Content-Type: multipart/mixed; boundary="-------------0203301608631" This is a multi-part message in MIME format. ---------------0203301608631 Content-Type: text/plain; name="02-160.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-160.comments" AMS-Code: 37H20, 34E15 (primary), 60H10 (secondary) ---------------0203301608631 Content-Type: text/plain; name="02-160.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-160.keywords" Singular perturbations, slow-fast systems, invariant manifolds, dynamic bifurcations, stochastic differential equations, first-exit times, concentration of measure. ---------------0203301608631 Content-Type: application/postscript; name="geom.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="geom.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: geom.dvi %%Pages: 41 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments 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5871 y(4)p eop %%Page: 5 5 5 4 bop 118 280 a FB(2)135 b(Results)118 487 y Fp(2.1)112 b(Preliminaries)118 658 y FA(Let)28 b Fx(D)j FA(b)s(e)c(an)g(op)s(en)g (subset)h(of)f Fr(R)1292 625 y Fw(n)1360 658 y Fx(\002)15 b Fr(R)1521 625 y Fw(m)1621 658 y FA(and)27 b Fz(")1837 672 y Fs(0)1904 658 y FA(a)i(constan)m(t.)41 b(W)-8 b(e)29 b(consider)d(slo)m(w{fast)j(sto)s(c)m(hastic)118 771 y(di\013eren)m(tial)g(equations)h(of)h(the)f(form)1041 1009 y(d)o Fz(x)1143 1023 y Fw(t)1198 1009 y FA(=)1304 947 y(1)p 1304 988 46 4 v 1306 1071 a Fz(")1360 1009 y(f)10 b FA(\()p Fz(x)1502 1023 y Fw(t)1531 1009 y Fz(;)15 b(y)1616 1023 y Fw(t)1646 1009 y Fz(;)g(")p FA(\))g(d)p Fz(t)20 b FA(+)2015 947 y Fz(\033)p 1983 988 119 4 v 1983 1006 a Fx(p)p 2059 1006 43 4 v 65 x Fz(")2111 1009 y(F)13 b FA(\()p Fz(x)2269 1023 y Fw(t)2299 1009 y Fz(;)i(y)2384 1023 y Fw(t)2414 1009 y Fz(;)g(")p FA(\))g(d)q Fz(W)2684 1023 y Fw(t)2713 1009 y Fz(;)1048 1205 y FA(d)p Fz(y)1144 1219 y Fw(t)1198 1205 y FA(=)25 b Fz(g)s FA(\()p 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b(and)e Fz(G)d Fx(2)f(C)2040 1495 y Fs(1)2080 1528 y FA(\()p Fx(D)e(\002)d FA([0)p Fz(;)15 b(")2451 1542 y Fs(0)2492 1528 y FA(\))p Fz(;)g Fr(R)2642 1495 y Fw(m)q Fv(\002)p Fw(k)2808 1528 y FA(\).)259 1640 y(W)-8 b(e)31 b(require)d(that)j Fz(f)10 b FA(,)29 b Fz(g)s FA(,)h(and)f(all)g(their)g(deriv)-5 b(ativ)m(es)29 b(up)f(to)i(order)g(2)g(are)g(uniformly)c(b)s(ounded)i(in)118 1753 y(norm)j(in)g Fx(D)24 b(\002)d FA([0)p Fz(;)15 b(")803 1767 y Fs(0)844 1753 y FA(\),)32 b(and)g(similarly)c(for)k Fz(F)13 b FA(,)33 b Fz(G)f FA(and)f(their)g(deriv)-5 b(ativ)m(es.)45 b(W)-8 b(e)33 b(also)f(assume)g(that)g Fz(f)118 1866 y FA(and)e Fz(g)k FA(satisfy)c(the)h(usual)e(\(lo)s (cal\))i(Lipsc)m(hitz)e(and)h(b)s(ounded-gro)m(wth)f(conditions)g(whic) m(h)h(guaran)m(tee)118 1979 y(existence)h(and)f(path)m(wise)g (uniqueness)e(of)i(a)h(strong)f(solution)f Fx(f)p FA(\()p Fz(x)2450 1993 y Fw(t)2481 1979 y Fz(;)15 b(y)2566 1993 y Fw(t)2595 1979 y FA(\))p Fx(g)2675 1993 y Fw(t)p Fo(>)p Fw(t)2780 2002 y Ft(0)2851 1979 y FA(of)38 b(\(2.1\))r(.)259 2092 y(The)26 b(sto)s(c)m(hastic)h(pro)s(cess)e Fx(f)p Fz(W)1301 2106 y Fw(t)1331 2092 y Fx(g)1376 2106 y Fw(t)p Fo(>)p Fs(0)1522 2092 y FA(is)g(a)h(standard)g Fz(k)s FA(-dimensional)d(Bro)m(wnian)i(motion)h(on)g(some)118 2205 y(probabilit)m(y)34 b(space)i(\(\012)p Fz(;)15 b Fx(F)9 b Fz(;)15 b Fr(P)p FA(\).)60 b(Initial)33 b(conditions)i(\()p Fz(x)2070 2219 y Fs(0)2110 2205 y Fz(;)15 b(y)2195 2219 y Fs(0)2234 2205 y FA(\))36 b(are)h(alw)m(a)m(ys)f(assumed)f(to)i(b)s (e)e(square-)118 2318 y(in)m(tegrable)g(with)f(resp)s(ect)i(to)g Fr(P)g FA(and)e(indep)s(enden)m(t)g(of)h Fx(f)p Fz(W)2222 2332 y Fw(t)2252 2318 y Fx(g)2297 2332 y Fw(t)p Fo(>)p Fs(0)2417 2318 y FA(.)56 b(Our)34 b(assumptions)g(on)h Fz(f)45 b FA(and)35 b Fz(g)118 2431 y FA(guaran)m(tee)29 b(the)f(existence)g(of)f(a)h(con)m(tin)m(uous)f(v)m(ersion)g(of)g Fx(f)p FA(\()p Fz(x)2233 2445 y Fw(t)2264 2431 y Fz(;)15 b(y)2349 2445 y Fw(t)2378 2431 y FA(\))p Fx(g)2458 2445 y Fw(t)p Fo(>)p 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FA(\()6 b(\026)-51 b Fz(x)p FA(\()p Fz(y)s(;)15 b(")p FA(\))p Fz(;)g(y)s FA(\))10 b(:)35 b Fz(y)i Fx(2)e(D)3058 3771 y Fs(0)3097 3757 y Fx(g)i FA(an)f Fy(adiab)-5 b(atic)118 3870 y(manifold)p FA(.)43 b(It)30 b(is)g(easy)h(to)g(see)g(from)e (\(2.3\))k(that)j(\026)-50 b Fz(x)p FA(\()p Fz(y)s(;)15 b(")p FA(\))31 b(m)m(ust)f(satisfy)g(the)h(PDE)1078 4071 y Fz("@)1168 4085 y Fw(y)1216 4071 y FA(\026)-51 b Fz(x)p FA(\()p Fz(y)s(;)15 b(")p FA(\))p Fz(g)s FA(\()6 b(\026)-51 b Fz(x)s FA(\()p Fz(y)s(;)15 b(")p FA(\))p Fz(;)g(y)s(;)g(")p FA(\))28 b(=)d Fz(f)10 b FA(\()c(\026)-51 b Fz(x)o FA(\()p Fz(y)s(;)15 b(")p FA(\))p Fz(;)g(y)s(;)g(")p FA(\))p Fz(:)777 b FA(\(2.5\))118 4272 y(The)39 b(lo)s(cal)f(existence)i(of)g (the)f(adiabatic)g(manifold)e(follo)m(ws)i(directly)f(from)h(the)g(cen) m(tre)h(manifold)118 4385 y(theorem.)h(Indeed,)30 b(w)m(e)h(can)f (rewrite)g(System)g(\(2.3\))i(in)d(the)i(form)1588 4583 y Fz(x)1640 4546 y Fv(0)1688 4583 y FA(=)25 b Fz(f)10 b FA(\()p Fz(x;)15 b(y)s(;)g(")p FA(\))p Fz(;)1592 4721 y(y)1640 4684 y Fv(0)1688 4721 y FA(=)25 b Fz("g)s FA(\()p Fz(x;)15 b(y)s(;)g(")p FA(\))p Fz(;)1597 4859 y(")1639 4821 y Fv(0)1688 4859 y FA(=)25 b(0)p Fz(;)3475 4722 y FA(\(2.6\))118 5058 y(where)32 b(prime)f(denotes)i(deriv)-5 b(ation)31 b(with)g(resp)s(ect)i(to)g(the)g(fast)g(time)f Fz(t=")p FA(.)47 b(An)m(y)33 b(p)s(oin)m(t)e(of)i(the)g(form)118 5171 y(\()p Fz(x)205 5138 y Fw(?)245 5171 y FA(\()p Fz(y)s FA(\))p Fz(;)15 b(y)s(;)g FA(0\))27 b(with)d Fz(y)k Fx(2)d(D)1029 5185 y Fs(0)1094 5171 y FA(is)f(an)i(equilibrium)20 b(p)s(oin)m(t)25 b(of)32 b(\(2.6\))r(.)39 b(The)25 b(linearization)f(of)32 b(\(2.6\))27 b(around)118 5283 y(suc)m(h)34 b(a)g(p)s(oin)m(t)f(admits) h(0)g(as)h(eigen)m(v)-5 b(alue)34 b(of)g(m)m(ultiplicit)m(y)d Fz(m)23 b FA(+)f(1,)36 b(the)e Fz(n)f FA(other)i(eigen)m(v)-5 b(alues)34 b(b)s(eing)118 5396 y(those)j(of)f Fz(A)539 5363 y Fw(?)579 5396 y FA(\()p Fz(y)s FA(\),)i(whic)m(h)d(are)h(b)s (ounded)e(a)m(w)m(a)m(y)k(from)e(the)g(imaginary)f(axis.)58 b(The)35 b(cen)m(tre)i(manifold)118 5509 y(theorem)d(implies)d(the)k (existence)f(of)g(a)g(lo)s(cal)f(in)m(v)-5 b(arian)m(t)33 b(manifold)f Fz(x)f FA(=)37 b(\026)-51 b Fz(x)p FA(\()p Fz(y)s(;)15 b(")p FA(\).)53 b(F)-8 b(enic)m(hel's)33 b(result)118 5622 y(sho)m(ws)d(that)h(this)e(manifold)g(actually)h (exists)g(for)g(all)f Fz(y)f Fx(2)d(D)2221 5636 y Fs(0)2261 5622 y FA(.)1867 5871 y(6)p eop %%Page: 7 7 7 6 bop 259 280 a FA(Being)35 b(a)g(cen)m(tre)g(manifold,)f(the)g (adiabatic)h(manifold)d(is)h(not)i(necessarily)f(unique)e(\(though)j (in)118 393 y(the)f(presen)m(t)g(case,)41 b(\026)-51 b Fz(x)p FA(\()p Fz(y)s(;)15 b FA(0\))32 b(=)f Fz(x)1260 360 y Fw(?)1299 393 y FA(\()p Fz(y)s FA(\))j(is)f(uniquely)e (de\014ned\).)50 b(Nev)m(ertheless,)41 b(\026)-51 b Fz(x)p FA(\()p Fz(y)s(;)15 b(")p FA(\))35 b(has)e(a)h(unique)118 506 y(T)-8 b(a)m(ylor)41 b(series)f(in)f Fz(y)k FA(and)d Fz(")p FA(,)k(whic)m(h)39 b(can)i(b)s(e)f(obtained)g(b)m(y)h(solving)f (\(2.5\))i(order)f(b)m(y)f(order.)71 b(The)118 619 y(dynamics)29 b(on)h(the)h(adiabatic)f(manifold)e(is)i(describ)s(ed)e(b)m(y)i(the)h (so-called)f Fy(r)-5 b(e)g(duc)g(e)g(d)34 b(e)-5 b(quation)1077 823 y FA(_)-41 b Fz(y)28 b FA(=)d Fz(g)s FA(\()6 b(\026)-51 b Fz(x)q FA(\()p Fz(y)s(;)15 b(")p FA(\))p Fz(;)g(y)s(;)g(")p FA(\))28 b(=)d Fz(g)s FA(\()p Fz(x)2026 786 y Fw(?)2066 823 y FA(\()p Fz(y)s FA(\))p Fz(;)15 b(y)s(;)g FA(0\))22 b(+)e Fx(O)s FA(\()p Fz(")p FA(\))p Fz(:)758 b FA(\(2.7\))118 1028 y(If)29 b Fz(x)260 995 y Fw(?)300 1028 y FA(\()p Fz(y)s FA(\))h(is)f(uniformly)e(asymptotically)i(stable,)36 b(\026)-51 b Fz(x)p FA(\()p Fz(y)s(;)15 b(")p FA(\))31 b(is)e(lo)s(cally)f(attractiv)m(e)k(and)d(th)m(us)h(an)m(y)g(solu-)118 1141 y(tion)e(of)36 b(\(2.3\))31 b(starting)d(su\016cien)m(tly)g(close) h(to)35 b(\026)-51 b Fz(x)p FA(\()p Fz(y)s(;)15 b(")p FA(\))30 b(con)m(v)m(erges)h(exp)s(onen)m(tially)c(fast)i(to)h(a)f (solution)118 1253 y(of)38 b(\(2.7\))r(.)118 1497 y Fp(2.3)112 b(Random)38 b(stable)f(case)118 1668 y FA(W)-8 b(e)29 b(turn)e(no)m(w)h(to)g(the)g(random)f(slo)m(w{fast)h(system)g(giv)m(en) g(b)m(y)g(the)g(sto)s(c)m(hastic)g(di\013eren)m(tial)f(equation)1041 1906 y(d)o Fz(x)1143 1920 y Fw(t)1198 1906 y FA(=)1304 1844 y(1)p 1304 1885 46 4 v 1306 1968 a Fz(")1360 1906 y(f)10 b FA(\()p Fz(x)1502 1920 y Fw(t)1531 1906 y Fz(;)15 b(y)1616 1920 y Fw(t)1646 1906 y Fz(;)g(")p FA(\))g(d)p Fz(t)20 b FA(+)2015 1844 y Fz(\033)p 1983 1885 119 4 v 1983 1903 a Fx(p)p 2059 1903 43 4 v 66 x Fz(")2111 1906 y(F)13 b FA(\()p Fz(x)2269 1920 y Fw(t)2299 1906 y Fz(;)i(y)2384 1920 y Fw(t)2414 1906 y Fz(;)g(")p FA(\))g(d)q Fz(W)2684 1920 y Fw(t)2713 1906 y Fz(;)1048 2103 y FA(d)p Fz(y)1144 2117 y Fw(t)1198 2103 y FA(=)25 b Fz(g)s FA(\()p Fz(x)1427 2117 y Fw(t)1458 2103 y Fz(;)15 b(y)1543 2117 y Fw(t)1572 2103 y Fz(;)g(")p FA(\))g(d)q Fz(t)20 b FA(+)g Fz(\033)1955 2065 y Fv(0)1979 2103 y Fz(G)p FA(\()p Fz(x)2137 2117 y Fw(t)2167 2103 y Fz(;)15 b(y)2252 2117 y Fw(t)2282 2103 y Fz(;)g(")p FA(\))g(d)q Fz(W)2552 2117 y Fw(t)2581 2103 y Fz(;)3475 1984 y FA(\(2.8\))118 2299 y(where)30 b(w)m(e)h(will)d(assume)i(the)g(follo)m(wing.)118 2487 y Fm(Assumption)38 b(2.2.)43 b FA(F)-8 b(or)34 b Fz(\033)e FA(=)d Fz(\033)1311 2454 y Fv(0)1364 2487 y FA(=)g(0,)34 b(System)f(\(2.8\))i(admits)c(a)j(uniformly)c(h)m(yp)s(erb)s(olic,)h (asymp-)118 2600 y(totically)f(stable)g(slo)m(w)g(manifold)f Fz(x)c FA(=)g Fz(x)1539 2567 y Fw(?)1578 2600 y FA(\()p Fz(y)s FA(\),)31 b Fz(y)d Fx(2)d(D)1981 2614 y Fs(0)2021 2600 y FA(.)259 2787 y(By)36 b(F)-8 b(enic)m(hel's)35 b(theorem,)h(there)g(exists)e(an)h(adiabatic)g(manifold)e Fz(x)g FA(=)38 b(\026)-51 b Fz(x)p FA(\()p Fz(y)s(;)15 b(")p FA(\))37 b(with)i(\026)-51 b Fz(x)p FA(\()p Fz(y)s(;)15 b FA(0\))35 b(=)118 2900 y Fz(x)170 2867 y Fw(?)210 2900 y FA(\()p Fz(y)s FA(\),)40 b Fz(y)h Fx(2)c(D)647 2914 y Fs(0)687 2900 y FA(.)63 b(W)-8 b(e)39 b(\014x)f(a)g(particular)e (solution)h(\()p Fz(x)2025 2867 y Fs(det)2025 2923 y Fw(t)2127 2900 y Fz(;)15 b(y)2215 2867 y Fs(det)2212 2923 y Fw(t)2317 2900 y FA(\))39 b(=)e(\()6 b(\026)-51 b Fz(x)q FA(\()p Fz(y)2670 2867 y Fs(det)2667 2923 y Fw(t)2772 2900 y Fz(;)15 b(")p FA(\))p Fz(;)g(y)2977 2867 y Fs(det)2974 2923 y Fw(t)3080 2900 y FA(\))39 b(of)f(the)g (deter-)118 3013 y(ministic)28 b(system.)41 b(\(That)31 b(is,)f Fz(y)1227 2980 y Fs(det)1224 3036 y Fw(t)1360 3013 y FA(satis\014es)g(the)g(reduced)g(equation)g(\(2.7\))r(.\))42 b(W)-8 b(e)31 b(w)m(an)m(t)h(to)f(describ)s(e)118 3126 y(the)39 b(noise-induced)d(deviations)h(of)i(the)g(sample)e(paths)h(\() p Fz(x)2228 3140 y Fw(t)2258 3126 y Fz(;)15 b(y)2343 3140 y Fw(t)2373 3126 y FA(\))2408 3140 y Fw(t)p Fo(>)p Fs(0)2566 3126 y FA(of)46 b(\(2.8\))40 b(from)e(the)h(adiabatic)118 3239 y(manifold.)259 3352 y(It)31 b(turns)e(out)h(to)i(b)s(e)d(con)m(v) m(enien)m(t)j(to)f(use)f(the)h(transformation)1415 3535 y Fz(x)1467 3549 y Fw(t)1522 3535 y FA(=)f(\026)-50 b Fz(x)p FA(\()p Fz(y)1753 3498 y Fs(det)1750 3558 y Fw(t)1875 3535 y FA(+)20 b Fz(\021)2011 3549 y Fw(t)2041 3535 y Fz(;)15 b(")p FA(\))21 b(+)f Fz(\030)2310 3549 y Fw(t)2339 3535 y Fz(;)1422 3686 y(y)1467 3700 y Fw(t)1522 3686 y FA(=)25 b Fz(y)1666 3648 y Fs(det)1663 3709 y Fw(t)1788 3686 y FA(+)20 b Fz(\021)1924 3700 y Fw(t)1953 3686 y Fz(;)3475 3609 y FA(\(2.9\))118 3883 y(whic)m(h)29 b(yields)g(a)i (system)f(of)h(the)f(form)988 4120 y(d)p Fz(\030)1079 4134 y Fw(t)1133 4120 y FA(=)1239 4059 y(1)p 1239 4099 46 4 v 1241 4182 a Fz(")1314 4096 y FA(^)1295 4120 y Fz(f)9 b FA(\()p Fz(\030)1424 4134 y Fw(t)1454 4120 y Fz(;)15 b(\021)1539 4134 y Fw(t)1569 4120 y Fz(;)g(t;)g(")p FA(\))g(d)q Fz(t)20 b FA(+)2011 4059 y Fz(\033)p 1980 4099 119 4 v 1980 4117 a Fx(p)p 2056 4117 43 4 v 66 x Fz(")2126 4097 y Fq(b)2108 4120 y Fz(F)13 b FA(\()p Fz(\030)2254 4134 y Fw(t)2284 4120 y Fz(;)i(\021)2369 4134 y Fw(t)2399 4120 y Fz(;)g(t;)g(")p FA(\))g(d)q Fz(W)2742 4134 y Fw(t)2771 4120 y Fz(;)983 4326 y FA(d)o Fz(\021)1078 4340 y 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2272 y Fs(0)1298 2258 y FA(\()p Fz(y)s(;)15 b FA(0\))1501 2221 y Fw(T)1572 2258 y FA(e)1613 2221 y Fw(sA)1699 2197 y Fl(?)1734 2221 y Fs(\()p Fw(y)r Fs(\))1825 2197 y Fl(T)1894 2258 y Fz(x)25 b FA(=)g(0)183 b Fx(8)p Fz(s)24 b Fn(>)h FA(0)p Fz(;)850 b FA(\(2.22\))118 2409 y(whic)m(h)29 b(o)s(ccurs)h(if)g(and)f(only)h(if)1021 2581 y Fz(F)1079 2595 y Fs(0)1119 2581 y FA(\()p Fz(y)s(;)15 b FA(0\))1322 2543 y Fw(T)1378 2581 y FA(\()p Fz(A)1481 2543 y Fw(?)1521 2581 y FA(\()p Fz(y)s FA(\))1639 2543 y Fw(T)1695 2581 y FA(\))1730 2543 y Fw(k)1773 2581 y Fz(x)25 b FA(=)g(0)182 b Fx(8)p Fz(k)28 b FA(=)d(0)p Fz(;)15 b FA(1)p Fz(;)g FA(2)p Fz(;)g(:)g(:)g(:)674 b FA(\(2.23\))118 2753 y(i.)15 b(e.,)31 b(when)e(the)i(k)m(ernel)f(of)g Fz(F)1128 2767 y Fs(0)1168 2753 y FA(\()p Fz(y)s(;)15 b FA(0\))1371 2720 y Fw(T)1458 2753 y FA(is)29 b(in)m(v)-5 b(arian)m(t)30 b(under)f Fz(A)2257 2720 y Fw(?)2296 2753 y FA(\()p Fz(y)s FA(\))2414 2720 y Fw(T)2470 2753 y FA(.)118 2913 y Fm(Theorem)48 b(2.4.)h 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Fz(\030)t(;)p 