 1400 y Fg(95)p Fu(,)g(367{392)j(\(1999\).)448 1587 y([21])47 b(L.)30 b(Rey-Bellet)h(and)e(L.)i(E.)f(Thomas,)39 b(Energy)30 b(deca)m(y)i(estimates)e(for)g(Hamilto-)635 1700 y(nian)f(systems)i (coupled)e(to)i(heat)g(reserv)m(oirs.)40 b(In)30 b(preparation.)448 1888 y([22])47 b(M.)38 b(Sc)m(hilder,)63 b(Some)37 b(asymptotic)h(form) m(ulae)g(for)f(Wiener)g(in)m(tegrals.)62 b(T)-8 b(rans.)635 2001 y(Amer.)31 b(Math.)g(So)s(c.)f Fg(125)p Fu(,)i(63{85)g(\(1966\).) 448 2188 y([23])47 b(H.)36 b(J.)g(Sussmann,)55 b(Lie)35 b(brac)m(k)m(ets,)j(real)e(analyticit)m(y)f(and)g(geometric)h(con)m (trol.)635 2301 y(In)28 b Fe(Di\013er)-5 b(ential)32 b(Ge)-5 b(ometric)33 b(Contr)-5 b(ol)33 b(The)-5 b(ory,)33 b(Pr)-5 b(o)g(c.)32 b(c)-5 b(onf.)32 b(Michigan)p Fu(,)d(1{116)635 2414 y(Birk\177)-45 b(auser,)30 b(\(1983\).)448 2602 y([24])47 b(D.W.)36 b(Stro)s(o)s(c)m(k)e(and)g(S.R.S.)g(V)-8 b(aradhan,)53 b(On)33 b(the)i(supp)s(ort)d(of)j(di\013usion)c(pro-)635 2715 y(cesses)26 b(with)d(applications)g(to)j(the)f(strong)g(maxim)m (um)f(principle.)k(In)c Fe(Pr)-5 b(o)g(c.)28 b(6-th)635 2828 y(Berkeley)33 b(Symp.)g(Math.)g(Stat.)h(Pr)-5 b(ob.)p Fu(,)31 b(v)m(ol)f(I)s(I)s(I,)f(333{368)34 b(\(1972\).)448 3015 y([25])47 b(S.R.S.)30 b(V)-8 b(aradhan,)42 b Fe(L)-5 b(ar)g(ge)34 b(Deviations)f(and)g(Applic)-5 b(ations)p Fu(.)44 b(SIAM,)30 b(Philadel-)635 3128 y(phia,)f(\(1984\).)1897 5225 y(40)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF 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