1911 1665 55 4 v 15 w(X)7 b FA(\()p Fz(y)s(;)15 b(")p FA(\))2169 1705 y Fv(\000)p Fs(1)2266 1738 y Fz(\030)t Fx(i)34 b Fz(<)f(h)2535 1705 y Fs(2)2575 1738 y Fx(g)i FA(than)h(through)e(the)i(\\sides")118 1851 y Fx(f)p Fz(y)44 b Fx(2)c FA(in)m(t)14 b Fx(D)546 1865 y Fs(0)586 1851 y Fz(;)h Fx(h)p Fz(\030)t(;)p 769 1778 V 15 w(X)7 b FA(\()p Fz(y)s(;)15 b(")p FA(\))1027 1818 y Fv(\000)p Fs(1)1124 1851 y Fz(\030)t Fx(i)41 b FA(=)f Fz(h)1407 1818 y Fs(2)1447 1851 y Fx(g)p FA(,)i(unless)c(w)m(e)i (w)m(ait)f(for)g(time)g(spans)g(exp)s(onen)m(tially)f(long)h(in)118 1963 y Fz(h)170 1930 y Fs(2)210 1963 y Fz(=\033)310 1930 y Fs(2)350 1963 y FA(.)h(Belo)m(w)28 b(w)m(e)h(discuss)d(ho)m(w)i(to)h (c)m(haracterize)h Fz(\034)1952 1977 y Fv(D)2007 1986 y Ft(0)2073 1963 y FA(more)e(precisely)-8 b(,)27 b(using)g(information) f(on)i(the)118 2076 y(reduced)34 b(dynamics)f(on)i(the)g(adiabatic)f (manifold.)51 b(If,)36 b(for)e(instance,)i(all)d(deterministic)g (solutions)118 2189 y(starting)d(in)f Fx(D)633 2203 y Fs(0)703 2189 y FA(remain)g(in)h(this)f(set,)i Fz(\034)1497 2203 y Fv(D)1552 2212 y Ft(0)1620 2189 y FA(will)d(t)m(ypically)h(b)s (e)h(v)m(ery)h(large.)259 2302 y(The)39 b(upp)s(er)e(b)s(ound)g (\(2.25\))k(has)e(b)s(een)g(designed)f(to)h(yield)f(the)h(b)s(est)g(p)s (ossible)e(exp)s(onen)m(t)i Fz(\024)3577 2269 y Fs(+)3636 2302 y FA(,)118 2415 y(while)h(the)j(prefactor)g Fx(C)988 2377 y Fs(+)983 2444 y Fw(n;m;\015)t(;)p Fs(\001)1292 2415 y FA(is)e(certainly)h(not)g(optimal.)76 b(Note)43 b(that)g(an)f(estimate)h(with)e(the)118 2528 y(same)35 b(exp)s(onen)m(t,)g(but)e(with)g(a)h(smaller)f(prefactor)h(holds)f(for) g(the)i(probabilit)m(y)c(that)k(the)f(endp)s(oin)m(t)118 2641 y(\()p Fz(x)205 2655 y Fw(t)235 2641 y Fz(;)15 b(y)320 2655 y Fw(t)350 2641 y FA(\))35 b(do)s(es)g(not)g(lie)f(in)f Fx(B)s FA(\()p Fz(h)p FA(\),)k(cf.)f(Corollary)d(3.10.)56 b(The)35 b(parameters)g(\001)g(and)f Fz(\015)40 b FA(can)35 b(b)s(e)g(c)m(hosen)118 2754 y(arbitrarily)24 b(within)f(their)j(in)m (terv)-5 b(als)25 b(of)h(de\014nition.)37 b(T)-8 b(aking)26 b(\001)g(small)f(and)g Fz(\015)32 b FA(close)26 b(to)h(1)p Fz(=)p FA(2)h(impro)m(v)m(es)118 2867 y(the)g(exp)s(onen)m(t)g(while)e (increasing)g(the)i(prefactor.)40 b(A)28 b(con)m(v)m(enien)m(t)h(c)m (hoice)f(is)f(to)i(tak)m(e)g(\001)e(and)g(1)p Fz(=)p FA(2)15 b Fx(\000)g Fz(\015)118 2980 y FA(of)34 b(order)f Fz(h)g FA(or)h Fz(")p FA(.)50 b(The)33 b(kind)e(of)j(time-dep)s (endence)e(of)i Fx(C)k FA(is)33 b(probably)e(not)j(optimal,)f(but)g (the)g(fact)118 3093 y(that)e Fx(C)k FA(increases)29 b(with)g(time)g(is)g(to)i(b)s(e)e(exp)s(ected,)i(since)e(it)g (re\015ects)h(the)h(fact)f(that)h(the)f(probabilit)m(y)118 3205 y(of)40 b(observing)e(paths)h(making)g(excursions)f(a)m(w)m(a)m(y) k(from)d(the)g(adiabatic)g(manifold)f(increases)h(with)118 3318 y(time.)i(As)30 b(for)g(the)h(dep)s(endence)e(of)h(the)h (prefactor)g(on)f(the)h(dimensions)c Fz(n)j FA(and)g Fz(m)p FA(,)g(it)g(is)f(due)h(to)h(the)118 3431 y(fact)37 b(that)f(the)f(tails)g(of)g(standard)g(Gaussian)g(random)f(v)-5 b(ariables)35 b(sho)m(w)g(their)g(t)m(ypical)g(deca)m(y)h(only)118 3544 y(outside)30 b(a)h(ball)d(of)j(radius)d(scaling)i(with)f(the)h (square-ro)s(ot)h(of)g(the)f(dimension.)259 3657 y(The)35 b(upp)s(er)e(b)s(ound)h(\(2.25\))j(and)e(lo)m(w)m(er)h(b)s(ound)d (\(2.28\))38 b(together)f(sho)m(w)e(that)h(the)f(exp)s(onen)m(tial)118 3770 y(rate)e(of)f(deca)m(y)i(of)e(the)g(probabilit)m(y)e(to)j(lea)m(v) m(e)g(the)g(set)f Fx(B)s FA(\()p Fz(h)p FA(\))h(b)s(efore)f(time)f Fz(t)h FA(b)s(eha)m(v)m(es)h(lik)m(e)e Fz(h)3365 3737 y Fs(2)3405 3770 y Fz(=)p FA(\(2)p Fz(\033)3585 3737 y Fs(2)3626 3770 y FA(\))118 3883 y(in)41 b(the)h(limit)d(of)j Fz(\033)s FA(,)j Fz(")d FA(and)f Fz(h)h FA(going)g(to)h(zero,)i(as)d (one)g(w)m(ould)f(exp)s(ect)h(from)g(other)g(approac)m(hes,)118 3996 y(based)31 b(for)h(instance)f(on)g(the)h(theory)g(of)f(large)h (deviations.)43 b(The)31 b(b)s(ounds)e(hold,)i(ho)m(w)m(ev)m(er,)j(in)c (a)i(full)118 4109 y(neigh)m(b)s(ourho)s(o)s(d)c(of)i Fz(\033)f FA(=)c Fz(")g FA(=)g Fz(h)h FA(=)f(0.)259 4222 y(Finally)-8 b(,)43 b(Estimate)e(\(2.31\))i(allo)m(ws)e(to)g(extend)g (these)h(results)e(to)i(all)d(initial)g(conditions)g(in)h(a)118 4335 y(neigh)m(b)s(ourho)s(o)s(d)30 b(of)k(order)e(1)i(of)f(the)g (adiabatic)g(manifold.)47 b(The)33 b(only)f(di\013erence)h(is)f(that)i (w)m(e)f(ha)m(v)m(e)118 4447 y(to)g(w)m(ait)f(for)f(a)i(time)e(of)h (order)g Fz(")p Fx(j)p FA(log)17 b Fz(h)p Fx(j)32 b FA(b)s(efore)g(the) g(path)f(is)g(lik)m(ely)g(to)h(ha)m(v)m(e)i(reac)m(hed)e(the)g(set)h Fx(B)s FA(\()p Fz(h)p FA(\).)118 4560 y(After)e(this)e(time,)h(t)m (ypical)g(paths)g(b)s(eha)m(v)m(e)h(as)g(if)e(they)i(had)f(started)g (on)h(the)f(adiabatic)g(manifold.)259 4673 y(W)-8 b(e)34 b(remark)e(in)f(passing)h(that)h(the)g(assumption)e(that)i Fx(k)p 2269 4600 V Fz(X)7 b FA(\()p Fz(y)s(;)15 b(")p FA(\))2528 4640 y Fv(\000)p Fs(1)2624 4673 y Fx(k)33 b FA(is)f(uniformly)d(b)s(ounded)i(in)118 4786 y Fx(D)188 4800 y Fs(0)258 4786 y FA(excludes)f(purely)e(m)m(ultiplicativ)m(e)g (noise.)259 4899 y(The)39 b(b)s(eha)m(viour)e(of)j(t)m(ypical)e(paths)h (dep)s(ends)e(essen)m(tially)h(on)h(the)g(dynamics)f(of)h(the)g (reduced)118 5012 y(deterministic)32 b(system)i(\(2.7\))r(.)52 b(In)33 b(fact,)j(in)d(the)h(pro)s(of)f(of)h(Theorem)g(2.4,)i(w)m(e)f (use)e(the)i(fact)g(that)f Fz(y)3632 5026 y Fw(t)118 5125 y FA(do)s(es)g(not)g(di\013er)e(to)s(o)j(m)m(uc)m(h)f(from)f Fz(y)1405 5092 y Fs(det)1402 5147 y Fw(t)1541 5125 y FA(on)h(timescales)g(of)g(order)f(1)h(\(see)h(Lemma)f(3.4\).)53 b(There)33 b(are)118 5238 y(th)m(us)d(t)m(w)m(o)i(main)d(p)s (ossibilities)d(to)31 b(b)s(e)f(considered:)181 5396 y Fx(\017)48 b FA(either)41 b(the)h(reduced)f(\015o)m(w)h(is)f(suc)m(h) h(that)g Fz(y)1838 5363 y Fs(det)1835 5419 y Fw(t)1982 5396 y FA(reac)m(hes)h(the)f(b)s(oundary)e(of)i Fx(D)3084 5410 y Fs(0)3165 5396 y FA(in)e(a)j(time)e(of)274 5509 y(order)30 b(1)g(\(for)g(instance,)h Fz(y)1189 5476 y Fs(det)1186 5532 y Fw(t)1321 5509 y FA(ma)m(y)f(approac)m(h)h(a)g (bifurcation)d(set)i(of)h(the)f(slo)m(w)g(manifold\);)f(then)274 5622 y Fz(y)319 5636 y Fw(t)378 5622 y FA(is)h(lik)m(ely)f(to)i(lea)m (v)m(e)h Fx(D)1116 5636 y Fs(0)1185 5622 y FA(as)f(w)m(ell;)1844 5871 y(10)p eop %%Page: 11 11 11 10 bop 181 280 a Fx(\017)48 b FA(or)34 b(the)g(reduced)g(\015o)m(w)g (is)f(suc)m(h)h(that)h Fz(y)1638 247 y Fs(det)1635 303 y Fw(t)1774 280 y FA(remains)e(in)g Fx(D)2297 294 y Fs(0)2370 280 y FA(for)h(all)f(times)h Fz(t)d Fn(>)h FA(0;)k(in)d(that)i(case,) 274 393 y(paths)26 b(can)g(only)f(lea)m(v)m(e)i Fx(B)s FA(\()p Fz(h)p FA(\))g(due)e(to)i(the)f(in\015uence)e(of)j(noise,)f (whic)m(h)f(w)m(e)h(exp)s(ect)h(to)f(b)s(e)g(unlik)m(ely)274 506 y(on)k(sub)s(exp)s(onen)m(tial)e(timescales.)259 668 y(W)-8 b(e)30 b(will)c(discuss)h(the)i(\014rst)f(situation)g(in)f (more)i(detail)f(in)g(Subsection)f(2.4.)41 b(In)28 b(b)s(oth)g (situations,)118 781 y(it)f(is)g(desirable)f(to)j(ha)m(v)m(e)g(a)f 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Fq(b)937 1898 y Fz(G)q FA(\(0)p Fz(;)15 b FA(0)p Fz(;)g(t;)g(")p FA(\))36 b(=)d Fz(G)p FA(\()6 b(\026)-51 b Fz(x)q FA(\()p Fz(y)1746 1865 y Fs(det)1743 1921 y Fw(t)1848 1898 y Fz(;)15 b(")p FA(\))p Fz(;)g(y)2053 1865 y Fs(det)2050 1921 y Fw(t)2156 1898 y Fz(;)g(")p FA(\))37 b(and)d(the)h(Jacobian)g(matrices)g Fz(B)40 b FA(and)118 2011 y Fz(C)d FA(are)31 b(giv)m(en)f(b)m(y)745 2215 y Fz(B)5 b FA(\()p Fz(y)902 2178 y Fs(det)899 2238 y Fw(t)1003 2215 y Fz(;)15 b(")p FA(\))27 b(=)e Fz(@)1291 2229 y Fw(\021)1336 2215 y FA(^)-48 b Fz(g)t FA(\(0)p Fz(;)15 b FA(0)p Fz(;)g(t;)g(")p FA(\))1147 2366 y(=)25 b Fz(C)7 b FA(\()p Fz(y)1398 2329 y Fs(det)1395 2389 y Fw(t)1500 2366 y Fz(;)15 b(")p FA(\))p Fz(@)1665 2380 y Fw(y)1713 2366 y FA(\026)-51 b Fz(x)p FA(\()p Fz(y)1842 2329 y Fs(det)1839 2389 y Fw(t)1945 2366 y Fz(;)15 b(")p FA(\))21 b(+)f Fz(@)2222 2380 y Fw(y)2264 2366 y Fz(g)s FA(\()6 b(\026)-51 b Fz(x)q FA(\()p Fz(y)2481 2329 y Fs(det)2478 2389 y Fw(t)2583 2366 y Fz(;)15 b(")p FA(\))p 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b(chosen)g(in)g(such)f(a)h(way)h(that)g Fx(k)p Fz(Y)20 b FA(\()p Fz(s)p FA(\))2696 360 y Fv(\000)p Fs(1)2791 393 y Fx(k)29 b FA(=)f Fx(O)s FA(\(1)p Fz(=)p FA(\()p Fz(\032)3246 360 y Fs(2)3309 393 y FA(+)21 b Fz(")p FA(\)\))36 b Fy(for)118 506 y FA(0)d Fn(6)e Fz(s)h Fn(6)g Fz(t)p Fy(.)53 b(Fix)37 b(an)f(initial)h(c)-5 b(ondition)38 b FA(\()p Fz(x)1648 520 y Fs(0)1688 506 y Fz(;)15 b(y)1773 520 y Fs(0)1813 506 y FA(\))36 b Fy(with)h Fz(y)2130 520 y Fs(0)2202 506 y Fx(2)31 b(D)2364 520 y Fs(0)2440 506 y Fy(and)38 b Fz(x)2673 520 y Fs(0)2744 506 y FA(=)g(\026)-51 b Fz(x)p FA(\()p Fz(y)2979 520 y Fs(0)3018 506 y Fz(;)15 b(")p FA(\))p Fy(,)38 b(and)g(let)e Fz(t)g Fy(b)-5 b(e)118 619 y(such)33 b(that)h Fz(y)559 586 y Fs(det)556 642 y Fw(s)686 619 y Fx(2)25 b(D)842 633 y Fs(0)914 619 y Fy(for)33 b(al)5 b(l)33 b Fz(s)25 b Fn(6)g Fz(t)p Fy(.)42 b(De\014ne)831 846 y Fz(R)q FA(\()p Fz(t)p FA(\))25 b(=)g Fx(k)p 1194 773 45 4 v(Z)7 b(k)1288 864 y Fs([0)p Fw(;t)p Fs(])1413 745 y Fq(h)1456 846 y FA(1)21 b(+)1613 745 y Fq(\020)1667 846 y FA(1)g(+)f Fx(k)p Fz(Y)1942 808 y Fv(\000)p Fs(1)2036 846 y Fx(k)2081 798 y Fs(1)p Fw(=)p Fs(2)2081 879 y([0)p Fw(;t)p Fs(])2205 745 y Fq(\021)2260 846 y Fz(\037)2317 808 y Fs(\(1\))2411 846 y FA(\()p Fz(t)p FA(\))h(+)f Fz(\037)2683 808 y Fs(\(2\))2777 846 y FA(\()p Fz(t)p FA(\))2880 745 y Fq(i)2923 846 y Fz(:)481 b FA(\(2.48\))118 1080 y Fy(Ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fz(h)1122 1094 y Fs(0)1187 1080 y Fz(>)25 b FA(0)p Fy(,)33 b(indep)-5 b(endent)34 b(of)f Fz(y)2045 1094 y Fs(0)2116 1080 y Fy(and)h Fz(t)p Fy(,)e(such)h(that)h(for)f(al)5 b(l)33 b Fz(h)26 b Fn(6)f Fz(h)3280 1094 y Fs(0)3319 1080 y Fz(R)q FA(\()p Fz(t)p FA(\))3492 1047 y Fv(\000)p Fs(1)3587 1080 y Fy(,)592 1307 y Fr(P)653 1270 y Fs(0)p Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))858 1206 y Fq(n)1034 1307 y FA(sup)918 1384 y Fs(0)p Fo(6)p Fw(s)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)1199 1395 y Fk(D)1247 1410 y Ft(0)1302 1234 y Fq(\012)1344 1307 y Fz(\020)1384 1321 y Fw(u)1429 1307 y Fz(;)p 1493 1234 V 15 w Fx(Z)7 b FA(\()p Fz(u)p FA(\))1664 1270 y Fv(\000)p Fs(1)1760 1307 y Fz(\020)1800 1321 y Fw(u)1845 1234 y Fq(\013)1913 1307 y Fn(>)25 b Fz(h)2061 1270 y Fs(2)2101 1206 y Fq(o)2186 1307 y Fn(6)g Fx(C)2330 1321 y Fw(n)p Fs(+)p Fw(m;\015)t(;)p Fs(\001)2633 1307 y FA(\()p Fz(t;)15 b(")p FA(\))g(e)2875 1270 y Fv(\000)p Fw(\024h)3012 1246 y Ft(2)3045 1270 y Fw(=\033)3122 1246 y Ft(2)3162 1307 y Fz(;)242 b FA(\(2.49\))118 1582 y Fy(with)809 1837 y Fx(C)857 1851 y Fw(n)p Fs(+)p Fw(m;\015)t(;)p Fs(\001)1159 1837 y FA(\()p Fz(t;)15 b(")p FA(\))27 b(=)1467 1709 y Fq(\030)1572 1775 y Fz(t)p 1530 1816 119 4 v 1530 1899 a FA(\001)p Fz(")1658 1709 y Fq(\031\024\022)1940 1775 y FA(1)p 1836 1816 255 4 v 1836 1899 a(1)20 b Fx(\000)g FA(2)p Fz(\015)2100 1709 y Fq(\023)2167 1731 y Fs(\()p Fw(n)p Fs(+)p Fw(m)p Fs(\))p Fw(=)p Fs(2)2477 1837 y FA(+)f(e)2608 1799 y Fs(\()p Fw(n)p Fs(+)p Fw(m)p Fs(\))p Fw(=)p Fs(4)2898 1709 y Fq(\025)2946 1837 y Fz(;)458 b FA(\(2.50\))1293 2112 y Fz(\024)26 b FA(=)f Fz(\015)1519 2011 y Fq(h)1562 2112 y FA(1)20 b Fx(\000)g(O)1793 2038 y Fq(\000)1835 2112 y Fz(")g FA(+)g(\001)g(+)g Fz(hR)q FA(\()p Fz(t)p FA(\))2400 2038 y Fq(\001)2442 2011 y(i)2485 2112 y Fz(:)919 b FA(\(2.51\))259 2381 y(Let)40 b(us)f(\014rst)f (consider)h(timescales)g(of)h(order)e(1.)69 b(Then)38 b(the)h(functions)f Fx(k)p 2935 2308 45 4 v(Z)7 b(k)3029 2399 y Fs([0)p Fw(;t)p Fs(])3154 2381 y FA(,)42 b Fz(\037)3278 2348 y Fs(\(1\))3372 2381 y FA(\()p Fz(t)p FA(\))e(and)118 2507 y Fz(\037)175 2474 y Fs(\(2\))269 2507 y FA(\()p Fz(t)p FA(\))d(are)f(at)h(most)g(of)f(order)g(1,)i(and)d Fx(k)p Fz(Y)21 b FA(\()p Fz(t)p FA(\))1777 2474 y Fv(\000)p Fs(1)1871 2507 y Fx(k)37 b FA(remains)e(of)h(the)g(same)h(order)e(as)i Fx(k)p Fz(Y)3282 2469 y Fv(\000)p Fs(1)3262 2533 y(0)3377 2507 y Fx(k)p FA(.)58 b(The)118 2620 y(probabilit)m(y)44 b(\(2.49\))j(b)s(ecomes)e(small)e(as)j(so)s(on)e(as)h Fz(h)50 b Fx(\035)f Fz(\033)s FA(.)85 b(Because)46 b(of)f(the)g (restriction)f Fz(h)50 b Fn(6)118 2733 y Fz(h)170 2747 y Fs(0)210 2733 y Fz(R)q FA(\()p Fz(t)p FA(\))383 2700 y Fv(\000)p Fs(1)478 2733 y FA(,)34 b(the)f(result)f(is)g(useful)g(pro) m(vided)f Fx(k)p Fz(Y)1805 2700 y Fv(\000)p Fs(1)1899 2733 y Fx(k)1944 2751 y Fs([0)p Fw(;t)p Fs(])2098 2733 y Fx(\034)f Fz(\033)2274 2700 y Fv(\000)p Fs(2)2368 2733 y FA(.)49 b(In)33 b(order)g(to)g(obtain)g(the)g(optimal)118 2846 y(concen)m(tration)e(result,)e(w)m(e)i(ha)m(v)m(e)g(to)f(c)m(ho)s (ose)h Fz(Y)1758 2860 y Fs(0)1827 2846 y FA(according)f(to)h(t)m(w)m(o) g(opp)s(osed)e(criteria.)40 b(On)29 b(the)h(one)118 2959 y(hand,)e(w)m(e)h(w)m(ould)e(lik)m(e)g(to)i(c)m(ho)s(ose)g Fz(Y)1376 2973 y Fs(0)1444 2959 y FA(as)f(small)f(as)h(p)s(ossible,)f (so)h(that)h(the)g(set)2862 2885 y Fq(\012)2905 2959 y Fz(\020)2945 2973 y Fw(u)2990 2959 y Fz(;)p 3054 2885 V 15 w Fx(Z)7 b FA(\()p Fz(u)p FA(\))3225 2926 y Fv(\000)p Fs(1)3321 2959 y Fz(\020)3361 2973 y Fw(u)3405 2885 y Fq(\013)3474 2959 y Fz(<)25 b(h)3622 2926 y Fs(2)118 3071 y FA(is)32 b(small.)47 b(On)32 b(the)h(other)h(hand,)e Fx(k)p Fz(Y)1421 3033 y Fv(\000)p Fs(1)1400 3097 y(0)1515 3071 y Fx(k)h FA(m)m(ust)g(not)h(exceed)g(certain)e(b)s(ounds)f(for)i (Theorem)g(2.5)h(to)118 3184 y(b)s(e)c(v)-5 b(alid.)39 b(Th)m(us)29 b(w)m(e)i(require)e(that)1421 3389 y Fz(Y)1474 3403 y Fs(0)1538 3389 y Fz(>)1634 3315 y Fq(\002)1672 3389 y Fz(\033)1727 3351 y Fs(2)1787 3389 y Fx(_)20 b FA(\()p Fz(\032)1950 3351 y Fs(2)2010 3389 y FA(+)g Fz(")p FA(\))2178 3315 y Fq(\003)2216 3389 y FA(1)-20 b(l)2267 3403 y Fw(m)2333 3389 y Fz(:)1071 b FA(\(2.52\))118 3593 y(Because)45 b(of)e(the)h(Gaussian)e(deca)m(y)j(of)f(the)f(probabilit)m (y)f(\(2.49\))k(in)c Fz(\033)s(=h)p FA(,)48 b(w)m(e)c(can)f(in)m (terpret)g(the)118 3706 y(theorem)21 b(b)m(y)g(sa)m(ying)g(that)g(the)g (t)m(ypical)f(spreading)f(of)i(paths)g(in)e(the)i Fz(y)s FA(-direction)f(is)g(of)g(order)h Fz(\033)s FA(\()p Fz(\032)q FA(+)3508 3640 y Fx(p)p 3584 3640 43 4 v 66 x Fz(")p FA(\))118 3819 y(if)29 b Fz(\033)g(<)c(\032)20 b FA(+)536 3753 y Fx(p)p 612 3753 V 66 x Fz(")30 b FA(and)g(of)h(order)f Fz(\033)1258 3786 y Fs(2)1328 3819 y FA(if)f Fz(\033)f(>)d(\032)c FA(+)1745 3753 y Fx(p)p 1821 3753 V 66 x Fz(")q FA(.)259 3932 y(The)40 b(term)f Fz(\032)h FA(is)f(clearly)g(due)g(to)i(the)f(in) m(tensit)m(y)f Fz(\033)2074 3899 y Fv(0)2138 3932 y FA(=)i Fz(\032\033)i FA(of)d(the)g(noise)f(acting)h(on)g(the)g(slo)m(w)118 4044 y(v)-5 b(ariable.)42 b(It)31 b(prev)-5 b(ails)29 b(if)h Fz(\032)d(>)f(\033)e Fx(_)1340 3979 y(p)p 1416 3979 V 65 x Fz(")p FA(.)43 b(The)31 b(term)1932 3979 y Fx(p)p 2008 3979 V 65 x Fz(")g FA(is)f(due)h(to)h(the)f(linear)f (part)h(of)g(the)g(coupling)118 4157 y(b)s(et)m(w)m(een)e(slo)m(w)e (and)h(fast)g(v)-5 b(ariables,)28 b(while)e(the)i(b)s(eha)m(viour)e(in) h Fz(\033)2379 4124 y Fs(2)2447 4157 y FA(observ)m(ed)h(when)f Fz(\033)h(>)d(\032)16 b FA(+)3379 4092 y Fx(p)p 3455 4092 V 65 x Fz(")28 b FA(can)118 4270 y(b)s(e)i(traced)h(bac)m(k)g(to)g (the)g Fy(nonline)-5 b(ar)32 b FA(coupling)c(b)s(et)m(w)m(een)k(slo)m (w)e(and)f(fast)i(v)-5 b(ariables.)259 4383 y(F)d(or)30 b(longer)e(timescales,)h(the)g(condition)f Fz(h)d Fn(6)g Fz(h)1924 4397 y Fs(0)1964 4383 y Fz(R)q FA(\()p Fz(t)p FA(\))2137 4350 y Fv(\000)p Fs(1)2260 4383 y FA(obliges)j(us)g(to)i (tak)m(e)g(a)f(larger)g Fz(Y)3361 4397 y Fs(0)3400 4383 y FA(,)g(while)118 4496 y Fz(Y)20 b FA(\()p Fz(t)p FA(\))31 b(t)m(ypically)e(gro)m(ws)i(with)e(time.)40 b(If)30 b(the)h(matrix)e Fz(B)35 b FA(alw)m(a)m(ys)c(has)f(eigen)m(v)-5 b(alues)30 b(with)f(p)s(ositiv)m(e)h(real)118 4609 y(parts)41 b(\(or,)j(more)c (precisely)-8 b(,)43 b(if)d(the)h(largest)g(Ly)m(apuno)m(v)g(exp)s (onen)m(t)g(is)e(p)s(ositiv)m(e\),)k(this)d(gro)m(wth)h(is)118 4722 y(exp)s(onen)m(tial)25 b(in)g(time,)i(so)f(that)h(the)f(spreading) e(of)j(paths)e(along)h(the)h(adiabatic)e(manifold)f(will)g(reac)m(h)118 4835 y(order)30 b(1)h(in)e(a)i(time)f(of)g(order)g(log)q Fx(j)p Fz(\033)24 b Fx(_)c FA(\()p Fz(\032)1543 4802 y Fs(2)1603 4835 y FA(+)g Fz(")p FA(\))p Fx(j)p FA(.)118 5022 y Fm(Remark)34 b(2.6.)43 b FA(Consider)28 b(the)j Fy(r)-5 b(e)g(duc)g(e)g(d)34 b(sto)-5 b(chastic)35 b(system)893 5227 y FA(d)o Fz(y)991 5189 y Fs(0)988 5249 y Fw(t)1056 5227 y FA(=)25 b Fz(g)s FA(\()6 b(\026)-51 b Fz(x)q FA(\()p Fz(y)1369 5189 y Fs(0)1366 5249 y Fw(t)1408 5227 y Fz(;)15 b(")p FA(\))p Fz(;)g(y)1613 5189 y Fs(0)1610 5249 y Fw(t)1654 5227 y Fz(;)g(")p FA(\))g(d)q Fz(t)20 b FA(+)g Fz(\033)2037 5189 y Fv(0)2060 5227 y Fz(G)p FA(\()6 b(\026)-51 b Fz(x)q FA(\()p Fz(y)2302 5189 y Fs(0)2299 5249 y Fw(t)2342 5227 y Fz(;)15 b(")p FA(\))p Fz(;)g(y)2547 5189 y Fs(0)2544 5249 y Fw(t)2588 5227 y Fz(;)g(")p FA(\))g(d)p Fz(W)2857 5241 y Fw(t)3429 5227 y FA(\(2.53\))118 5431 y(obtained)27 b(b)m(y)h(setting)g Fz(x)f FA(equal)h(to)34 b(\026)-51 b Fz(x)p FA(\()p Fz(y)s(;)15 b(")p FA(\))29 b(in)d(\(2.8\))r(.)40 b(One)27 b(ma)m(y)i(w)m(onder)e(whether)g Fz(y)3061 5398 y Fs(0)3058 5453 y Fw(t)3128 5431 y FA(giv)m(es)h(a)g(b)s(etter)1844 5871 y(13)p eop %%Page: 14 14 14 13 bop 118 280 a FA(appro)m(ximation)29 b(of)i Fz(y)873 294 y Fw(t)932 280 y FA(than)g Fz(y)1193 247 y Fs(det)1190 303 y Fw(t)1295 280 y FA(.)40 b(In)30 b(fact,)h(one)g(can)g(sho)m(w)f (that)208 547 y Fr(P)269 509 y Fs(0)p Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))474 446 y Fq(n)650 547 y FA(sup)535 624 y Fs(0)p Fo(6)p Fw(s)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)816 635 y Fk(D)864 650 y Ft(0)903 470 y Fq(\015)903 524 y(\015)953 547 y Fz(y)1001 509 y Fs(0)998 569 y Fw(s)1061 547 y Fx(\000)20 b Fz(y)1200 509 y Fs(det)1197 569 y Fw(s)1302 470 y Fq(\015)1302 524 y(\015)1377 547 y Fn(>)25 b Fz(h)1525 561 y Fs(1)1565 446 y Fq(o)1651 547 y Fn(6)g Fz(c)1786 446 y Fq(\020)1841 547 y FA(1)20 b(+)g Fz(t)2030 446 y Fq(\021)2099 547 y FA(e)2140 509 y Fw(m=)p Fs(4)2292 547 y FA(exp)2431 446 y Fq(n)2492 547 y Fx(\000)2573 485 y Fz(\024)2625 499 y Fs(1)2664 485 y Fz(h)2716 452 y Fs(2)2716 510 y(1)2756 485 y FA(\(1)h Fx(\000)f(O)s FA(\()p Fz(h)3110 499 y Fs(1)3150 485 y Fz(\037)3207 452 y 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Fs(\(1\))3301 775 y FA(\()p Fz(t)p FA(\)\)\))p 2573 815 V 2802 903 a(\()p Fz(\033)2892 876 y Fv(0)2916 903 y FA(\))2951 876 y Fs(2)2991 903 y Fz(\037)3048 876 y Fs(\(2\))3142 903 y FA(\()p Fz(t)p FA(\))3485 735 y Fq(o)1396 1127 y FA(+)g Fz(c)1526 1026 y Fq(\020)1580 1127 y FA(1)h(+)1752 1065 y Fz(t)p 1747 1106 43 4 v 1747 1189 a(")1799 1026 y Fq(\021)1869 1127 y FA(e)1909 1089 y Fw(m=)p Fs(4)2061 1127 y FA(exp)2200 1026 y Fq(n)2261 1127 y Fx(\000)2342 1065 y Fz(\024)2394 1079 y Fs(2)2433 1065 y Fz(h)2485 1032 y Fs(2)2525 1065 y FA(\(1)g Fx(\000)f(O)s FA(\(\()p Fz(h)h FA(+)f Fz(h)3078 1079 y Fs(1)3118 1065 y FA(\))p Fz(\037)3210 1032 y Fs(\(1\))3304 1065 y FA(\()p Fz(t)p FA(\)\)\))p 2342 1106 1138 4 v 2493 1193 a([\()p Fz(\033)2608 1167 y Fv(0)2632 1193 y FA(\))2667 1167 y Fs(2)2707 1193 y Fz(h)2759 1167 y Fs(2)2819 1193 y FA(+)g Fz(\033)2965 1167 y Fs(2)3005 1193 y Fz(")p FA(])p Fz(\037)3129 1167 y Fs(\(2\))3223 1193 y FA(\()p Fz(t)p FA(\))3489 1026 y Fq(o)3429 1303 y FA(\(2.54\))118 1519 y(holds)28 b(for)h(all)g Fz(h;)15 b(h)763 1533 y Fs(1)832 1519 y FA(up)29 b(to)h(order)f Fz(\037)1367 1486 y Fs(\(1\))1461 1519 y FA(\()p Fz(t)p FA(\))1564 1486 y Fv(\000)p Fs(1)1688 1519 y FA(and)g(some)h(p)s(ositiv)m(e)e(constan)m(ts)j Fz(c;)15 b(\024)2959 1533 y Fs(1)2999 1519 y Fz(;)g(\024)3091 1533 y Fs(2)3132 1519 y FA(.)40 b(\(The)29 b(pro)s(ofs)118 1632 y(can)d(b)s(e)f(adapted)h(from)f(the)h(pro)s(of)f(of)h(Lemma)g (3.4\).)41 b(This)24 b(sho)m(ws)h(that)i(the)f(t)m(ypical)f(spreading)f (of)i Fz(y)3622 1599 y Fs(0)3619 1655 y Fw(t)118 1745 y FA(around)k Fz(y)475 1712 y Fs(det)472 1768 y Fw(t)608 1745 y FA(is)g(of)h(order)f Fz(\033)1097 1712 y Fv(0)1121 1745 y Fz(\037)1178 1712 y Fs(\(2\))1272 1745 y FA(\()p Fz(t)p FA(\))1375 1712 y Fs(1)p Fw(=)p Fs(2)1511 1745 y FA(=)c Fz(\032\033)s(\037)1767 1712 y Fs(\(2\))1862 1745 y FA(\()p Fz(t)p FA(\))1965 1712 y Fs(1)p Fw(=)p Fs(2)2075 1745 y FA(,)31 b(while)e(the)i(t)m(ypical)g(deviation)f(of)h (paths)f Fz(y)3622 1712 y Fs(0)3619 1768 y Fw(t)118 1858 y FA(of)d(the)g(reduced)f(system)h(from)f(paths)h Fz(y)1505 1872 y Fw(t)1561 1858 y FA(of)g(the)g(original)e(system)i(is)f(of)h (order)f Fz(\033)2914 1793 y Fx(p)p 2990 1793 43 4 v 65 x Fz("\037)3089 1825 y Fs(\(2\))3183 1858 y FA(\()p Fz(t)p FA(\))3286 1825 y Fs(1)p Fw(=)p Fs(2)3397 1858 y FA(.)39 b(Th)m(us)118 1971 y(for)33 b Fz(\032)c(>)436 1905 y Fx(p)p 512 1905 V 66 x Fz(")p FA(,)34 b(the)f(reduced)f(sto)s(c) m(hastic)i(system)f(giv)m(es)g(a)g(b)s(etter)g(appro)m(ximation)f(of)h (the)g(dynamics)118 2084 y(than)d(the)h(deterministic)d(one.)259 2272 y(If)k Fz(B)38 b FA(has)32 b(no)h(eigen)m(v)-5 b(alue)32 b(with)g(p)s(ositiv)m(e)f(real)i(part,)g(the)g(spreading)e(of)i(paths)f (will)e(gro)m(w)j(more)118 2384 y(slo)m(wly)-8 b(.)67 b(As)39 b(an)h(imp)s(ortan)m(t)e(particular)g(case,)43 b(let)c(us)g(consider)f(the)i(situation)e(where)h Fz(y)3324 2351 y Fs(det)3321 2407 y Fw(t)3465 2384 y FA(is)f(an)118 2497 y(asymptotically)30 b(stable)f(p)s(erio)s(dic)f(orbit)h(with)g(p)s (erio)s(d)f Fz(T)13 b FA(,)30 b(en)m(tirely)f(con)m(tained)i(in)e Fx(D)3090 2511 y Fs(0)3159 2497 y FA(\(and)h(not)h(to)s(o)118 2610 y(close)e(to)f(its)g(b)s(oundary\).)38 b(Then)27 b(all)g(co)s(e\016cien)m(ts)i(in)e(\(2.33\))j(dep)s(end)d(p)s(erio)s (dically)d(on)k(time,)g(and,)h(in)118 2723 y(particular,)g(Flo)s (quet's)h(theorem)h(allo)m(ws)f(us)f(to)j(write)1574 2927 y Fz(V)20 b FA(\()p Fz(t)p FA(\))26 b(=)f Fz(P)13 b FA(\()p Fz(t)p FA(\))i(e)2102 2890 y Fs(\003)p Fw(t)2180 2927 y Fz(;)1224 b FA(\(2.55\))118 3132 y(where)34 b Fz(P)13 b FA(\()p Fz(t)p FA(\))36 b(is)d(a)j Fz(T)13 b FA(-p)s(erio)s(dic)32 b(matrix.)53 b(The)34 b(asymptotic)h(stabilit)m (y)f(of)h(the)g(orbit)e(means)i(that)g(all)118 3245 y(eigen)m(v)-5 b(alues)28 b(but)g(one)h(of)g(the)f(mono)s(drom)m(y)g(matrix)g(\003)h (ha)m(v)m(e)h(strictly)d(negativ)m(e)j(real)e(parts,)h(the)f(last)118 3357 y(eigen)m(v)-5 b(alue,)26 b(whic)m(h)d(corresp)s(onds)g(to)j (translations)d(along)i(the)f(orbit,)h(b)s(eing)f(0.)39 b(In)24 b(that)h(case,)i Fz(\037)3464 3324 y Fs(\(1\))3558 3357 y FA(\()p Fz(t)p FA(\))118 3470 y(and)i Fz(\037)351 3437 y Fs(\(2\))445 3470 y FA(\()p Fz(t)p FA(\))g(gro)m(w)h(only)e (linearly)f(with)h(time,)h(so)g(that)h(the)f(spreading)f(of)h(paths)g (in)e(the)j Fz(y)s FA(-direction)118 3583 y(remains)f(small)g(on)h (timescales)h(of)f(order)g(1)p Fz(=)p FA(\()p Fz(\033)25 b Fx(_)20 b FA(\()p Fz(\032)1958 3550 y Fs(2)2018 3583 y FA(+)f Fz(")p FA(\)\).)259 3696 y(In)30 b(fact,)j(w)m(e)e(ev)m(en)h (exp)s(ect)f(this)f(spreading)g(to)h(o)s(ccur)g(mainly)e(along)i(the)h (p)s(erio)s(dic)c(orbit,)i(while)118 3809 y(the)h(paths)g(remain)f (con\014ned)g(to)i(a)f(neigh)m(b)s(ourho)s(o)s(d)d(of)k(the)f(orbit)f (on)h(sub)s(exp)s(onen)m(tial)d(timescales.)118 3922 y(T)-8 b(o)38 b(see)f(that)h(this)e(is)g(true,)j(w)m(e)f(can)f(use)g(a) g(new)g(set)h(of)f(v)-5 b(ariables)36 b(in)f(the)j(neigh)m(b)s(ourho)s (o)s(d)c(of)j(the)118 4035 y(orbit.)73 b(In)41 b(order)g(not)g(to)h(in) m(tro)s(duce)f(to)s(o)h(man)m(y)f(new)g(notations,)k(w)m(e)c(will)e (replace)j Fz(y)i FA(b)m(y)d(\()p Fz(y)s(;)15 b(z)t FA(\),)118 4148 y(where)38 b Fz(y)j Fx(2)d Fr(R)649 4115 y Fw(m)p Fv(\000)p Fs(1)850 4148 y FA(describ)s(es)e(the)j(degrees)g(of)f (freedom)g(transv)m(ersal)g(to)h(the)g(orbit,)g(and)f Fz(z)43 b Fx(2)38 b Fr(R)118 4261 y FA(parametrizes)23 b(the)f(motion)h(along)f(the)h(orbit.)37 b(In)22 b(fact,)j(w)m(e)e(can) g(use)f(an)g(equal-time)h(parametrization)118 4374 y(of)34 b(the)f(orbit,)g(so)h(that)50 b(_)-41 b Fz(z)34 b FA(=)c(1)k(on)f(the)h (orbit,)f(i.)15 b(e.,)35 b(w)m(e)e(ha)m(v)m(e)i Fz(z)2332 4341 y Fs(det)2328 4396 y Fw(t)2464 4374 y FA(=)30 b Fz(t)41 b FA(\(mo)s(d)29 b Fz(T)13 b FA(\).)50 b(The)33 b(SDE)g(tak)m(es)118 4487 y(the)e(form)862 4706 y(d)p Fz(x)965 4720 y Fw(t)1020 4706 y FA(=)1126 4645 y(1)p 1126 4685 46 4 v 1128 4769 a Fz(")1181 4706 y(f)10 b FA(\()p Fz(x)1323 4720 y Fw(t)1352 4706 y Fz(;)15 b(y)1437 4720 y Fw(t)1467 4706 y Fz(;)g(z)1549 4720 y Fw(t)1579 4706 y Fz(;)g(")p FA(\))g(d)q Fz(t)20 b FA(+)1948 4645 y Fz(\033)p 1917 4685 119 4 v 1917 4704 a Fx(p)p 1993 4704 43 4 v 65 x Fz(")2045 4706 y(F)13 b FA(\()p Fz(x)2203 4720 y Fw(t)2233 4706 y Fz(;)i(y)2318 4720 y Fw(t)2348 4706 y Fz(;)g(z)2430 4720 y Fw(t)2460 4706 y Fz(;)g(")p FA(\))g(d)q Fz(W)2730 4720 y Fw(t)2759 4706 y Fz(;)870 4925 y FA(d)o Fz(y)965 4939 y Fw(t)1020 4925 y FA(=)25 b Fz(g)s FA(\()p Fz(x)1249 4939 y Fw(t)1279 4925 y Fz(;)15 b(y)1364 4939 y Fw(t)1394 4925 y Fz(;)g(z)1476 4939 y Fw(t)1506 4925 y Fz(;)g(")p FA(\))g(d)q Fz(t)20 b FA(+)g Fz(\033)1889 4887 y Fv(0)1912 4925 y Fz(G)p FA(\()p Fz(x)2070 4939 y Fw(t)2101 4925 y Fz(;)15 b(y)2186 4939 y Fw(t)2216 4925 y Fz(;)g(z)2298 4939 y Fw(t)2328 4925 y Fz(;)g(")p FA(\))g(d)q Fz(W)2598 4939 y Fw(t)2627 4925 y Fz(;)872 5143 y FA(d)p Fz(z)965 5157 y Fw(t)1020 5143 y FA(=)1116 5069 y Fq(\002)1154 5143 y FA(1)20 b(+)g Fz(h)p FA(\()p Fz(x)1449 5157 y Fw(t)1480 5143 y Fz(;)15 b(y)1565 5157 y Fw(t)1594 5143 y Fz(;)g(z)1676 5157 y Fw(t)1706 5143 y Fz(;)g(")p FA(\))1823 5069 y Fq(\003)1878 5143 y FA(d)o Fz(t)20 b FA(+)g Fz(\033)2127 5106 y Fv(0)2151 5143 y Fz(H)7 b FA(\()p Fz(x)2321 5157 y Fw(t)2350 5143 y Fz(;)15 b(y)2435 5157 y Fw(t)2465 5143 y Fz(;)g(z)2547 5157 y Fw(t)2577 5143 y Fz(;)g(")p FA(\))g(d)q Fz(W)2847 5157 y Fw(t)2877 5143 y Fz(;)3429 4930 y FA(\(2.56\))118 5406 y(where)33 b Fz(h)f FA(=)e Fx(O)s FA(\()p Fx(k)p Fz(y)769 5420 y Fw(t)799 5406 y Fx(k)844 5373 y Fs(2)907 5406 y FA(+)22 b Fx(k)p Fz(x)1097 5420 y Fw(t)1150 5406 y Fx(\000)g Fz(x)1295 5373 y Fs(det)1295 5429 y Fw(t)1397 5406 y Fx(k)1442 5373 y Fs(2)1482 5406 y FA(\))34 b(and)f Fz(@)1779 5420 y Fw(y)1821 5406 y Fz(g)s FA(\()p Fz(x)1954 5373 y Fs(det)1954 5429 y Fw(t)2057 5406 y Fz(;)15 b FA(0)p Fz(;)g(z)2228 5373 y Fs(det)2224 5429 y Fw(t)2332 5406 y Fz(;)g(")p FA(\))34 b(has)g(eigen)m(v)-5 b(alues)34 b(with)e(negativ)m(e)118 5519 y(real)g(parts,)h(uniformly)c(b)s(ounded) h(a)m(w)m(a)m(y)k(from)e(zero.)48 b(As)32 b(linear)f(appro)m(ximation)g (of)h(the)h(dynamics)1844 5871 y(14)p eop %%Page: 15 15 15 14 bop 118 280 a FA(of)31 b(\()p Fz(\030)297 294 y Fw(t)326 280 y Fz(;)15 b(\021)411 294 y Fw(t)442 280 y FA(\))25 b(=)g(\()p Fz(x)685 294 y Fw(t)735 280 y Fx(\000)20 b Fz(x)878 247 y Fs(det)878 303 y Fw(t)980 280 y Fz(;)15 b(y)1065 294 y Fw(t)1115 280 y Fx(\000)20 b Fz(y)1254 247 y Fs(det)1251 303 y Fw(t)1356 280 y FA(\))26 b(=)f(\()p Fz(x)1600 294 y Fw(t)1650 280 y Fx(\000)20 b Fz(x)1793 247 y Fs(det)1793 303 y Fw(t)1895 280 y Fz(;)15 b(y)1980 294 y Fw(t)2009 280 y FA(\))31 b(w)m(e)g(tak)m(e)765 523 y(d)o Fz(\030)859 485 y Fs(0)855 545 y Fw(t)924 523 y FA(=)1030 461 y(1)p 1030 502 46 4 v 1032 585 a Fz(")1085 523 y(A)p FA(\()p Fz(z)1234 485 y Fs(det)1230 545 y Fw(t)1337 523 y Fz(;)15 b(")p FA(\))p Fz(\030)1498 485 y Fs(0)1494 545 y Fw(t)1554 523 y FA(d)p Fz(t)20 b FA(+)1790 461 y Fz(\033)p 1758 502 119 4 v 1758 520 a Fx(p)p 1834 520 43 4 v 66 x Fz(")1887 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Fz(;)15 b(")p FA(\))g(d)p Fz(W)1861 974 y Fw(t)1891 960 y Fz(;)3429 747 y FA(\(2.57\))118 1207 y(whic)m(h)28 b(dep)s(ends)e(p)s(erio)s(dically)f(on)k(time.)40 b(One)28 b(can)h(again)g(compute)g(the)f(co)m(v)-5 b(ariance)30 b(matrix)e(of)h(the)118 1320 y(Gaussian)k(pro)s(cess)h(\()p Fz(\030)912 1287 y Fs(0)908 1342 y Fw(t)952 1320 y Fz(;)15 b(\021)1040 1287 y Fs(0)1037 1342 y Fw(t)1080 1320 y Fz(;)g(z)1166 1287 y Fs(0)1162 1342 y Fw(t)1206 1320 y FA(\))35 b(as)f(a)h(function)e(of)h(the)g(principal)d(solutions)i Fz(U)44 b FA(and)34 b Fz(V)54 b FA(asso)s(ciated)118 1433 y(with)29 b Fz(A)i FA(and)e Fz(B)5 b FA(.)41 b(In)29 b(particular,)g(the)i(co)m(v)-5 b(ariance)31 b(matrix)f Fz(Y)20 b FA(\()p Fz(t)p FA(\))31 b(of)f Fz(\021)2548 1400 y Fs(0)2545 1455 y Fw(t)2619 1433 y FA(still)e(ob)s(eys)i(the)g (ODE)311 1614 y(_)287 1637 y Fz(Y)45 b FA(=)25 b Fz(B)5 b FA(\()p Fz(y)s(;)15 b(")p FA(\))p Fz(Y)41 b FA(+)20 b Fz(Y)g(B)5 b FA(\()p Fz(y)s(;)15 b(")p FA(\))1287 1599 y Fw(T)1363 1637 y FA(+)20 b Fz(C)7 b FA(\()p Fz(y)s(;)15 b(")p FA(\))p 1750 1564 41 4 v Fz(Z)28 b FA(+)p 1930 1564 V 20 w Fz(Z)1976 1599 y Fw(T)2030 1637 y Fz(C)7 b FA(\()p Fz(y)s(;)15 b(")p FA(\))2302 1599 y Fw(T)2378 1637 y FA(+)20 b Fz(\032)2516 1599 y Fs(2)2556 1637 y Fz(G)2627 1651 y Fs(0)2666 1637 y FA(\()p Fz(y)s(;)15 b(")p FA(\))p Fz(G)2937 1651 y Fs(0)2979 1637 y FA(\()p Fz(y)s(;)g(")p FA(\))3179 1599 y Fw(T)3235 1637 y Fz(:)169 b FA(\(2.58\))118 1841 y(This)24 b(is)h(no)m(w)h(a)h(linear,)e (inhomogeneous)h(ODE)g(with)e(time-p)s(erio)s(dic)g(co)s(e\016cien)m (ts.)40 b(It)26 b(is)f(w)m(ell)g(kno)m(wn)118 1954 y(that)32 b(suc)m(h)f(a)g(system)h(admits)e(a)h(unique)f(p)s(erio)s(dic)e (solution)i Fz(Y)2340 1910 y Fs(p)r(er)2320 1978 y Fw(t)2444 1954 y FA(,)i(whic)m(h)e(is)g(of)h(order)g Fz(\032)3244 1921 y Fs(2)3304 1954 y FA(+)20 b Fz(")32 b FA(since)p 141 1994 V 118 2067 a Fz(Z)47 b FA(is)40 b(of)h(order)f Fz(\032)738 2002 y Fx(p)p 813 2002 43 4 v 813 2067 a Fz(")28 b FA(+)e Fz(")41 b FA(and)f Fz(\032)1297 2034 y Fs(2)1337 2067 y Fz(G)1408 2081 y Fs(0)1448 2067 y Fz(G)1519 2034 y Fw(T)1519 2091 y Fs(0)1615 2067 y FA(is)f(of)i(order)f Fz(\032)2125 2034 y Fs(2)2165 2067 y FA(.)71 b(W)-8 b(e)42 b(can)f(th)m(us)f(de\014ne)g(an)g(asymptotic)118 2180 y(co)m(v)-5 b(ariance)39 b(matrix)p 892 2107 45 4 v 37 w Fx(Z)7 b FA(\()p Fz(t)p FA(\))39 b(of)f(\()p Fz(\030)1273 2147 y Fs(0)1269 2202 y Fw(t)1313 2180 y Fz(;)15 b(\021)1401 2147 y Fs(0)1398 2202 y Fw(t)1441 2180 y FA(\),)40 b(whic)m(h)d(dep)s (ends)f(p)s(erio)s(dically)e(on)k(time.)63 b(If)37 b Fz(\020)3205 2194 y Fw(t)3272 2180 y FA(=)h(\()p Fz(\030)3456 2194 y Fw(t)3486 2180 y Fz(;)15 b(\021)3571 2194 y Fw(t)3601 2180 y FA(\),)118 2293 y(Theorem)30 b(2.5)h(sho)m(ws)f(that)h(on)f (timescales)g(of)h(order)e(1)i(\(at)g(least\),)g(the)g(paths)e Fz(\020)2942 2307 y Fw(t)3002 2293 y FA(are)i(concen)m(trated)118 2406 y(in)e(a)i(set)g(of)f(the)h(form)f Fx(h)p Fz(\020)992 2420 y Fw(t)1022 2406 y Fz(;)p 1085 2333 V 15 w Fx(Z)7 b FA(\()p Fz(t)p FA(\))1238 2373 y Fv(\000)p Fs(1)1333 2406 y Fz(\020)1373 2420 y Fw(t)1402 2406 y Fx(i)26 b Fz(<)f(h)1611 2373 y Fs(2)1651 2406 y FA(,)30 b(while)f Fz(z)1986 2420 y Fw(t)2046 2406 y FA(remains)g Fz(h)p FA(-close)i(to)h Fz(z)2843 2373 y Fs(det)2839 2428 y Fw(t)2945 2406 y FA(.)259 2519 y(On)d(longer)g(timescales,)h(the)g (distribution)c(of)j(paths)h(will)c(b)s(e)j(smeared)h(out)g(along)f (the)h(p)s(erio)s(dic)118 2632 y(orbit.)42 b(Ho)m(w)m(ev)m(er,)33 b(the)e(same)g(line)f(of)h(reasoning)f(as)h(in)f(Section)g(3.2,)j (based)d(on)h(a)h(comparison)e(with)118 2744 y(di\013eren)m(t)39 b(deterministic)e(solutions)h(on)h(successiv)m(e)h(time)f(in)m(terv)-5 b(als)38 b(of)i(order)f(1,)j(can)d(b)s(e)g(used)g(to)118 2857 y(sho)m(w)31 b(that)g Fz(\020)581 2871 y Fw(t)641 2857 y FA(remains)f(concen)m(trated)i(in)e(the)h(set)g Fx(h)p Fz(\020)2000 2871 y Fw(t)2030 2857 y Fz(;)p 2093 2784 V 15 w Fx(Z)7 b FA(\()p Fz(t)p FA(\))2246 2824 y Fv(\000)p Fs(1)2341 2857 y Fz(\020)2381 2871 y Fw(t)2410 2857 y Fx(i)27 b Fz(<)e(h)2620 2824 y Fs(2)2691 2857 y FA(up)k(to)j(exp)s(onen)m(tially)d(long)118 2970 y(timescales.)118 3214 y Fp(2.4)112 b(Bifurcations)118 3385 y FA(In)33 b(the)h(previous)e(section,)j(w)m(e)f(ha)m(v)m(e)h(assumed)e(that)i (the)f(slo)m(w)f(manifold)f Fz(x)e FA(=)h Fz(x)2991 3352 y Fw(?)3030 3385 y FA(\()p Fz(y)s FA(\))k(is)d(uniformly)118 3498 y(asymptotically)26 b(stable)h(for)g Fz(y)h Fx(2)d(D)1351 3512 y Fs(0)1391 3498 y FA(.)39 b(W)-8 b(e)29 b(consider)d(no)m(w)h (the)g(situation)f(arising)g(when)g(the)h(reduced)118 3611 y(deterministic)f(\015o)m(w)i(causes)g Fz(y)1173 3578 y Fs(det)1170 3634 y Fw(t)1303 3611 y FA(to)g(lea)m(v)m(e)i Fx(D)1704 3625 y Fs(0)1743 3611 y FA(,)f(and)e(to)i(approac)m(h)f(a)g (bifurcation)e(p)s(oin)m(t)h(of)h(the)g(slo)m(w)118 3724 y(manifold.)259 3837 y(W)-8 b(e)32 b(call)d(\()6 b(^)-51 b Fz(x)q(;)21 b FA(^)-51 b Fz(y)s FA(\))31 b(a)f(bifurcation)f(p)s(oin) m(t)g(of)i(the)f(deterministic)f(system)1595 4038 y Fz(")17 b FA(_)-42 b Fz(x)26 b FA(=)f Fz(f)10 b FA(\()p Fz(x;)15 b(y)s(;)g(")p FA(\))p Fz(;)1658 4176 y FA(_)-41 b Fz(y)28 b FA(=)d Fz(g)s FA(\()p Fz(x;)15 b(y)s(;)g(")p FA(\))p Fz(;)3429 4108 y FA(\(2.59\))118 4378 y(if)33 b Fz(f)10 b FA(\()c(^)-51 b Fz(x;)21 b FA(^)-51 b Fz(y)s(;)15 b FA(0\))32 b(=)f(0)j(and)f Fz(@)996 4392 y Fw(x)1040 4378 y Fz(f)10 b FA(\()c(^)-51 b Fz(x;)22 b FA(^)-52 b Fz(y)s(;)15 b FA(0\))35 b(has)f Fz(q)i FA(eigen)m(v)-5 b(alues)34 b(on)g(the)g(imaginary)e(axis,)j Fz(q)f Fx(2)c(f)p FA(1)p Fz(;)15 b(:)g(:)g(:)j(;)d(n)p Fx(g)p FA(.)118 4491 y(W)-8 b(e)33 b(consider)d(here)h(the)g(situation)f(where)h Fz(q)e(<)e(n)k FA(and)f(the)i(other)f Fz(n)20 b Fx(\000)h Fz(q)34 b FA(eigen)m(v)-5 b(alues)31 b(ha)m(v)m(e)h(strictly)118 4604 y(negativ)m(e)g(real)e(parts.)259 4717 y(The)h(most)i(generic)e (cases)i(are)f(the)g(saddle{no)s(de)e(bifurcation)g(\(where)i Fz(q)e FA(=)d(1\),)33 b(corresp)s(onding)118 4830 y(to)e(a)g(fold)f(in) 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Fz(x)1455 5474 y Fv(\000)1542 5507 y Fx(2)d Fr(R)1705 5474 y Fw(n)p Fv(\000)p Fw(q)1878 5507 y FA(and)k Fz(z)h Fx(2)27 b Fr(R)2293 5474 y Fw(q)2337 5507 y FA(,)32 b(in)f(whic)m(h)f (the)i(matrix)f Fz(@)3266 5521 y Fw(x)3310 5507 y Fz(f)10 b FA(\()c(^)-51 b Fz(x;)22 b FA(^)-52 b Fz(y)s(;)15 b FA(0\))118 5620 y(b)s(ecomes)38 b(blo)s(c)m(k-diagonal,)g(with)e(a)i (blo)s(c)m(k)f Fz(A)1736 5587 y Fv(\000)1832 5620 y Fx(2)f Fr(R)2004 5587 y Fs(\()p Fw(n)p Fv(\000)p Fw(q)r Fs(\))p Fv(\002)p Fs(\()q Fw(n)p Fv(\000)p Fw(q)r Fs(\))2479 5620 y FA(ha)m(ving)h(eigen)m(v)-5 b(alues)37 b(in)f(the)i(left)1844 5871 y(15)p eop %%Page: 16 16 16 15 bop 118 280 a FA(half-plane,)29 b(and)h(a)h(blo)s(c)m(k)f Fz(A)1126 247 y Fs(0)1191 280 y Fx(2)25 b Fr(R)1352 247 y Fw(q)r Fv(\002)p Fw(q)1515 280 y FA(ha)m(ving)30 b(eigen)m(v)-5 b(alues)30 b(on)h(the)f(imaginary)f(axis.)41 b(On)29 b(the)i(fast)118 393 y(timescale)g Fz(t=")p FA(,)g(\(2.59\))h(can)f(b)s (e)f(rewritten)f(as)1438 565 y(\()p Fz(x)1525 527 y Fv(\000)1584 565 y FA(\))1619 527 y Fv(0)1668 565 y FA(=)c Fz(f)1819 527 y Fv(\000)1877 565 y FA(\()p Fz(x)1964 527 y Fv(\000)2024 565 y Fz(;)15 b(z)t(;)g(y)s(;)g(")p FA(\))p Fz(;)1573 712 y(z)1619 675 y Fv(0)1668 712 y FA(=)25 b Fz(f)1819 675 y Fs(0)1858 712 y FA(\()p Fz(x)1945 675 y Fv(\000)2004 712 y Fz(;)15 b(z)t(;)g(y)s(;)g(")p FA(\))p Fz(;)1572 850 y(y)1620 812 y Fv(0)1668 850 y FA(=)25 b Fz("g)s FA(\()p Fz(x)1939 812 y Fv(\000)1999 850 y Fz(;)15 b(z)t(;)g(y)s(;)g(") p FA(\))p Fz(;)1577 988 y(")1619 950 y Fv(0)1668 988 y FA(=)25 b(0)p Fz(;)3429 777 y FA(\(2.60\))118 1164 y(whic)m(h)40 b(admits)h(\()6 b(^)-51 b Fz(x)786 1131 y Fv(\000)845 1164 y Fz(;)21 b FA(^)-51 b Fz(z)5 b(;)21 b FA(^)-51 b Fz(y)s(;)15 b FA(0\))42 b(as)g(an)f(equilibrium)c(p)s(oin) m(t.)73 b(The)40 b(linearization)g(at)i(this)e(p)s(oin)m(t)h(has)118 1277 y Fz(q)23 b FA(+)d Fz(m)f FA(+)h(1)31 b(eigen)m(v)-5 b(alues)30 b(on)g(the)h(imaginary)d(axis)i(\(coun)m(ting)h(m)m 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FA(pla)m(ys)33 b(the)g(same)h(r^)-45 b(ole)33 b(the)g(adiabatic)g(manifold)e(pla)m(y)m(ed)i(in)f(the)i (stable)118 1955 y(case,)e(and)e(the)g(dynamics)f(on)36 b(\026)-51 b Fz(x)1249 1922 y Fv(\000)1339 1955 y FA(is)29 b(go)m(v)m(erned)j(b)m(y)e(the)h(reduced)e(equation)1362 2122 y Fz(")16 b FA(_)-41 b Fz(z)30 b FA(=)24 b Fz(f)1626 2085 y Fs(0)1665 2122 y FA(\()6 b(\026)-51 b Fz(x)1752 2085 y Fv(\000)1812 2122 y FA(\()p Fz(z)t(;)15 b(y)s(;)g(")p FA(\))p Fz(;)g(z)t(;)g(y)s(;)g(")p FA(\))p Fz(;)1419 2260 y FA(_)-42 b Fz(y)29 b FA(=)24 b Fz(g)s FA(\()6 b(\026)-51 b Fz(x)1704 2223 y Fv(\000)1764 2260 y FA(\()p Fz(z)t(;)15 b(y)s(;)g(")p FA(\))p Fz(;)g(z)t(;)g(y)s(;)g(")p FA(\))p Fz(:)3429 2191 y FA(\(2.61\))118 2437 y(The)30 b(function)35 b(\026)-51 b Fz(x)713 2404 y Fv(\000)772 2437 y FA(\()p Fz(z)t(;)15 b(y)s(;)g(")p FA(\))32 b(solv)m(es)f(the)g (PDE)688 2611 y Fz(f)743 2574 y Fv(\000)801 2611 y FA(\()6 b(\026)-51 b Fz(x)888 2574 y Fv(\000)948 2611 y FA(\()p Fz(z)t(;)15 b(y)s(;)g(")p 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3249 y FA(=)1118 3187 y(1)p 1118 3228 46 4 v 1120 3311 a Fz(")1174 3249 y(f)1229 3211 y Fv(\000)1287 3249 y FA(\()p Fz(x)1374 3210 y Fv(\000)1374 3273 y Fw(t)1433 3249 y Fz(;)15 b(z)1515 3263 y Fw(t)1546 3249 y Fz(;)g(y)1631 3263 y Fw(t)1660 3249 y Fz(;)g(")p FA(\))g(d)q Fz(t)20 b FA(+)2029 3187 y Fz(\033)p 1998 3228 119 4 v 1998 3246 a Fx(p)p 2074 3246 43 4 v 65 x Fz(")2126 3249 y(F)2197 3211 y Fv(\000)2256 3249 y FA(\()p Fz(x)2343 3210 y Fv(\000)2343 3273 y Fw(t)2403 3249 y Fz(;)15 b(z)2485 3263 y Fw(t)2515 3249 y Fz(;)g(y)2600 3263 y Fw(t)2629 3249 y Fz(;)g(")p FA(\))g(d)q Fz(W)2899 3263 y Fw(t)2929 3249 y Fz(;)865 3486 y FA(d)o Fz(z)957 3500 y Fw(t)1012 3486 y FA(=)1118 3425 y(1)p 1118 3465 46 4 v 1120 3549 a Fz(")1174 3486 y(f)1229 3449 y Fs(0)1267 3486 y FA(\()p Fz(x)1354 3448 y Fv(\000)1354 3510 y Fw(t)1414 3486 y Fz(;)g(z)1496 3500 y Fw(t)1526 3486 y Fz(;)g(y)1611 3500 y Fw(t)1641 3486 y Fz(;)g(")p FA(\))g(d)p Fz(t)20 b FA(+)2010 3425 y Fz(\033)p 1978 3465 119 4 v 1978 3484 a 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4464 y(Z)1237 4490 y Fw(t)1197 4670 y Fs(0)1282 4587 y Fz(U)10 b FA(\()p Fz(s;)15 b(v)s FA(\))p Fz(F)1612 4601 y Fs(0)1653 4587 y FA(\()p Fz(y)1736 4550 y Fs(det)1733 4610 y Fw(v)1839 4587 y Fz(;)g(")p FA(\))p Fz(F)2014 4601 y Fs(0)2055 4587 y FA(\()p Fz(y)2138 4550 y Fs(det)2135 4610 y Fw(v)2240 4587 y Fz(;)g(")p FA(\))2357 4550 y Fw(T)2413 4587 y Fz(U)10 b FA(\()p Fz(s;)15 b(v)s FA(\))2685 4550 y Fw(T)2756 4587 y FA(d)p Fz(v)2854 4459 y Fq(\023)2921 4587 y Fz(Q)p FA(\()p Fz(t)p FA(\))3096 4550 y Fw(T)3152 4587 y Fz(:)252 b FA(\(3.22\))118 4837 y(Th)m(us,)30 b(using)f(completion)g(of)i (squares)f(to)h(compute)f(the)h(Gaussian)f(in)m(tegral,)g(w)m(e)h (\014nd)1054 5104 y Fr(E)1115 5066 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1228 5075 y Ft(0)1269 5066 y Fw(;)p Fs(0\))1355 5030 y Fq(\010)1408 5104 y FA(e)1449 5066 y Ff(b)-39 b Fw(\015)t Fv(k)p Fw(Q)p Fs(\()p Fw(t)p Fs(\)\007)1714 5043 y Ft(0)1714 5085 y Fl(t)1750 5066 y Fv(k)1785 5043 y Ft(2)1824 5030 y Fq(\011)1902 5104 y FA(=)2036 5042 y(e)2077 5009 y Ff(b)f Fw(\015)t Fv(h)p Fw(E)t(;)p Fs(\(1)-18 b(l)q Fv(\000)p Fs(2)q Ff(b)-40 b Fw(\015)t Fs(\006\))2492 4986 y Fk(\000)p Ft(1)2575 5009 y Fw(E)t Fv(i)p 2008 5083 683 4 v 2008 5170 a FA(\(det[1)-20 b(l)20 b Fx(\000)g FA(2)q Fq(b)-52 b Fz(\015)6 b FA(\006]\))2580 5144 y Fs(1)p Fw(=)p Fs(2)2700 5104 y Fz(:)704 b FA(\(3.23\))118 5361 y(By)32 b(\(2.19\))r(,)f(w)m(e)g(can)f(write)208 5565 y(\006)25 b(=)g Fz(\033)450 5527 y Fs(2)489 5565 y Fz(Q)p FA(\()p Fz(t)p FA(\))p Fz(U)10 b FA(\()p Fz(s;)15 b(t)p FA(\))922 5491 y Fq(\002)p 984 5492 55 4 v 961 5565 a Fz(X)7 b FA(\()p Fz(y)1126 5527 y Fs(det)1123 5587 y Fw(t)1229 5565 y FA(\))20 b Fx(\000)g Fz(U)10 b FA(\()p Fz(t)p FA(\))p 1574 5492 V Fz(X)d FA(\()p Fz(y)1715 5527 y Fs(det)1712 5587 y(0)1819 5565 y FA(\))p Fz(U)j FA(\()p Fz(t)p FA(\))2029 5527 y Fw(T)2085 5491 y Fq(\003)2122 5565 y Fz(U)g FA(\()p Fz(s;)15 b(t)p FA(\))2380 5527 y Fw(T)2436 5565 y Fz(Q)p FA(\()p Fz(t)p FA(\))2611 5527 y Fw(T)2692 5565 y FA(=)25 b Fz(\033)2843 5527 y Fs(2)2882 5491 y Fq(\002)2920 5565 y FA(1)-20 b(l)20 b Fx(\000)g Fz(R)q(R)3222 5527 y Fw(T)3276 5491 y Fq(\003)3314 5565 y Fz(;)90 b FA(\(3.24\))1844 5871 y(22)p eop %%Page: 23 23 23 22 bop 118 280 a FA(where)32 b Fz(R)d FA(=)g Fz(Q)p FA(\()p Fz(t)p FA(\))p Fz(U)10 b FA(\()p Fz(s)p FA(\))p 965 207 55 4 v Fz(X)d FA(\()p Fz(y)1106 247 y Fs(det)1103 305 y(0)1209 280 y FA(\))1244 247 y Fs(1)p Fw(=)p Fs(2)1354 280 y FA(,)33 b(and)f(w)m(e)h(ha)m(v)m(e)h(used)d(the)i(fact)g(that)g Fz(U)10 b FA(\()p Fz(s;)15 b(t)p FA(\))p 2969 207 V Fz(X)7 b FA(\()p Fz(y)3110 247 y Fs(det)3107 303 y Fw(t)3213 280 y FA(\))p Fz(U)j FA(\()p Fz(s;)15 b(t)p FA(\))3506 247 y Fw(T)3591 280 y FA(=)118 393 y Fz(Q)p FA(\()p Fz(t)p FA(\))293 360 y Fv(\000)p Fs(2)388 393 y FA(.)40 b(This)29 b(sho)m(ws)h(in)f(particular)g(that)1332 598 y(det)q([1)-20 b(l)20 b Fx(\000)g FA(2)q Fq(b)-52 b Fz(\015)5 b FA(\006])25 b Fn(>)g FA(\(1)c Fx(\000)f FA(2)q Fq(b)-52 b Fz(\015)6 b(\033)2300 560 y Fs(2)2339 598 y FA(\))2374 560 y Fw(n)2422 598 y Fz(:)982 b FA(\(3.25\))118 802 y(Moreo)m(v)m(er,)33 b(since)d Fx(k)p Fz(R)q(R)951 769 y Fw(T)1006 802 y Fx(k)25 b FA(=)g Fx(k)p Fz(R)1287 769 y Fw(T)1342 802 y Fz(R)q Fx(k)h(2)f FA(\(0)p Fz(;)15 b FA(1\),)32 b(w)m(e)f(also)f(ha)m(v)m(e) 511 1006 y Fx(h)p Fz(E)5 b(;)15 b FA(\(1)-20 b(l)21 b Fx(\000)f FA(2)q Fq(b)-52 b Fz(\015)6 b FA(\006\))1055 968 y Fv(\000)p Fs(1)1149 1006 y Fz(E)f Fx(i)26 b FA(=)f Fx(h)p 1437 933 V Fz(X)7 b FA(\()p Fz(y)1578 968 y Fs(det)1575 1028 y(0)1681 1006 y FA(\))1716 968 y Fv(\000)p Fs(1)p Fw(=)p Fs(2)1882 1006 y Fz(\030)1922 1020 y Fs(0)1961 1006 y Fz(;)15 b(R)2071 968 y Fw(T)2126 1006 y FA(\(1)-20 b(l)20 b Fx(\000)g FA(2)q Fq(b)-52 b Fz(\015)6 b FA(\006\))2522 968 y Fv(\000)p Fs(1)2616 1006 y Fz(R)p 2709 933 V 1 w(X)h FA(\()p Fz(y)2851 968 y Fs(det)2848 1028 y(0)2953 1006 y FA(\))2988 968 y Fv(\000)p Fs(1)p Fw(=)p Fs(2)3154 1006 y Fz(\030)3194 1020 y Fs(0)3233 1006 y Fx(i)1282 1167 y Fn(6)25 b Fz(\013)1436 1129 y Fs(2)1476 1167 y Fz(h)1528 1129 y Fs(2)1568 1090 y Fq(\015)1568 1144 y(\015)1618 1167 y Fz(R)1688 1129 y Fw(T)1743 1093 y Fq(\000)1785 1167 y FA(1)-20 b(l)20 b Fx(\000)g FA(2)q Fq(b)-52 b Fz(\015)5 b(\033)2099 1129 y Fs(2)2139 1167 y FA([1)-20 b(l)20 b Fx(\000)g Fz(R)q(R)2466 1129 y Fw(T)2520 1167 y FA(])2545 1093 y Fq(\001)2587 1116 y Fv(\000)p Fs(1)2681 1167 y Fz(R)2751 1090 y Fq(\015)2751 1144 y(\015)1282 1328 y Fn(6)25 b Fz(\013)1436 1290 y Fs(2)1476 1328 y Fz(h)1528 1290 y Fs(2)1568 1254 y Fq(\000)1610 1328 y FA([1)c Fx(\000)e FA(2)q Fq(b)-52 b Fz(\015)6 b(\033)1944 1290 y Fs(2)1984 1328 y FA(])p Fx(k)p Fz(R)2124 1290 y Fw(T)2179 1328 y Fz(R)q Fx(k)2294 1290 y Fv(\000)p Fs(1)2409 1328 y FA(+)20 b(2)q Fq(b)-52 b Fz(\015)5 b(\033)2652 1290 y Fs(2)2692 1254 y Fq(\001)2734 1277 y Fv(\000)p Fs(1)2853 1328 y Fn(6)25 b Fz(\013)3007 1290 y Fs(2)3047 1328 y Fz(h)3099 1290 y Fs(2)3429 1328 y FA(\(3.26\))118 1542 y(for)33 b(all)f Fz(\030)429 1556 y Fs(0)502 1542 y FA(satisfying)g Fx(h)p Fz(\030)985 1556 y Fs(0)1025 1542 y Fz(;)p 1088 1469 V 15 w(X)7 b FA(\()p Fz(y)1227 1556 y Fs(0)1267 1542 y FA(\))1302 1509 y Fv(\000)p Fs(1)1396 1542 y Fz(\030)1436 1556 y Fs(0)1476 1542 y Fx(i)30 b Fn(6)g Fz(\013)1700 1509 y Fs(2)1740 1542 y Fz(h)1792 1509 y Fs(2)1832 1542 y FA(.)50 b(No)m(w,)35 b(\(3.20\))g(follo)m(ws)e (from)g(\(3.23\))i(b)m(y)f(c)m(ho)s(osing)119 1654 y Fq(b)-52 b Fz(\015)30 b FA(=)25 b Fz(\015)5 b(=\033)443 1621 y Fs(2)484 1654 y FA(.)p 3595 1654 4 62 v 3599 1596 55 4 v 3599 1654 V 3653 1654 4 62 v 118 1842 a Fm(Lemma)33 b(3.3.)42 b Fy(Under)33 b(the)g(hyp)-5 b(otheses)35 b(of)e(Pr)-5 b(op)g(osition)35 b(3.1,)133 2120 y Fz(P)191 2134 y Fs(1)256 2120 y FA(=)25 b Fr(P)413 2083 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)526 2092 y Ft(0)561 2083 y Fw(;)p Fs(0\))648 1992 y Fq(\032)866 2120 y FA(sup)716 2197 y Fw(s)p Fo(6)p Fw(u)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)1003 2209 y Fl(\030)1037 2197 y Fv(^)p Fw(\034)1115 2205 y Fl(\021)1153 2120 y Fx(k)p Fz(Q)p FA(\()p Fz(t)p FA(\)\007)1444 2083 y Fs(1)1444 2143 y Fw(u)1490 2120 y Fx(k)h Fn(>)f Fz(H)1733 2134 y Fs(1)1772 1992 y Fq(\033)1850 2120 y Fn(6)10 b FA(exp)2085 1992 y Fq(\032)2153 2120 y Fx(\000)2234 1980 y Fq(\000)2275 2053 y Fz(H)2358 2020 y Fs(2)2351 2078 y(1)2418 2053 y Fx(\000)20 b Fz(\033)2564 2020 y Fs(2)2603 2053 y Fz(M)2701 2020 y Fs(2)2691 2078 y(1)2741 2053 y FA(\()2776 1986 y Fx(p)p 2852 1986 137 4 v 67 x Fz(K)2929 2067 y Fs(+)2988 2053 y Fz(h)h FA(+)f Fz(h)3204 2067 y Fs(1)3244 2053 y FA(\))3279 2020 y Fs(2)3318 2053 y FA(\010\()p Fz(t)p FA(\))3487 1980 y Fq(\001)3529 2002 y Fs(2)p 2234 2100 1336 4 v 2325 2185 a FA(8)p Fz(\033)2425 2158 y Fs(2)2465 2185 y Fz(M)2563 2153 y Fs(2)2553 2211 y(1)2603 2185 y FA(\()2638 2118 y Fx(p)p 2714 2118 137 4 v 67 x Fz(K)2791 2199 y Fs(+)2850 2185 y Fz(h)h FA(+)f Fz(h)3066 2199 y Fs(1)3106 2185 y FA(\))3141 2158 y Fs(2)3180 2185 y Fz(H)3263 2153 y Fs(2)3256 2211 y(1)3303 2185 y FA(\011\()p Fz(t)p FA(\))3579 1992 y Fq(\033)3429 2304 y FA(\(3.27\))118 2417 y Fy(holds)34 b(uniformly)g(for)f(al)5 b(l)33 b Fz(\030)1080 2431 y Fs(0)1152 2417 y Fy(such)g(that)h Fx(h)p Fz(\030)1620 2431 y Fs(0)1659 2417 y Fz(;)p 1723 2344 55 4 v 15 w(X)7 b FA(\()p Fz(y)1861 2431 y Fs(0)1902 2417 y FA(\))1937 2384 y Fv(\000)p Fs(1)2031 2417 y Fz(\030)2071 2431 y Fs(0)2110 2417 y Fx(i)26 b Fn(6)f Fz(h)2319 2384 y Fs(2)2359 2417 y Fy(.)118 2605 y Fg(Pr)n(oof:)45 b FA(Let)31 b Fz(\034)40 b FA(denote)31 b(the)g(stopping)e(time)1059 2809 y Fz(\034)35 b FA(=)25 b Fz(\034)1270 2824 y Fw(\030)1328 2809 y Fx(^)20 b Fz(\034)1449 2823 y Fw(\021)1510 2809 y Fx(^)g FA(inf)6 b Fx(f)p Fz(u)26 b Fn(>)f FA(0)10 b(:)31 b Fx(k)p Fz(Q)p FA(\()p Fz(t)p FA(\)\007)2323 2772 y Fs(1)2323 2832 y Fw(u)2368 2809 y Fx(k)26 b Fn(>)f Fz(H)2611 2823 y Fs(1)2650 2809 y Fx(g)p Fz(;)709 b FA(\(3.28\))118 3013 y(and)30 b(de\014ne,)g(for)g(a)h(giv)m(en)f Fz(\015)1082 3027 y Fs(1)1122 3013 y FA(,)g(the)h(sto)s(c)m(hastic)g(pro)s(cess)1530 3218 y(\004)1591 3232 y Fw(u)1661 3218 y FA(=)25 b(e)1797 3180 y Fw(\015)1833 3189 y Ft(1)1868 3180 y Fv(k)p Fw(Q)p Fs(\()p Fw(t)p Fs(\)\007)2093 3157 y Ft(1)2093 3197 y Fl(u)2135 3180 y Fv(k)2170 3157 y Ft(2)2224 3218 y Fz(:)1180 b FA(\(3.29\))118 3422 y(\(\004)214 3436 y Fw(u)259 3422 y FA(\))294 3436 y Fw(u)380 3422 y FA(b)s(eing)40 b(a)h(p)s(ositiv)m(e) f(submartingale,)i(another)f(application)f(of)h(Do)s(ob's)g (submartingale)e(in-)118 3535 y(equalit)m(y)30 b(yields)1334 3648 y Fz(P)1392 3662 y Fs(1)1457 3648 y Fn(6)25 b FA(e)1594 3610 y Fv(\000)p Fw(\015)1685 3619 y Ft(1)1720 3610 y Fw(H)1783 3587 y Ft(2)1778 3631 y(1)1837 3648 y Fr(E)1897 3610 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)2010 3619 y Ft(0)2051 3610 y Fw(;)p Fs(0\))2138 3574 y Fq(\010)2191 3648 y FA(\004)2252 3662 y Fw(t)p Fv(^)p Fw(\034)2367 3574 y Fq(\011)2420 3648 y Fz(:)984 b FA(\(3.30\))118 3814 y(It^)-45 b(o's)28 b(form)m(ula)e(\(together)j(with)d(the)i(fact)g(that)f(\(d)p Fz(W)1940 3828 y Fw(u)1985 3814 y FA(\))2020 3781 y Fw(T)2076 3814 y Fz(R)2146 3781 y Fw(T)2200 3814 y Fz(R)16 b FA(d)p Fz(W)2422 3828 y Fw(u)2492 3814 y FA(=)25 b(T)-8 b(r\()p Fz(R)2787 3781 y Fw(T)2842 3814 y Fz(R)q FA(\))15 b(d)o Fz(u)28 b FA(for)e(an)m(y)i(matrix)118 3927 y Fz(R)e Fx(2)f Fr(R)374 3894 y Fw(n)p Fv(\002)p Fw(k)520 3927 y FA(\))31 b(sho)m(ws)f(that)h(\004)1105 3941 y Fw(u)1180 3927 y FA(ob)s(eys)f(the)g(SDE)118 4187 y(d\004)230 4201 y Fw(u)299 4187 y FA(=)25 b(2)p Fz(\015)487 4201 y Fs(1)569 4126 y Fz(\033)p 537 4166 119 4 v 537 4184 a Fx(p)p 613 4184 43 4 v 66 x Fz(")665 4187 y FA(\004)726 4201 y Fw(u)771 4187 y FA(\(\007)877 4149 y Fs(1)877 4209 y Fw(u)922 4187 y FA(\))957 4149 y Fw(T)1013 4187 y Fz(Q)p FA(\()p Fz(t)p FA(\))1188 4149 y Fs(2)1227 4187 y Fz(U)10 b FA(\()p Fz(s;)15 b(u)p FA(\))p Fz(F)1562 4201 y Fs(1)1603 4187 y FA(\()p Fz(\030)1678 4201 y Fw(u)1723 4187 y Fz(;)g(\021)1808 4201 y Fw(u)1854 4187 y Fz(;)g(u;)g(")p FA(\))g(d)q Fz(W)2216 4201 y Fw(u)2267 4187 y FA(+)7 b Fz(\015)2392 4201 y Fs(1)2441 4126 y Fz(\033)2496 4093 y Fs(2)p 2441 4166 95 4 v 2467 4249 a Fz(")2546 4187 y FA(\004)2607 4201 y Fw(u)2667 4187 y FA(T)-8 b(r)2760 4113 y Fq(\002)2798 4187 y Fz(R)2868 4149 y Fw(T)2867 4209 y Fs(1)2923 4187 y Fz(R)2992 4201 y Fs(1)3038 4187 y FA(+)7 b(2)p Fz(\015)3208 4201 y Fs(1)3247 4187 y Fz(R)3317 4149 y Fw(T)3316 4209 y Fs(2)3372 4187 y Fz(R)3441 4201 y Fs(2)3481 4113 y Fq(\003)3534 4187 y FA(d)o Fz(u;)3429 4348 y FA(\(3.31\))118 4461 y(where)1116 4640 y Fz(R)1185 4654 y Fs(1)1249 4640 y FA(=)25 b Fz(Q)p FA(\()p Fz(t)p FA(\))p Fz(U)10 b FA(\()p Fz(s;)15 b(u)p FA(\))p Fz(F)1855 4654 y Fs(1)1896 4640 y FA(\()p Fz(\030)1971 4654 y Fw(u)2017 4640 y Fz(;)g(\021)2102 4654 y Fw(u)2147 4640 y Fz(;)g(u;)g(")p FA(\))p Fz(;)1116 4790 y(R)1185 4804 y Fs(2)1249 4790 y FA(=)25 b(\(\007)1451 4752 y Fs(1)1451 4812 y Fw(u)1496 4790 y FA(\))1531 4752 y Fw(T)1587 4790 y Fz(Q)p FA(\()p Fz(t)p FA(\))1762 4752 y Fs(2)1802 4790 y Fz(U)10 b FA(\()p Fz(s;)15 b(u)p FA(\))p Fz(F)2137 4804 y Fs(1)2177 4790 y FA(\()p Fz(\030)2252 4804 y Fw(u)2297 4790 y Fz(;)g(\021)2382 4804 y Fw(u)2428 4790 y Fz(;)g(u;)g(")p FA(\))p Fz(:)3429 4715 y FA(\(3.32\))118 4986 y(The)30 b(\014rst)g(term)g(in)f(the)i(trace)g(can)g(b)s(e)f (estimated)g(as)286 5190 y(T)-8 b(r)380 5117 y Fq(\002)418 5190 y Fz(R)488 5153 y Fw(T)487 5213 y Fs(1)543 5190 y Fz(R)612 5204 y Fs(1)651 5117 y Fq(\003)714 5190 y FA(=)25 b(T)-8 b(r)904 5117 y Fq(\002)942 5190 y Fz(R)1011 5204 y Fs(1)1050 5190 y Fz(R)1120 5153 y Fw(T)1119 5213 y Fs(1)1175 5117 y Fq(\003)1238 5190 y Fn(6)25 b Fz(M)1432 5153 y Fs(2)1422 5213 y(1)1472 5117 y Fq(\000)1513 5190 y Fx(k)p Fz(\030)1598 5204 y Fw(u)1644 5190 y Fx(k)20 b FA(+)g Fx(k)p Fz(\021)1890 5204 y Fw(u)1936 5190 y Fx(k)1981 5117 y Fq(\001)2023 5139 y Fs(2)2078 5190 y FA(T)-8 b(r)2171 5117 y Fq(\002)2209 5190 y Fz(Q)p FA(\()p Fz(t)p FA(\))2384 5153 y Fw(T)2440 5190 y Fz(U)10 b FA(\()p Fz(s;)15 b(u)p FA(\))p Fz(U)10 b FA(\()p Fz(s;)15 b(u)p FA(\))2994 5153 y Fw(T)3050 5190 y Fz(Q)p FA(\()p Fz(t)p FA(\))3225 5117 y Fq(\003)1238 5351 y Fn(6)25 b Fz(M)1432 5314 y Fs(2)1422 5374 y(1)1472 5278 y Fq(\000)1513 5351 y Fx(k)p Fz(\030)1598 5365 y Fw(u)1644 5351 y Fx(k)20 b FA(+)g Fx(k)p Fz(\021)1890 5365 y Fw(u)1936 5351 y Fx(k)1981 5278 y Fq(\001)2023 5300 y Fs(2)2078 5351 y FA(T)-8 b(r)2171 5278 y Fq(\002)2209 5351 y Fz(U)10 b FA(\()p Fz(t;)15 b(u)p FA(\))2476 5314 y Fw(T)p 2555 5278 55 4 v 2532 5351 a Fz(X)7 b FA(\()p Fz(y)2697 5314 y Fs(det)2694 5374 y Fw(t)2800 5351 y FA(\))2835 5314 y Fv(\000)p Fs(1)2930 5351 y Fz(U)j FA(\()p Fz(t;)15 b(u)p FA(\))3197 5278 y Fq(\003)3235 5351 y Fz(;)169 b FA(\(3.33\))1844 5871 y(23)p eop %%Page: 24 24 24 23 bop 118 280 a FA(while)28 b(the)j(second)f(term)h(satis\014es)f (the)g(b)s(ound)379 485 y(T)-8 b(r)473 411 y Fq(\002)511 485 y Fz(R)581 447 y Fw(T)580 507 y Fs(2)636 485 y Fz(R)705 499 y Fs(2)744 411 y Fq(\003)807 485 y FA(=)903 407 y Fq(\015)903 462 y(\015)954 485 y Fz(F)1012 499 y Fs(1)1052 485 y FA(\()p Fz(\030)1127 499 y Fw(u)1172 485 y Fz(;)15 b(\021)1257 499 y Fw(u)1302 485 y Fz(;)g(u;)g(")p FA(\))1511 447 y Fw(T)1568 485 y Fz(U)10 b FA(\()p Fz(s;)15 b(u)p FA(\))1845 447 y Fw(T)1901 485 y Fz(Q)p FA(\()p Fz(t)p FA(\))2076 447 y Fs(2)2116 485 y FA(\007)2187 447 y Fs(1)2187 507 y Fw(u)2231 407 y Fq(\015)2231 462 y(\015)2282 434 y Fs(2)807 645 y Fn(6)25 b Fz(M)1001 608 y Fs(2)991 668 y(1)1041 572 y Fq(\000)1083 645 y Fx(k)p Fz(\030)1168 659 y Fw(u)1213 645 y Fx(k)20 b FA(+)g Fx(k)p Fz(\021)1459 659 y Fw(u)1505 645 y Fx(k)1550 572 y Fq(\001)1592 595 y Fs(2)1632 568 y Fq(\015)1632 623 y(\015)1682 645 y Fz(U)10 b FA(\()p Fz(s;)15 b(u)p FA(\))1959 608 y Fw(T)2015 645 y Fz(Q)p FA(\()p Fz(t)p FA(\))2190 568 y Fq(\015)2190 623 y(\015)2241 595 y Fs(2)2280 645 y Fx(k)p Fz(Q)p FA(\()p Fz(t)p FA(\)\007)2571 608 y Fs(1)2571 668 y Fw(u)2617 645 y Fx(k)2662 608 y Fs(2)807 806 y FA(=)25 b Fz(M)1001 769 y Fs(2)991 829 y(1)1041 733 y Fq(\000)1083 806 y Fx(k)p Fz(\030)1168 820 y Fw(u)1213 806 y Fx(k)20 b FA(+)g Fx(k)p Fz(\021)1459 820 y Fw(u)1505 806 y Fx(k)1550 733 y 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Fs(det)1214 1775 y Fw(t)1320 1752 y FA(\))1355 1715 y Fv(\000)p Fs(1)1449 1752 y Fz(U)j FA(\()p Fz(t;)15 b(v)s FA(\))1711 1679 y Fq(\003)1771 1752 y FA(+)20 b(2)p Fz(\015)1954 1766 y Fs(1)1993 1752 y Fz(H)2076 1715 y Fs(2)2069 1775 y(1)2116 1675 y Fq(\015)2116 1730 y(\015)2166 1752 y Fz(U)10 b FA(\()p Fz(t;)15 b(v)s FA(\))2428 1715 y Fw(T)p 2508 1679 V 2485 1752 a Fz(X)7 b FA(\()p Fz(y)2650 1715 y Fs(det)2647 1775 y Fw(t)2752 1752 y FA(\))2787 1715 y Fv(\000)p Fs(1)2882 1752 y Fz(U)j FA(\()p Fz(t;)15 b(v)s FA(\))3144 1675 y Fq(\015)3144 1730 y(\015)3196 1651 y(i)3254 1752 y FA(d)o Fz(v)s(;)53 b FA(\(3.35\))118 1974 y(and)30 b(Gron)m(w)m(all's)g(inequalit)m(y)e(yields)424 2196 y Fr(E)485 2158 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)598 2167 y Ft(0)639 2158 y Fw(;)p Fs(0\))725 2122 y Fq(\010)778 2196 y FA(\004)839 2210 y Fw(t)p Fv(^)p Fw(\034)955 2122 y Fq(\011)1033 2196 y Fn(6)d FA(exp)1268 2095 y Fq(n)1328 2196 y Fz(\015)1375 2210 y Fs(1)1415 2196 y Fz(\033)1470 2158 y Fs(2)1510 2196 y 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5529 y Fs(0)467 5515 y Fy(,)e Fz(M)625 5482 y Fv(0)615 5539 y Fs(0)687 5515 y Fy(ar)-5 b(e)34 b(c)-5 b(onstants)34 b(dep)-5 b(ending)34 b(only)g(on)f Fx(k)2055 5492 y Fq(b)2037 5515 y Fz(F)13 b Fx(k)2153 5529 y Fv(1)2228 5515 y Fy(,)33 b Fx(k)2352 5492 y Fq(b)2334 5515 y Fz(G)q Fx(k)2451 5529 y Fv(1)2526 5515 y Fy(,)f Fz(M)10 b Fy(,)33 b Fx(k)p Fz(C)7 b Fx(k)2907 5529 y Fv(1)3014 5515 y Fy(and)34 b Fz(U)10 b Fy(.)1844 5871 y FA(24)p eop %%Page: 25 25 25 24 bop 118 280 a Fg(Pr)n(oof:)45 b FA(W)-8 b(e)26 b(\014rst)e(consider)g(a)h(time)g(in)m(terv)-5 b(al)24 b([)p Fz(s;)15 b(t)p FA(])25 b(with)f Fz(t)9 b Fx(\000)g Fz(s)25 b FA(=)g(\001)p Fz(")p FA(.)39 b(Let)25 b Fz(u)h Fx(2)e FA([)p Fz(s;)15 b(t)p FA(])26 b(and)e(recall)g(the)118 393 y(de\014ning)i(SDE)i(\(3.3\))i(for)d Fz(\021)1065 407 y Fw(u)1110 393 y FA(.)40 b(Its)28 b(solution)f(can)h(b)s(e)f (split)f(in)m(to)i(four)g(parts,)g Fz(\021)2805 407 y Fw(u)2875 393 y FA(=)d Fz(\021)3019 360 y Fs(0)3016 416 y Fw(u)3077 393 y FA(+)15 b Fz(\021)3211 360 y Fs(1)3208 416 y Fw(u)3268 393 y FA(+)g Fz(\021)3402 360 y Fs(2)3399 416 y Fw(u)3460 393 y FA(+)g Fz(\021)3594 360 y Fs(3)3591 416 y Fw(u)3636 393 y FA(,)118 506 y(where)1128 732 y Fz(\021)1176 694 y Fs(0)1173 754 y Fw(u)1243 732 y FA(=)25 b Fz(\033)1394 694 y Fv(0)1433 608 y Fq(Z)1524 634 y Fw(u)1483 814 y Fs(0)1584 732 y Fz(V)20 b FA(\()p Fz(u;)15 b(v)s FA(\))1886 709 y Fq(b)1866 732 y Fz(G)r FA(\()p Fz(\030)2014 746 y Fw(v)2055 732 y Fz(;)g(\021)2140 746 y Fw(v)2181 732 y Fz(;)g(v)s(;)g(")p FA(\))g(d)r Fz(W)2539 746 y Fw(v)2580 732 y Fz(;)1128 974 y(\021)1176 936 y Fs(1)1173 996 y Fw(u)1243 974 y FA(=)1380 912 y Fz(\033)p 1349 953 119 4 v 1349 971 a Fx(p)p 1425 971 43 4 v 65 x Fz(")1492 850 y Fq(Z)1583 876 y Fw(u)1543 1056 y Fs(0)1643 974 y Fz(S)5 b FA(\()p Fz(u;)15 b(v)s FA(\))1933 951 y Fq(b)1913 974 y Fz(F)g FA(\()p Fz(\030)2061 988 y Fw(v)2102 974 y Fz(;)g(\021)2187 988 y Fw(v)2228 974 y Fz(;)g(v)s(;)g(")p FA(\))g(d)r Fz(W)2586 988 y Fw(v)2627 974 y Fz(;)1128 1217 y(\021)1176 1180 y Fs(2)1173 1240 y Fw(u)1243 1217 y FA(=)1339 1094 y Fq(Z)1430 1120 y Fw(u)1390 1300 y Fs(0)1490 1217 y Fz(V)20 b FA(\()p Fz(u;)15 b(v)s FA(\))p Fz(c)p FA(\()p Fz(\030)1886 1231 y Fw(v)1929 1217 y Fz(;)g(\021)2014 1231 y Fw(v)2056 1217 y Fz(;)g(v)s(;)g(")p FA(\))g(d)q Fz(v)s(;)1128 1459 y(\021)1176 1422 y Fs(3)1173 1482 y Fw(u)1243 1459 y FA(=)1349 1398 y(1)p 1349 1438 46 4 v 1351 1521 a Fz(")1420 1335 y Fq(Z)1510 1362 y Fw(u)1470 1541 y Fs(0)1571 1459 y Fz(S)5 b FA(\()p Fz(u;)15 b(v)s FA(\))p Fz(b)p FA(\()p Fz(\030)1955 1473 y Fw(v)1997 1459 y Fz(;)g(\021)2082 1473 y Fw(v)2124 1459 y Fz(;)g(v)s(;)g(")p FA(\))g(d)q Fz(v)s(;)3429 1097 y FA(\(3.41\))118 1704 y(with)1050 1839 y Fz(S)5 b FA(\()p Fz(u;)15 b(v)s FA(\))27 b(=)1443 1715 y Fq(Z)1534 1741 y Fw(u)1493 1921 y(v)1594 1839 y Fz(V)20 b FA(\()p Fz(u;)15 b(w)r FA(\))p Fz(C)7 b FA(\()p Fz(y)2051 1801 y Fs(det)2048 1861 y Fw(w)2155 1839 y Fz(;)15 b(")p FA(\))p Fz(U)10 b FA(\()p Fz(w)r(;)15 b(v)s FA(\))g(d)r Fz(w)r(:)701 b FA(\(3.42\))118 2046 y(Let)46 b Fz(\034)61 b FA(=)50 b Fz(\034)558 2065 y Fv(B)r Fs(\()p Fw(h)p Fs(\))735 2046 y Fx(^)30 b Fz(\034)866 2060 y Fw(\021)908 2046 y FA(.)86 b(It)46 b(follo)m(ws)e(immediately)g (from)h(the)h(de\014nitions)d(of)i Fz(\034)2994 2065 y Fv(B)r Fs(\()p Fw(h)p Fs(\))3142 2046 y FA(,)k Fz(\034)3256 2060 y Fw(\021)3343 2046 y FA(and)c(the)118 2159 y(b)s(ounds)28 b(\(3.4\))k(that)1005 2292 y Fq(\015)1005 2346 y(\015)1056 2369 y Fz(\021)1104 2332 y Fs(2)1101 2392 y Fw(u)p Fv(^)p Fw(\034)1232 2292 y Fq(\015)1232 2346 y(\015)1308 2369 y Fn(6)25 b Fz(M)10 b FA(\(1)21 b(+)f Fx(O)s FA(\(\001)p Fz(")p FA(\)\))p Fz(\037)2049 2332 y Fs(\(1\))2144 2369 y FA(\()p Fz(t)p FA(\)\()p Fz(K)2359 2383 y Fs(+)2419 2369 y Fz(h)2471 2332 y Fs(2)2531 2369 y FA(+)g Fz(h)2674 2332 y Fs(2)2674 2392 y(1)2714 2369 y FA(\))p Fz(;)1005 2446 y Fq(\015)1005 2501 y(\015)1056 2524 y Fz(\021)1104 2486 y Fs(3)1101 2546 y Fw(u)p Fv(^)p Fw(\034)1232 2446 y Fq(\015)1232 2501 y(\015)1308 2524 y Fn(6)25 b Fz(M)1502 2486 y Fv(0)1525 2524 y Fz(\037)1582 2486 y Fs(\(1\))1676 2524 y FA(\()p Fz(t)p FA(\)\()p Fz(K)1891 2538 y Fs(+)1951 2524 y Fz(h)2003 2486 y Fs(2)2063 2524 y FA(+)2154 2443 y Fq(p)p 2245 2443 137 4 v 81 x Fz(K)2322 2538 y Fs(+)2381 2524 y Fz(hh)2485 2538 y Fs(1)2526 2524 y FA(\))3429 2443 y(\(3.43\))118 2727 y(for)32 b(all)g Fz(u)c Fx(2)h FA([)p Fz(s;)15 b(t)p FA(].)47 b(Here)33 b Fz(M)1111 2694 y Fv(0)1166 2727 y FA(dep)s(ends)e(only)g(on)i Fz(M)10 b FA(,)33 b Fz(U)42 b FA(and)32 b Fx(k)p Fz(C)7 b Fx(k)2451 2741 y Fv(1)2526 2727 y FA(.)47 b(F)-8 b(urthermore,)33 b(using)e(similar)118 2840 y(ideas)k(as)h(in)e(the)h(pro)s(of)g(of)g (Lemma)h(3.3,)i(it)d(is)f(straigh)m(tforw)m(ard)h(to)h(establish)e(for) h(all)g Fz(H)3284 2854 y Fs(0)3323 2840 y Fz(;)15 b(H)3439 2854 y Fs(1)3512 2840 y Fz(>)33 b FA(0)118 2953 y(that)583 3190 y Fr(P)644 3152 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)757 3161 y Ft(0)792 3152 y Fw(;)p Fs(0\))879 3062 y Fq(\032)1026 3190 y FA(sup)947 3267 y Fw(s)p Fo(6)p Fw(u)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)1242 3190 y Fx(k)p Fz(\021)1335 3152 y Fs(0)1332 3212 y Fw(u)1377 3190 y Fx(k)26 b Fn(>)f Fz(H)1620 3204 y Fs(0)1659 3062 y Fq(\033)1753 3190 y Fn(6)g FA(e)1889 3152 y Fw(m=)p Fs(4)2041 3190 y FA(exp)2180 3062 y Fq(\032)2248 3190 y Fx(\000)2385 3128 y Fz(H)2468 3095 y Fs(2)2461 3153 y(0)2508 3128 y FA(\(1)c Fx(\000)f(O)s FA(\(\001)p Fz(")p FA(\)\))p 2329 3169 726 4 v 2329 3268 a(8\()p Fz(\033)2464 3242 y Fv(0)2488 3268 y FA(\))2523 3242 y Fs(2)2563 3268 y Fx(k)2627 3245 y Fq(b)2608 3268 y Fz(G)q Fx(k)2725 3242 y Fs(2)2725 3291 y Fv(1)2800 3268 y Fz(\037)2857 3242 y Fs(\(2\))2951 3268 y FA(\()p Fz(t)p FA(\))3065 3062 y Fq(\033)3133 3190 y Fz(;)583 3462 y Fr(P)644 3424 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)757 3433 y Ft(0)792 3424 y Fw(;)p Fs(0\))879 3333 y Fq(\032)1026 3462 y FA(sup)947 3538 y Fw(s)p Fo(6)p Fw(u)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)1242 3462 y Fx(k)p Fz(\021)1335 3424 y Fs(1)1332 3484 y Fw(u)1377 3462 y Fx(k)26 b Fn(>)f Fz(H)1620 3476 y Fs(1)1659 3333 y Fq(\033)1753 3462 y Fn(6)g FA(e)1889 3424 y Fw(m=)p Fs(4)2041 3462 y FA(exp)2180 3333 y Fq(\032)2248 3462 y Fx(\000)2404 3400 y Fz(H)2487 3367 y Fs(2)2480 3424 y(1)2527 3400 y FA(\(1)c Fx(\000)f(O)s FA(\(\001)p Fz(")p FA(\)\))p 2329 3441 765 4 v 2329 3540 a(8)p Fz(\033)2429 3514 y Fs(2)2469 3540 y Fz("c)2550 3554 y Fw(S)2602 3540 y Fx(k)2665 3517 y Fq(b)2647 3540 y Fz(F)13 b Fx(k)2763 3514 y Fs(2)2763 3563 y Fv(1)2838 3540 y Fz(\037)2895 3514 y Fs(\(2\))2990 3540 y FA(\()p Fz(t)p FA(\))3103 3333 y Fq(\033)3171 3462 y Fz(;)3429 3331 y FA(\(3.44\))118 3731 y(where)33 b Fz(c)423 3745 y Fw(S)507 3731 y FA(is)g(a)g(constan)m(t)i(dep)s(ending)c(only)h(on)h Fz(S)5 b FA(.)50 b(Then)32 b(the)h(lo)s(cal)g(analogue)h(of)f(estimate) h(\(3.40\))118 3844 y(\(without)44 b(the)i Fz(t)p FA(-dep)s(enden)m(t)e (prefactor\))h(is)f(obtained)h(b)m(y)f(taking,)49 b(for)c(instance,)j Fz(H)3217 3858 y Fs(0)3306 3844 y FA(=)h Fz(H)3502 3858 y Fs(1)3591 3844 y FA(=)128 3921 y Fs(1)p 128 3936 36 4 v 128 3988 a(2)173 3957 y Fz(h)225 3971 y Fs(1)285 3957 y Fx(\000)20 b FA(2\()p Fz(M)31 b FA(+)20 b Fz(M)764 3924 y Fv(0)788 3957 y FA(\))p Fz(\037)880 3924 y Fs(\(1\))974 3957 y FA(\()p Fz(t)p FA(\)\()p Fz(K)1189 3971 y Fs(+)1249 3957 y Fz(h)1301 3924 y Fs(2)1361 3957 y FA(+)g Fz(h)1504 3924 y Fs(2)1504 3981 y(1)1544 3957 y FA(\).)259 4070 y(It)36 b(remains)e(to)i(extend)f(\(3.40\))j(to)e(a)g(general)f(time)g (in)m(terv)-5 b(al)35 b([0)p Fz(;)15 b(t)p FA(])36 b(for)g Fz(t)f FA(of)g(order)g(1.)56 b(F)-8 b(or)36 b(this)118 4182 y(purp)s(ose,)31 b(w)m(e)i(c)m(ho)s(ose)h(a)e(partition)f(0)e(=)g Fz(u)1598 4196 y Fs(0)1666 4182 y Fz(<)f(u)1817 4196 y Fs(1)1885 4182 y Fz(<)g Fx(\001)15 b(\001)g(\001)30 b Fz(<)e(u)2270 4196 y Fw(K)2367 4182 y FA(=)g Fz(t)k FA(of)g([0)p Fz(;)15 b(t)p FA(],)35 b(satisfying)c Fz(u)3323 4197 y Fw(k)3394 4182 y FA(=)d Fz(k)s FA(\001)p Fz(")118 4295 y FA(for)h(0)d Fn(6)f Fz(k)j(<)d(K)32 b FA(=)25 b Fx(d)p Fz(t=)p FA(\(\001)p Fz(")p FA(\))p Fx(e)p FA(.)42 b(Applying)27 b(the)i(lo)s(cal)f(v)m(ersion)g(of)37 b(\(3.40\))31 b(to)f(eac)m(h)g(in)m(terv)-5 b(al)28 b([)p Fz(u)3368 4310 y Fw(k)3411 4295 y Fz(;)15 b(u)3503 4310 y Fw(k)r Fs(+1)3636 4295 y FA(])118 4408 y(and)30 b(using)f(the)h(monotonicit)m (y)h(of)f Fz(\037)1400 4375 y Fs(\(2\))1495 4408 y FA(\()p Fz(u)p FA(\),)h(the)f(claimed)g(estimate)h(follo)m(ws)e(from)298 4699 y Fr(P)359 4661 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)472 4670 y Ft(0)507 4661 y Fw(;)p Fs(0\))594 4598 y Fq(n)796 4699 y FA(sup)654 4776 y Fs(0)p Fo(6)p Fw(u)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)943 4790 y Fk(B)q Ft(\()p Fl(h)p Ft(\))1074 4699 y Fx(k)p Fz(\021)1164 4713 y Fw(u)1209 4699 y Fx(k)d Fn(>)f Fz(h)1428 4713 y Fs(1)1468 4598 y Fq(o)1554 4699 y Fn(6)1650 4585 y Fw(K)5 b Fv(\000)p Fs(1)1661 4612 y Fq(X)1663 4810 y Fw(k)r Fs(=0)1819 4699 y Fr(P)1880 4661 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1993 4670 y Ft(0)2029 4661 y Fw(;)p Fs(0\))2115 4598 y Fq(n)2404 4699 y FA(sup)2176 4776 y Fw(u)2217 4788 y Fl(k)2255 4776 y Fo(6)p Fw(u)p Fo(6)p Fw(u)2447 4788 y Fl(k)q Ft(+1)2561 4776 y Fv(^)p Fw(\034)2639 4790 y Fk(B)q Ft(\()p Fl(h)p Ft(\))2770 4699 y Fx(k)p Fz(\021)2860 4713 y Fw(u)2905 4699 y Fx(k)26 b Fn(>)f Fz(h)3124 4713 y Fs(1)3164 4598 y Fq(o)3224 4699 y Fz(:)180 b FA(\(3.45\))p 3595 4980 4 62 v 3599 4922 55 4 v 3599 4980 V 3653 4980 4 62 v 259 5168 a(Prop)s(osition)42 b(3.1)i(and)f(Lemma)g(3.4)h(together)h (are)f(su\016cien)m(t)e(to)i(pro)m(v)m(e)g(Theorem)f(2.4)i(on)e(a)118 5281 y(timescale)34 b(of)f(order)g(1.)51 b(W)-8 b(e)35 b(con)m(tin)m(ue)e(to)i(assume)e(that)h Fz(y)2187 5295 y Fs(0)2256 5281 y Fx(2)c(D)2417 5295 y Fs(0)2490 5281 y FA(but)j(w)m(e)h(will)d(no)i(longer)h(assume)118 5394 y(that)h Fz(y)367 5361 y Fs(det)364 5416 y Fw(u)500 5394 y Fx(2)c(D)662 5408 y Fs(0)736 5394 y FA(automatically)i(holds)g(for)h (all)f Fz(u)e Fn(6)g Fz(t)p FA(.)52 b(Instead,)35 b(w)m(e)f(will)e (emplo)m(y)h(Lemma)h(3.4)i(to)118 5506 y(compare)d Fz(y)529 5520 y Fw(u)602 5506 y Fx(2)28 b(D)761 5520 y Fs(0)833 5506 y FA(and)k Fz(y)1060 5473 y Fs(det)1057 5529 y Fw(u)1162 5506 y FA(,)h(taking)f(adv)-5 b(an)m(tage)34 b(of)f(the)f(fact)h(that)g (on)g(timescales)f(of)g(order)g(1,)i Fz(\021)3632 5520 y Fw(t)118 5619 y FA(is)28 b(lik)m(ely)f(to)i(remain)f(small.)39 b(Note)30 b(that)f(if)f(the)h(uniform-h)m(yp)s(erb)s(olicit)m(y)24 b(Assumption)j(2.2)i(holds)f(for)1844 5871 y(25)p eop %%Page: 26 26 26 25 bop 118 280 a Fx(D)188 294 y Fs(0)228 280 y FA(,)27 b(then)f(there)g(exists)f(a)i Fz(\016)i(>)c FA(0)h(of)h(order)e(1)i (suc)m(h)e(that)i(the)f Fz(\016)s FA(-neigh)m(b)s(ourho)s(o)s(d)f Fx(D)2980 242 y Fs(+)2977 306 y(0)3038 280 y FA(\()p Fz(\016)s FA(\))j(also)e(satis\014es)118 393 y(this)37 b(assumption.)61 b(W)-8 b(e)39 b(in)m(tro)s(duce)d(the)i(\014rst-exit)g (time)f Fz(\034)2219 360 y Fs(det)2209 420 y Fv(D)2264 429 y Ft(0)2359 393 y FA(of)h(the)g(deterministic)d(pro)s(cess)i Fz(y)3559 360 y Fs(det)3556 416 y Fw(u)118 516 y FA(from)30 b Fx(D)406 478 y Fs(+)403 542 y(0)465 516 y FA(\()p Fz(\016)s FA(\))i(as)1256 629 y Fz(\034)1306 592 y Fs(det)1296 652 y Fv(D)1351 661 y Ft(0)1434 629 y FA(=)25 b(inf)5 b Fx(f)p Fz(u)26 b Fn(>)f FA(0)10 b(:)31 b Fz(y)2018 592 y Fs(det)2015 652 y Fw(u)2146 629 y Fx(62)24 b(D)2304 591 y Fs(+)2301 655 y(0)2363 629 y FA(\()p Fz(\016)s FA(\))p Fx(g)908 b FA(\(3.46\))118 790 y(and)30 b(remark)g(in)f (passing)g(that)i Fz(\034)1268 809 y Fv(B)r Fs(\()p Fw(h)p Fs(\))1436 790 y Fx(^)20 b Fz(\034)1557 804 y Fw(\021)1624 790 y Fn(6)k Fz(\034)1769 757 y Fs(det)1759 817 y Fv(D)1814 826 y Ft(0)1902 790 y FA(holds)29 b(whenev)m(er)h Fz(h)2591 804 y Fs(1)2656 790 y Fn(6)25 b Fz(\016)s FA(.)118 983 y Fm(Corollary)31 b(3.5.)39 b Fy(Fix)29 b(a)g(time)f Fz(t)d(>)g FA(0)k Fy(and)g Fz(h)d(>)f FA(0)k Fy(in)f(such)h(a)g(way)g (that)h Fz(h)25 b Fn(6)g Fz(c)2821 997 y Fs(1)2861 983 y Fz(\037)2918 950 y Fs(\(1\))3012 983 y FA(\()p Fz(t)11 b Fx(^)g Fz(\034)3213 950 y Fs(det)3203 1010 y Fv(D)3258 1019 y Ft(0)3316 983 y FA(\))3351 950 y Fv(\000)p Fs(1)3474 983 y Fy(for)29 b(a)118 1111 y(su\016ciently)j(smal)5 b(l)32 b(c)-5 b(onstant)33 b Fz(c)1216 1125 y Fs(1)1281 1111 y Fz(>)25 b FA(0)31 b Fy(and)i Fz(\037)1686 1078 y Fs(\(2\))1780 1111 y FA(\()p Fz(t)17 b Fx(^)g Fz(\034)1993 1078 y Fs(det)1983 1138 y Fv(D)2038 1147 y Ft(0)2095 1111 y FA(\))26 b Fn(6)f FA(\()p Fz(\032)2334 1078 y Fs(2)2391 1111 y FA(+)16 b Fz(")p FA(\))2555 1078 y Fv(\000)p Fs(1)2651 1111 y Fy(.)41 b(Then)31 b(for)h(any)g Fz(\013)26 b Fx(2)f FA([0)p Fz(;)15 b FA(1\))p Fy(,)118 1224 y(any)33 b Fz(\015)e Fx(2)25 b FA(\(0)p Fz(;)15 b FA(1)p Fz(=)p FA(2\))35 b Fy(and)f(any)f(su\016ciently)g(smal)5 b(l)33 b FA(\001)p Fy(,)381 1428 y Fz(C)453 1391 y Fv(\000)446 1451 y Fw(n;m)575 1428 y FA(\()p Fz(t;)15 b(")p FA(\))g(e)817 1391 y Fv(\000)p Fw(\024)913 1367 y Fk(\000)964 1391 y Fs(\(0\))p Fw(h)1094 1367 y Ft(2)1130 1391 y Fw(=\033)1207 1367 y Ft(2)1272 1428 y Fn(6)25 b Fr(P)1429 1391 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1542 1400 y Ft(0)1577 1391 y Fw(;)p Fs(0\))1664 1354 y Fq(\010)1717 1428 y Fz(\034)1757 1447 y Fv(B)r Fs(\()p Fw(h)p Fs(\))1929 1428 y Fz(<)g(t)2058 1354 y Fq(\011)2136 1428 y Fn(6)g Fz(C)2304 1391 y Fs(+)2297 1451 y Fw(n;m;\015)2486 1428 y FA(\()p Fz(t;)15 b(")p FA(\))g(e)2728 1391 y Fv(\000)p Fw(\024)2824 1367 y Ft(+)2874 1391 y Fs(\()p Fw(\013)p Fs(\))p Fw(h)3014 1367 y Ft(2)3050 1391 y Fw(=\033)3127 1367 y Ft(2)3429 1428 y FA(\(3.47\))118 1633 y Fy(holds)34 b(uniformly)g(for)f(al)5 b(l)33 b Fz(\030)1080 1647 y Fs(0)1152 1633 y Fy(satisfying)g Fx(h)p Fz(\030)1631 1647 y Fs(0)1671 1633 y Fz(;)p 1735 1560 55 4 v 15 w(X)7 b FA(\()p Fz(y)1873 1647 y Fs(0)1913 1633 y FA(\))1948 1600 y Fv(\000)p Fs(1)2043 1633 y Fz(\030)2083 1647 y Fs(0)2122 1633 y Fx(i)26 b Fn(6)f Fz(\013)2337 1600 y Fs(2)2377 1633 y Fz(h)2429 1600 y Fs(2)2468 1633 y Fy(.)42 b(Her)-5 b(e)455 1876 y Fz(\024)507 1838 y Fs(+)567 1876 y FA(\()p Fz(\013)p FA(\))26 b(=)f Fz(\015)869 1802 y Fq(\002)907 1876 y FA(1)c Fx(\000)f Fz(\013)1122 1838 y Fs(2)1182 1876 y Fx(\000)g(O)s FA(\(\001\))g Fx(\000)g(O)s FA(\(\(1)i(+)d Fz(\037)1964 1838 y Fs(\(1\))2059 1876 y FA(\()p Fz(t)h Fx(^)g Fz(\034)2278 1838 y Fs(det)2268 1898 y Fv(D)2323 1907 y Ft(0)2380 1876 y FA(\)\))p Fz(h)p FA(\))2537 1802 y Fq(\003)2576 1876 y Fz(;)828 b FA(\(3.48\))468 2115 y Fz(\024)520 2078 y Fv(\000)580 2115 y FA(\(0\))26 b(=)827 2054 y(1)p 827 2095 46 4 v 827 2178 a(2)883 2042 y Fq(\002)920 2115 y FA(1)21 b(+)f Fx(O)s FA(\()p Fz(h)p FA(\))h(+)f Fx(O)1461 2042 y Fq(\000)1503 2115 y FA(e)1543 2078 y Fv(\000)p Fw(K)1658 2087 y Ft(0)1692 2078 y Fw(t=")1790 2042 y Fq(\001)o(\003)1869 2115 y Fz(;)1535 b FA(\(3.49\))256 2343 y Fz(C)328 2305 y Fs(+)321 2365 y Fw(n;m;\015)510 2343 y FA(\()p Fz(t;)15 b(")p FA(\))26 b(=)817 2215 y Fq(\030)923 2281 y Fz(t)p 880 2322 119 4 v 880 2405 a FA(\001)p Fz(")1008 2215 y Fq(\031\024)1318 2281 y FA(1)p 1119 2322 443 4 v 1119 2409 a(\(1)21 b Fx(\000)f FA(2)p Fz(\015)5 b FA(\))1443 2383 y Fw(n=)p Fs(2)1592 2343 y FA(+)1683 2269 y Fq(\000)1724 2343 y FA(e)1765 2305 y Fw(n=)p Fs(4)1898 2343 y FA(+2)15 b(e)2069 2305 y Fw(m=)p Fs(4)2206 2269 y Fq(\001)2263 2343 y FA(e)2304 2305 y Fv(\000)p Fw(\024)2400 2282 y Ft(+)2450 2305 y Fs(\(0\))p Fw(h)2580 2282 y Ft(2)2615 2305 y Fw(=\033)2692 2282 y Ft(2)2732 2215 y Fq(\025)2780 2343 y Fz(;)624 b FA(\(3.50\))316 2615 y Fz(C)388 2578 y Fv(\000)381 2638 y Fw(n;m)510 2615 y FA(\()p Fz(t;)15 b(")p FA(\))26 b(=)817 2487 y Fq(\022)884 2472 y(r)p 975 2472 76 4 v 990 2554 a FA(2)p 985 2594 56 4 v 985 2677 a Fz(\031)1061 2554 y(h)p 1060 2594 V 1060 2677 a(\033)1145 2615 y Fx(^)20 b FA(1)1271 2487 y Fq(\023)1359 2615 y Fx(\000)1450 2541 y Fq(\000)1491 2615 y FA(e)1532 2578 y Fw(n=)p Fs(4)1664 2615 y FA(+4)15 b(e)1836 2578 y Fw(m=)p Fs(4)1973 2541 y Fq(\001)2030 2615 y FA(e)2071 2568 y Fv(\000)2151 2541 y Fl(h)2187 2520 y Ft(2)p 2135 2553 103 3 v 2135 2602 a(2)p Fl(\033)2203 2588 y Ft(2)2248 2568 y Fs([1)p Fv(\000O)r Fs(\(e)2474 2544 y Fk(\000)p Fl(K)2573 2559 y Ft(0)2608 2544 y Fl(t=")2697 2568 y Fs(\))p Fv(\000O)r Fs(\(\(1+)p Fw(\037)3025 2544 y Ft(\(1\))3109 2568 y Fs(\()p Fw(t)p Fv(^)p Fw(\034)3247 2544 y Ft(det)3239 2590 y Fk(D)3287 2605 y Ft(0)3337 2568 y Fs(\)\))p Fw(h)p Fs(\)])3498 2615 y Fz(:)3429 2789 y FA(\(3.51\))118 2980 y Fg(Pr)n(oof:)45 b FA(W)-8 b(e)42 b(\014rst)f(establish)f(the)h(upp)s(er)e(b)s(ound.)71 b(Fix)41 b(an)g(initial)e(condition)h(\()p Fz(\030)3085 2994 y Fs(0)3124 2980 y Fz(;)15 b FA(0\))43 b(satisfying)118 3093 y Fx(h)p Fz(\030)193 3107 y Fs(0)233 3093 y Fz(;)p 296 3020 55 4 v 15 w(X)7 b FA(\()p Fz(y)435 3107 y Fs(0)475 3093 y FA(\))510 3060 y Fv(\000)p Fs(1)605 3093 y Fz(\030)645 3107 y Fs(0)684 3093 y Fx(i)25 b Fn(6)g Fz(\013)898 3060 y Fs(2)938 3093 y Fz(h)990 3060 y Fs(2)1030 3093 y FA(,)31 b(and)e(observ)m(e)i(that)210 3286 y Fr(P)271 3248 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)384 3257 y Ft(0)419 3248 y Fw(;)p Fs(0\))506 3212 y Fq(\010)559 3286 y Fz(\034)599 3304 y Fv(B)r Fs(\()p Fw(h)p Fs(\))772 3286 y Fz(<)25 b(t)901 3212 y Fq(\011)979 3286 y Fn(6)g Fr(P)1136 3248 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1249 3257 y Ft(0)1284 3248 y Fw(;)p Fs(0\))1370 3212 y Fq(\010)1424 3286 y Fz(\034)1464 3304 y Fv(B)r Fs(\()p Fw(h)p Fs(\))1636 3286 y Fz(<)g(t)20 b Fx(^)g Fz(\034)1906 3300 y Fw(\021)1948 3212 y Fq(\011)2021 3286 y FA(+)g Fr(P)2173 3248 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)2286 3257 y Ft(0)2321 3248 y Fw(;)p Fs(0\))2408 3212 y Fq(\010)2461 3286 y Fz(\034)2501 3300 y Fw(\021)2568 3286 y Fz(<)25 b(t)20 b Fx(^)f Fz(\034)2837 3304 y Fv(B)r Fs(\()p Fw(h)p Fs(\))2985 3212 y Fq(\011)3429 3286 y FA(\(3.52\))979 3442 y(=)25 b Fr(P)1136 3404 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1249 3413 y Ft(0)1284 3404 y Fw(;)p Fs(0\))1370 3368 y Fq(\010)1424 3442 y Fz(\034)1464 3460 y Fv(B)r Fs(\()p Fw(h)p Fs(\))1636 3442 y Fz(<)g(t)20 b Fx(^)g Fz(\034)1916 3404 y Fs(det)1906 3464 y Fv(D)1961 3473 y Ft(0)2038 3442 y Fx(^)g Fz(\034)2159 3456 y Fw(\021)2201 3368 y Fq(\011)2274 3442 y FA(+)g Fr(P)2426 3404 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)2539 3413 y Ft(0)2574 3404 y Fw(;)p Fs(0\))2661 3368 y Fq(\010)2714 3442 y Fz(\034)2754 3456 y Fw(\021)2821 3442 y Fz(<)25 b(t)20 b Fx(^)g Fz(\034)3101 3404 y Fs(det)3091 3464 y Fv(D)3146 3473 y Ft(0)3223 3442 y Fx(^)g Fz(\034)3344 3460 y Fv(B)r Fs(\()p Fw(h)p Fs(\))3491 3368 y Fq(\011)3544 3442 y Fz(:)118 3635 y FA(T)-8 b(o)28 b(estimate)g(the)f(\014rst)g(term)g(on)g(the)h(righ)m (t-hand)e(side,)h(w)m(e)g(again)h(in)m(tro)s(duce)e(a)h(partition)f(0)g (=)f Fz(u)3526 3649 y Fs(0)3591 3635 y Fz(<)118 3748 y(u)170 3762 y Fs(1)249 3748 y Fz(<)39 b Fx(\001)15 b(\001)g(\001)40 b Fz(<)f(u)666 3762 y Fw(K)774 3748 y FA(=)f Fz(t)h FA(of)g(the)g(time) f(in)m(terv)-5 b(al)38 b([0)p Fz(;)15 b(t)p FA(],)42 b(de\014ned)c(b)m(y)h Fz(u)2528 3763 y Fw(k)2609 3748 y FA(=)g Fz(k)s FA(\001)p Fz(")g FA(for)g(0)h Fn(6)e Fz(k)43 b(<)c(K)46 b FA(=)118 3861 y Fx(d)p Fz(t=)p FA(\(\001)p Fz(")p FA(\))p Fx(e)p FA(.)d(Th)m(us)29 b(w)m(e)i(obtain)264 4140 y Fr(P)325 4103 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)438 4112 y Ft(0)473 4103 y Fw(;)p Fs(0\))560 4067 y Fq(\010)613 4140 y Fz(\034)653 4159 y Fv(B)r Fs(\()p Fw(h)p Fs(\))825 4140 y Fz(<)25 b(t)20 b Fx(^)g Fz(\034)1105 4103 y Fs(det)1095 4163 y Fv(D)1150 4172 y Ft(0)1228 4140 y Fx(^)f Fz(\034)1348 4154 y Fw(\021)1390 4067 y Fq(\011)1468 4140 y Fn(6)1598 4027 y Fw(K)1564 4054 y Fq(X)1566 4252 y Fw(k)r Fs(=1)1711 4140 y Fr(P)1772 4103 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1885 4112 y Ft(0)1920 4103 y Fw(;)p Fs(0\))2007 4067 y Fq(\010)2060 4140 y Fz(u)2112 4155 y Fw(k)r Fv(\000)p Fs(1)2270 4140 y Fn(6)25 b Fz(\034)2406 4159 y Fv(B)r Fs(\()p Fw(h)p Fs(\))2578 4140 y Fz(<)g(u)2726 4155 y Fw(k)2789 4140 y Fx(^)20 b Fz(\034)2920 4103 y Fs(det)2910 4163 y Fv(D)2965 4172 y Ft(0)3043 4140 y Fx(^)f Fz(\034)3163 4154 y Fw(\021)3205 4067 y Fq(\011)3258 4140 y Fz(:)146 b FA(\(3.53\))118 4418 y(Before)45 b(w)m(e)f(estimate)h(the)f(summands)e(on)h(the)h(righ) m(t-hand)f(side)g(of)51 b(\(3.53\))r(,)d(note)c(that)g(b)m(y)g(the)118 4530 y(b)s(oundedness)21 b(assumption)g(on)i Fx(k)p 1306 4457 V Fz(X)7 b FA(\()p Fz(y)s FA(\))p Fx(k)25 b FA(and)e Fx(k)p 1790 4457 V Fz(X)1849 4498 y Fv(\000)p Fs(1)1944 4530 y FA(\()p Fz(y)s FA(\))p Fx(k)p FA(,)i(w)m(e)f(ha)m(v)m(e)p 2510 4457 V 24 w Fz(X)7 b FA(\()p Fz(y)2648 4544 y Fw(u)2694 4530 y FA(\))2729 4498 y Fv(\000)p Fs(1)2849 4530 y FA(=)p 2968 4457 V 25 w Fz(X)g FA(\()p Fz(y)3110 4498 y Fs(det)3107 4553 y Fw(u)3212 4530 y FA(\))3247 4498 y Fv(\000)p Fs(1)3348 4530 y FA(+)f Fx(O)s FA(\()p Fz(h)3587 4544 y Fs(1)3626 4530 y FA(\))118 4643 y(for)39 b Fz(u)h Fn(6)g Fz(\034)519 4610 y Fs(det)509 4670 y Fv(D)564 4679 y Ft(0)648 4643 y Fx(^)25 b Fz(\034)774 4657 y Fw(\021)816 4643 y FA(.)68 b(Th)m(us)38 b(the)h(b)s(ound)f(obtained)g(in)g(Prop)s(osition)g(3.1)i (can)g(also)f(b)s(e)f(applied)g(to)118 4756 y(estimate)31 b(\014rst-exit)f(times)g(from)g Fx(B)s FA(\()p Fz(h)p FA(\))h(itself:)369 4950 y Fr(P)430 4912 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)543 4921 y Ft(0)578 4912 y Fw(;)p Fs(0\))664 4876 y Fq(\010)718 4950 y Fz(u)770 4965 y Fw(k)r Fv(\000)p Fs(1)928 4950 y Fn(6)25 b Fz(\034)1064 4968 y Fv(B)r Fs(\()p Fw(h)p Fs(\))1236 4950 y Fz(<)g(u)1384 4965 y Fw(k)1447 4950 y Fx(^)20 b Fz(\034)1578 4912 y Fs(det)1568 4972 y Fv(D)1623 4981 y Ft(0)1700 4950 y Fx(^)g Fz(\034)1821 4964 y Fw(\021)1863 4876 y Fq(\011)576 5123 y Fn(6)25 b Fr(P)733 5085 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)846 5094 y Ft(0)881 5085 y Fw(;)p Fs(0\))968 5022 y Fq(n)1299 5123 y FA(sup)1028 5213 y Fw(u)1069 5225 y Fl(k)q Fk(\000)p Ft(1)1186 5213 y Fo(6)p Fw(u)25 b Fz(h)2588 5085 y Fs(2)2628 5123 y FA(\(1)c Fx(\000)f(O)s FA(\()p Fz(h)2982 5137 y Fs(1)3022 5123 y FA(\)\))3092 5022 y Fq(o)3153 5123 y Fz(:)251 b FA(\(3.54\))118 5417 y(The)30 b(second)g(term)h(on)f(the)h(righ)m(t-hand)e(side)g(of)38 b(\(3.52\))32 b(can)f(b)s(e)f(estimated)g(b)m(y)h(Lemma)f(3.4)i(with) 357 5622 y Fz(\026)412 5585 y Fs(2)476 5622 y FA(=)25 b(8)p Fz(M)715 5585 y Fs(2)705 5645 y(1)755 5549 y Fq(\002)793 5542 y(p)p 884 5542 137 4 v 80 x Fz(K)961 5636 y Fs(+)1040 5622 y FA(+)20 b Fz(h)1183 5636 y Fs(1)1223 5622 y Fz(=)p FA(\()p Fz(h)p FA(\(1)i Fx(\000)e(O)s FA(\()p Fz(h)1710 5636 y Fs(1)1750 5622 y FA(\)\)\))1855 5549 y Fq(\003)1894 5571 y Fs(2)1933 5622 y FA(\011\()p Fz(t)h Fx(^)e Fz(\034)2223 5585 y Fs(det)2213 5645 y Fv(D)2268 5654 y Ft(0)2326 5622 y FA(\))p Fz(=)2406 5549 y Fq(\002)2444 5622 y FA(1)i Fx(\000)f(O)s FA(\()p Fz(h)2763 5636 y Fs(1)2803 5622 y FA(\))g Fx(\000)g Fz(M)3037 5636 y Fs(0)3077 5622 y FA(\001)3153 5549 y Fq(\003)3429 5622 y FA(\(3.55\))1844 5871 y(26)p eop %%Page: 27 27 27 26 bop 118 280 a FA(and)35 b Fz(h)352 294 y Fs(1)424 280 y FA(=)e Fz(h=)625 222 y Fx(p)p 702 222 92 4 v 702 280 a Fz(\024)754 294 y Fs(0)829 280 y FA(whic)m(h)g(are)j(c)m(hosen)f (in)f(suc)m(h)h(a)g(w)m(a)m(y)h(that)g(the)f(Gaussian)f(part)h(of)g Fz(\030)3243 294 y Fw(t)3308 280 y FA(giv)m(es)g(the)118 393 y(ma)5 b(jor)37 b(con)m(tribution)f(to)i(the)g(probabilit)m(y)-8 b(.)60 b(Th)m(us)36 b(w)m(e)i(obtain)e(that)i(the)g(probabilit)m(y)d (in)h(\(3.52\))j(is)118 506 y(b)s(ounded)28 b(b)m(y)485 633 y Fq(\030)590 699 y Fz(t)p 548 740 119 4 v 548 823 a FA(\001)p Fz(")676 633 y Fq(\031\024)985 699 y FA(1)p 787 740 443 4 v 787 827 a(\(1)21 b Fx(\000)f FA(2)p Fz(\015)5 b FA(\))1111 801 y Fw(n=)p Fs(2)1254 761 y FA(exp)1393 660 y Fq(n)1454 761 y Fx(\000)p Fz(\015)1588 699 y(h)1640 666 y Fs(2)p 1586 740 95 4 v 1586 823 a Fz(\033)1641 797 y Fs(2)1691 687 y Fq(\002)1729 761 y FA(1)21 b Fx(\000)e Fz(\013)1943 723 y Fs(2)2004 761 y Fx(\000)h(O)s FA(\(\001\))g Fx(\000)g(O)s FA(\()p Fz(h)p FA(\))2624 687 y Fq(\003)2663 660 y(o)2744 761 y FA(+)g(e)2875 723 y Fw(n=)p Fs(4)3008 761 y FA(e)3048 723 y Fv(\000)p Fw(h)3144 700 y Ft(2)3178 723 y Fw(=\033)3255 700 y Ft(2)778 1037 y FA(+)f(2)c(e)969 1000 y Fw(m=)p Fs(4)1122 1037 y FA(exp)1261 909 y Fq(\032)1329 1037 y Fx(\000)1410 966 y Fz(h)1462 933 y Fs(2)1501 966 y FA(\(1)21 b Fx(\000)f(O)s FA(\()p Fz(\037)1860 933 y Fs(\(1\))1955 966 y FA(\()p Fz(t)g Fx(^)g Fz(\034)2174 933 y Fs(det)2164 993 y Fv(D)2219 1002 y Ft(0)2276 966 y FA(\))p Fz(h)p FA(\))h Fx(\000)f(O)s FA(\(\001)p Fz(")p FA(\)\))p 1410 1016 1400 4 v 1653 1105 a Fz(\033)1708 1079 y Fs(2)1747 1105 y FA(\()p Fz(\032)1829 1079 y Fs(2)1889 1105 y FA(+)g Fz(")p FA(\))p Fz(\037)2114 1078 y Fs(\(2\))2209 1105 y FA(\()p Fz(t)h Fx(^)e Fz(\034)2428 1073 y Fs(det)2418 1133 y Fv(D)2473 1142 y Ft(0)2531 1105 y FA(\))2819 909 y Fq(\033\025)2935 1037 y Fz(;)469 b FA(\(3.56\))118 1310 y(where)24 b(w)m(e)h(ha)m(v)m(e)h(used)d(the)i(fact)g(that)h (\010\()p Fz(t)p FA(\))f Fn(6)g Fz(n)p FA(\011\()p Fz(t)p FA(\),)h(while)c(\011\()p Fz(t)p FA(\))j(and)f(\002\()p Fz(t)p FA(\))g(are)h(at)g(most)g(of)g(order)f(1.)118 1423 y(This)29 b(completes)h(the)h(pro)s(of)f(of)g(the)h(upp)s(er)d(b)s (ound)g(in)h(\(3.47\))s(.)259 1536 y(The)35 b(lo)m(w)m(er)h(b)s(ound)e (is)g(a)j(consequence)f(of)g(the)g(fact)g(that)g(the)g(Gaussian)f(part) g(of)h Fz(\030)3242 1550 y Fw(t)3307 1536 y FA(giv)m(es)g(the)118 1649 y(ma)5 b(jor)31 b(con)m(tribution)e(to)i(the)g(probabilit)m(y)d (in)h(\(3.47\))r(.)42 b(T)-8 b(o)31 b(c)m(hec)m(k)h(this,)e(w)m(e)h (split)d(the)j(probabilit)m(y)d(as)118 1762 y(follo)m(ws:)253 1966 y Fr(P)314 1929 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)427 1938 y Ft(0)462 1929 y Fw(;)p Fs(0\))549 1893 y Fq(\010)602 1966 y Fz(\034)642 1985 y Fv(B)r Fs(\()p Fw(h)p Fs(\))815 1966 y Fz(<)d(t)944 1893 y Fq(\011)1022 1966 y FA(=)g Fr(P)1179 1929 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1292 1938 y Ft(0)1327 1929 y Fw(;)p Fs(0\))1413 1893 y Fq(\010)1466 1966 y Fz(\034)1506 1985 y Fv(B)r Fs(\()p Fw(h)p Fs(\))1679 1966 y Fz(<)g(t;)15 b(\034)1888 1980 y Fw(\021)1955 1966 y Fn(>)25 b Fz(t;)20 b FA(~)-50 b Fz(\034)2164 1981 y Fw(\030)2227 1966 y Fn(>)25 b Fz(t)2356 1893 y Fq(\011)1138 2122 y FA(+)20 b Fr(P)1290 2084 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1403 2093 y Ft(0)1438 2084 y Fw(;)p Fs(0\))1525 2048 y Fq(\010)1578 2122 y Fz(\034)1618 2136 y Fw(\021)1685 2122 y Fn(>)25 b Fz(t;)20 b FA(~)-50 b Fz(\034)1894 2137 y Fw(\030)1956 2122 y Fz(<)25 b(t)2085 2048 y Fq(\011)2159 2122 y FA(+)19 b Fr(P)2310 2084 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)2423 2093 y Ft(0)2459 2084 y Fw(;)p Fs(0\))2545 2048 y Fq(\010)2598 2122 y Fz(\034)2638 2140 y Fv(B)r Fs(\()p Fw(h)p Fs(\))2811 2122 y Fz(<)25 b(t;)15 b(\034)3020 2136 y Fw(\021)3087 2122 y Fz(<)25 b(t)3216 2048 y Fq(\011)3269 2122 y Fz(;)135 b FA(\(3.57\))118 2326 y(where)854 2439 y(~)-50 b Fz(\034)889 2454 y Fw(\030)952 2439 y FA(=)25 b(inf)6 b Fx(f)p Fz(u)25 b Fn(>)g FA(0)10 b(:)1488 2366 y Fq(\012)1531 2439 y Fz(\030)1571 2453 y Fw(u)1616 2439 y Fz(;)p 1680 2366 55 4 v 15 w(X)d FA(\()p Fz(y)1821 2402 y Fs(det)1818 2462 y Fw(u)1924 2439 y FA(\))1959 2402 y Fv(\000)p Fs(1)2054 2439 y Fz(\030)2094 2453 y Fw(u)2139 2366 y Fq(\013)2207 2439 y Fn(>)25 b Fz(h)2355 2402 y Fs(2)2395 2439 y FA(\(1)c(+)f Fx(O)s FA(\()p Fz(h)2749 2453 y Fs(1)2789 2439 y FA(\)\))p Fx(g)p Fz(;)500 b FA(\(3.58\))118 2606 y(and)30 b(the)g Fx(O)s FA(\()p Fz(h)613 2620 y Fs(1)653 2606 y FA(\)-term)h(stems)f(from)g(estimating)p 1872 2533 V 30 w Fz(X)7 b FA(\()p Fz(y)2011 2620 y Fw(u)2056 2606 y FA(\))2091 2573 y Fv(\000)p Fs(1)2216 2606 y FA(b)m(y)p 2365 2533 V 30 w Fz(X)g FA(\()p Fz(y)2507 2573 y Fs(det)2504 2628 y Fw(u)2610 2606 y FA(\))2645 2573 y Fv(\000)p Fs(1)2770 2606 y FA(as)30 b(in)f(\(3.54\))r(.)41 b(Also)30 b(note)118 2719 y(that)f Fz(\034)353 2737 y Fv(B)r Fs(\()p Fw(h)p Fs(\))526 2719 y Fn(6)g FA(~)-49 b Fz(\034)662 2734 y Fw(\030)699 2719 y FA(.)40 b(The)28 b(\014rst)f(term)i(on)f(the)g(righ) m(t-hand)g(side)f(of)35 b(\(3.57\))c(can)d(b)s(e)g(estimated)h(as)f(in) f(the)118 2832 y(pro)s(of)37 b(of)g(Prop)s(osition)f(3.1.)62 b(\(Note)39 b(that)f(a)g(lo)m(w)m(er)g(b)s(ound)d(is)i(obtained)f (trivially)f(b)m(y)i(considering)118 2945 y(the)f(endp)s(oin)m(t)f (instead)g(of)h(the)g(whole)f(path.\))58 b(Instead)35 b(of)i(applying)c(Lemma)j(3.2,)j(the)d(Gaussian)118 3058 y(con)m(tribution)29 b(can)i(b)s(e)f(estimated)h(b)s(elo)m(w)e(b)m(y)i (a)g(straigh)m(tforw)m(ard)f(calculation.)40 b(The)30 b(non-Gaussian)118 3171 y(parts)k(are)g(estimated)g Fy(ab)-5 b(ove)41 b FA(as)34 b(b)s(efore)g(and)f(are)h(of)g(smaller)f(order.)50 b(Finally)-8 b(,)34 b(w)m(e)g(need)g(an)f(upp)s(er)118 3283 y(b)s(ound)25 b(for)j(the)f(probabilit)m(y)e(that)j Fz(\034)1381 3297 y Fw(\021)1448 3283 y Fz(<)d(t)g Fn(6)g Fz(\034)1738 3302 y Fv(B)r Fs(\()p Fw(h)p Fs(\))1913 3283 y FA(whic)m(h)h(can)h(b)s(e)g(obtained)g(from)g(Lemma)g(3.4.)p 3595 3283 4 62 v 3599 3225 55 4 v 3599 3283 V 3653 3283 4 62 v 118 3527 a Fp(3.2)112 b(Longer)38 b(timescales)118 3699 y FA(Corollary)k(3.5)j(describ)s(es)d(the)h(dynamics)g(on)g(a)h (timescale)g(of)f(order)g(1,)48 b(or)43 b(ev)m(en)i(on)e(a)h(sligh)m (tly)118 3811 y(longer)38 b(timescale)h(if)f Fz(\037)951 3778 y Fs(\(1\))1045 3811 y FA(\()p Fz(t)p FA(\),)j Fz(\037)1271 3778 y Fs(\(2\))1365 3811 y FA(\()p Fz(t)p FA(\))e(do)g(not)g(gro)m(w)g (to)s(o)g(fast.)66 b(It)39 b(ma)m(y)g(happ)s(en,)g(ho)m(w)m(ev)m(er,)k (that)118 3924 y Fz(y)166 3891 y Fs(det)163 3947 y Fw(t)303 3924 y FA(remains)34 b(in)g Fx(D)828 3938 y Fs(0)903 3924 y FA(for)h(all)f(p)s(ositiv)m(e)g(times)h(\(e.)16 b(g.)36 b(when)e Fx(D)2301 3938 y Fs(0)2376 3924 y FA(is)g(p)s(ositiv)m (ely)f(in)m(v)-5 b(arian)m(t)35 b(under)e(the)118 4037 y(reduced)e(deterministic)f(\015o)m(w\).)46 b(In)32 b(suc)m(h)g(a)g (case,)i(one)e(w)m(ould)f(exp)s(ect)i(the)f(v)-5 b(ast)32 b(ma)5 b(jorit)m(y)33 b(of)f(paths)118 4150 y(to)f(remain)f(concen)m (trated)i(in)d Fx(B)s FA(\()p Fz(h)p FA(\))i(for)f(a)g(rather)h(long)f (p)s(erio)s(d)e(of)i(time.)259 4263 y(The)37 b(approac)m(h)h(used)e(in) g(Subsection)g(3.1)j(fails)d(to)i(con)m(trol)f(the)h(dynamics)e(on)h (timescales)g(on)118 4376 y(whic)m(h)29 b Fz(\037)435 4343 y Fs(\()p Fw(i)p Fs(\))518 4376 y FA(\()p Fz(t)p FA(\))d Fx(\035)f FA(1,)31 b(b)s(ecause)f(it)f(uses)h(in)f(an)h(essen)m (tial)f(w)m(a)m(y)j(the)e(fact)h(that)f Fz(\021)2837 4390 y Fw(t)2892 4376 y FA(=)25 b Fz(y)3033 4390 y Fw(t)3082 4376 y Fx(\000)20 b Fz(y)3221 4343 y Fs(det)3218 4398 y Fw(t)3353 4376 y FA(remains)118 4489 y(small.)68 b(Our)39 b(strategy)i(in)e(order)h(to)g(describ)s(e)f(the)h(paths)g(on)f(longer) h(timescales)g(is)f(to)i(compare)118 4602 y(them)g(to)h(di\013eren)m(t) e(deterministic)f(solutions)h(on)h(time)f(in)m(terv)-5 b(als)40 b([0)p Fz(;)15 b(T)e FA(],)45 b([)p Fz(T)8 b(;)15 b FA(2)p Fz(T)e FA(],)45 b(.)15 b(.)h(.)f(,)44 b(where)d Fz(T)118 4715 y FA(is)35 b(a)h(p)s(ossibly)d(large)i(constan)m(t)i(suc) m(h)f(that)g(Corollary)e(3.5)j(holds)d(on)i(time)f(in)m(terv)-5 b(als)35 b(of)g(length)g Fz(T)13 b FA(,)118 4828 y(pro)m(vided)22 b Fz(y)530 4842 y Fw(t)582 4828 y FA(remains)g(in)f Fx(D)1082 4842 y Fs(0)1122 4828 y FA(.)38 b(Essen)m(tial)22 b(ingredien)m(ts)g (for)h(this)e(approac)m(h)j(are)f(the)g(Mark)m(o)m(v)i(prop)s(ert)m(y) 118 4941 y(and)30 b(the)g(follo)m(wing)f(tec)m(hnical)i(lemma,)f(whic)m (h)f(is)g(based)h(on)h(in)m(tegration)f(b)m(y)g(parts.)118 5128 y Fm(Lemma)j(3.6.)43 b Fy(Fix)33 b(c)-5 b(onstants)35 b Fz(s)1299 5142 y Fs(1)1364 5128 y Fn(6)26 b Fz(s)1504 5142 y Fs(2)1576 5128 y Fy(in)33 b FA([0)p Fz(;)15 b Fx(1)p FA(])p Fy(,)34 b(and)g(assume)f(we)h(ar)-5 b(e)34 b(given)e(two)i(c)-5 b(ontinuously)118 5241 y(di\013er)g(entiable)34 b(functions)182 5378 y Fx(\017)48 b Fz(')25 b FA(:)h([0)p Fz(;)15 b Fx(1)p FA(\))27 b Fx(!)e FA([0)p Fz(;)15 b Fx(1)p FA(\))p Fy(,)34 b(which)f(is)g(monotonously)i(incr)-5 b(e)g(asing)34 b(and)f(satis\014es)h Fz(')p FA(\()p Fz(s)3102 5392 y Fs(2)3142 5378 y FA(\))25 b(=)g(1)p Fy(,)182 5491 y Fx(\017)48 b Fz(')334 5505 y Fs(0)399 5491 y FA(:)25 b([0)p Fz(;)15 b Fx(1)p FA(\))27 b Fx(!)e Fr(R)57 b Fy(which)33 b(satis\014es)h Fz(')1598 5505 y Fs(0)1638 5491 y FA(\()p Fz(s)p FA(\))25 b Fn(6)g FA(0)33 b Fy(for)g(al)5 b(l)33 b Fz(s)25 b Fn(6)g Fz(s)2432 5505 y Fs(1)2471 5491 y Fy(.)1844 5871 y FA(27)p eop %%Page: 28 28 28 27 bop 118 280 a Fy(L)-5 b(et)30 b Fz(X)j Fn(>)25 b FA(0)30 b Fy(b)-5 b(e)29 b(a)h(r)-5 b(andom)33 b(variable)d(such)g (that)h Fr(P)p Fx(f)p Fz(X)j(<)25 b(s)p Fx(g)g Fn(>)g Fz(')2361 294 y Fs(0)2401 280 y FA(\()p Fz(s)p FA(\))30 b Fy(for)g(al)5 b(l)30 b Fz(s)25 b Fn(>)g FA(0)p Fy(.)41 b(Then)30 b(we)g(have,)118 393 y(for)j(al)5 b(l)33 b Fz(t)25 b Fn(>)g FA(0)p Fy(,)750 621 y Fr(E)805 547 y Fq(\010)864 621 y FA(1)909 639 y Fs([0)p Fw(;t)p Fs(\))1041 621 y FA(\()p Fz(X)7 b FA(\))15 b(^)-60 b Fz(')q FA(\()p Fz(X)7 b FA(\))1405 547 y Fq(\011)1485 621 y Fn(6)39 b FA(^)-59 b Fz(')p FA(\()p Fz(t)p FA(\))p Fr(P)p Fx(f)p Fz(X)34 b(<)25 b(t)p Fx(g)c(\000)2238 497 y Fq(Z)2329 523 y Fw(s)2362 532 y Ft(2)2396 523 y Fv(^)p Fw(t)2288 703 y(s)2321 712 y Ft(1)2355 703 y Fv(^)p Fw(t)2488 621 y Fz(')2547 583 y Fv(0)2570 621 y FA(\()p Fz(s)p FA(\))p Fz(')2742 635 y Fs(0)2783 621 y FA(\()p Fz(s)p FA(\))15 b(d)p Fz(s;)399 b FA(\(3.59\))118 852 y Fy(wher)-5 b(e)48 b FA(^)-59 b Fz(')p FA(\()p Fz(s)p FA(\))26 b(=)f Fz(')p FA(\()p Fz(s)p FA(\))c Fx(^)f FA(1)p Fy(.)259 1016 y FA(W)-8 b(e)46 b(omit)e(the)h(pro)s(of)f(of)h(this)f(result,)j(whic)m (h)c(is)h(rather)g(standard.)83 b(See,)49 b(for)44 b(instance,)k([7)q (,)118 1129 y(Lemma)31 b(A.1])g(for)f(a)h(v)m(ery)g(similar)c(result.) 259 1242 y(When)h(applying)f(the)h(preceding)f(lemma,)i(w)m(e)g(will)c (also)k(need)f(an)g(estimate)h(on)f(the)h(probabilit)m(y)118 1355 y(that)h Fx(h)p Fz(\030)389 1369 y Fw(T)444 1355 y Fz(;)p 508 1282 55 4 v 15 w(X)7 b FA(\()p Fz(y)646 1369 y Fw(T)702 1355 y FA(\))737 1322 y Fv(\000)p Fs(1)832 1355 y Fz(\030)872 1369 y Fw(T)926 1355 y Fx(i)30 b FA(exceeds)g Fz(h)1368 1322 y Fs(2)1408 1355 y FA(.)40 b(Corollary)28 b(3.5)i(pro)m(vides,)f(of)g(course,)h(suc)m(h)f(an)g(estimate,)h(but) 118 1468 y(since)i(it)g(applies)f(to)i(the)g(whole)f(path,)h(it)f(do)s (es)h(not)g(giv)m(e)g(optimal)e(b)s(ounds)g(for)h(the)h(endp)s(oin)m (t.)46 b(An)118 1581 y(impro)m(v)m(ed)37 b(b)s(ound)f(is)h(giv)m(en)h (b)m(y)f(the)h(follo)m(wing)e(lemma.)63 b(Recall)37 b(the)h (de\014nition)e(of)i(the)g(\014rst-exit)118 1694 y(time)30 b Fz(\034)365 1708 y Fv(D)420 1717 y Ft(0)489 1694 y FA(of)g Fz(y)637 1708 y Fw(t)697 1694 y FA(from)f Fx(D)981 1708 y Fs(0)1051 1694 y FA(from)h(\(2.24\))s(.)118 1858 y Fm(Lemma)35 b(3.7.)43 b Fy(If)35 b Fz(T)47 b Fy(and)35 b Fz(h)g Fy(satisfy)g Fz(h)29 b Fn(6)g Fz(c)1664 1872 y Fs(1)1704 1858 y Fz(\037)1761 1825 y Fs(\(1\))1855 1858 y FA(\()p Fz(T)34 b Fx(^)22 b Fz(\034)2110 1825 y Fs(det)2100 1885 y Fv(D)2155 1894 y Ft(0)2212 1858 y FA(\))2247 1825 y Fv(\000)p Fs(1)2376 1858 y Fy(and)35 b Fz(\037)2611 1825 y Fs(\(2\))2705 1858 y FA(\()p Fz(T)g Fx(^)21 b Fz(\034)2960 1825 y Fs(det)2950 1885 y Fv(D)3005 1894 y Ft(0)3063 1858 y FA(\))29 b Fn(6)f FA(\()p Fz(\032)3308 1825 y Fs(2)3369 1858 y FA(+)21 b Fz(")p FA(\))3538 1825 y Fv(\000)p Fs(1)3634 1858 y Fy(,)118 1971 y(we)33 b(have,)g(for)g (every)f Fz(\015)f Fx(2)25 b FA(\(0)p Fz(;)15 b FA(1)p Fz(=)p FA(2\))p Fy(,)505 2170 y FA(sup)200 2259 y Fw(\030)231 2268 y Ft(0)285 2259 y Fs(:)35 b Fv(h)p Fw(\030)398 2268 y Ft(0)433 2259 y Fw(;)p 470 2207 43 3 v(X)5 b Fs(\()p Fw(y)578 2268 y Ft(0)613 2259 y Fs(\))640 2240 y Fk(\000)p Ft(1)723 2259 y Fw(\030)754 2268 y Ft(0)788 2259 y Fv(i)p Fo(6)p Fw(h)911 2240 y Ft(2)961 2170 y Fr(P)1022 2132 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1135 2141 y Ft(0)1171 2132 y Fw(;)p Fs(0\))1257 2069 y Fq(n)1318 2170 y Fx(h)p Fz(\030)1393 2184 y Fw(T)1448 2170 y Fz(;)p 1512 2097 55 4 v 15 w(X)i FA(\()p Fz(y)1650 2184 y Fw(T)1706 2170 y FA(\))1741 2132 y Fv(\000)p Fs(1)1835 2170 y Fz(\030)1875 2184 y Fw(T)1930 2170 y Fx(i)26 b Fn(>)f Fz(h)2139 2132 y Fs(2)2179 2170 y Fz(;)15 b(\034)2259 2184 y Fv(D)2314 2193 y Ft(0)2377 2170 y Fn(>)25 b Fz(T)2539 2069 y Fq(o)2625 2170 y Fn(6)2739 2147 y Fq(b)2721 2170 y Fz(C)2786 2184 y Fw(n;m;\015)2975 2170 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))g(e)3244 2132 y Fv(\000)p Fw(\024)3340 2109 y Fk(0)3362 2132 y Fw(h)3403 2109 y Ft(2)3438 2132 y Fw(=\033)3515 2109 y Ft(2)3554 2170 y Fz(;)3429 2361 y FA(\(3.60\))118 2474 y Fy(wher)-5 b(e)1067 2650 y Fz(\024)1119 2612 y Fv(0)1168 2650 y FA(=)25 b Fz(\015)1316 2576 y Fq(\002)1354 2650 y FA(1)c Fx(\000)f(O)s FA(\(\001\))g Fx(\000)g(O)s FA(\()p Fz(h)p FA(\))h Fx(\000)f(O)2227 2576 y Fq(\000)2269 2650 y FA(e)2309 2612 y Fv(\000)p Fs(2)p Fw(K)2459 2621 y Ft(0)2493 2612 y Fw(T)8 b(=")2630 2650 y Fz(=)p FA(\(1)21 b Fx(\000)f FA(2)p Fz(\015)5 b FA(\))2999 2576 y Fq(\001)q(\003)3079 2650 y Fz(;)325 b FA(\(3.61\))693 2814 y Fq(b)675 2837 y Fz(C)740 2851 y Fw(n;m;\015)929 2837 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))26 b(=)1473 2775 y(1)p 1274 2816 443 4 v 1274 2903 a(\(1)21 b Fx(\000)f FA(2)p Fz(\015)5 b FA(\))1598 2877 y Fw(n=)p Fs(2)1747 2837 y FA(+)20 b(4)p Fz(C)1955 2799 y Fs(+)1948 2859 y Fw(n;m;\015)2137 2837 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))g(e)2407 2799 y Fv(\000)p Fs(2)p Fw(\024)2538 2776 y Ft(+)2588 2799 y Fs(\(0\))p Fw(h)2718 2776 y Ft(2)2754 2799 y Fw(=\033)2831 2776 y Ft(2)2885 2837 y Fz(:)519 b FA(\(3.62\))118 3073 y Fg(Pr)n(oof:)45 b FA(W)-8 b(e)32 b(decomp)s(ose)e Fz(\030)1131 3087 y Fw(t)1191 3073 y FA(as)g Fz(\030)1342 3087 y Fw(t)1397 3073 y FA(=)25 b Fz(\030)1537 3040 y Fs(0)1533 3096 y Fw(t)1597 3073 y FA(+)20 b Fz(\030)1732 3040 y Fs(1)1728 3096 y Fw(t)1791 3073 y FA(+)g Fz(\030)1926 3040 y Fs(2)1922 3096 y Fw(t)1965 3073 y FA(,)31 b(where)1015 3306 y Fz(\030)1059 3268 y Fs(0)1055 3328 y Fw(t)1123 3306 y FA(=)25 b Fz(U)10 b FA(\()p Fz(t)p FA(\))p Fz(\030)1434 3320 y Fs(0)1494 3306 y FA(+)1626 3244 y Fz(\033)p 1595 3285 119 4 v 1595 3303 a Fx(p)p 1671 3303 43 4 v 65 x Fz(")1738 3182 y Fq(Z)1829 3208 y Fw(t)1789 3388 y Fs(0)1874 3306 y Fz(U)g FA(\()p Fz(t;)15 b(u)p FA(\))p Fz(F)2199 3320 y Fs(0)2240 3306 y FA(\()p Fz(y)2323 3268 y Fs(det)2320 3328 y Fw(u)2425 3306 y Fz(;)g(")p FA(\))g(d)q Fz(W)2695 3320 y Fw(u)2740 3306 y Fz(;)1015 3562 y(\030)1059 3524 y Fs(1)1055 3584 y Fw(t)1123 3562 y FA(=)1261 3500 y Fz(\033)p 1229 3541 119 4 v 1229 3559 a Fx(p)p 1305 3559 43 4 v 65 x Fz(")1372 3438 y Fq(Z)1463 3464 y Fw(t)1423 3644 y Fs(0)1508 3562 y Fz(U)10 b FA(\()p Fz(t;)15 b(u)p FA(\))p Fz(F)1833 3576 y Fs(1)1874 3562 y FA(\()p Fz(\030)1949 3576 y Fw(u)1994 3562 y Fz(;)g(\021)2079 3576 y Fw(u)2125 3562 y Fz(;)g(u;)g(")p FA(\))g(d)q Fz(W)2487 3576 y Fw(u)2532 3562 y Fz(;)1015 3818 y(\030)1059 3780 y Fs(2)1055 3840 y Fw(t)1123 3818 y FA(=)1229 3756 y(1)p 1229 3797 46 4 v 1231 3880 a Fz(")1300 3694 y Fq(Z)1391 3720 y Fw(t)1350 3900 y Fs(0)1435 3818 y Fz(U)10 b FA(\()p Fz(t;)15 b(u)p FA(\))p Fz(b)p FA(\()p Fz(\030)1816 3832 y Fw(u)1862 3818 y Fz(;)g(\021)1947 3832 y Fw(u)1993 3818 y Fz(;)g(u;)g(")p FA(\))g(d)q Fz(u;)1083 b FA(\(3.63\))118 4035 y(and)39 b(in)m(tro)s(duce)f(the)h(notations)44 b(~)-49 b Fz(\034)1331 4050 y Fw(\030)1407 4035 y FA(and)44 b(~)-50 b Fz(\034)1633 4049 y Fw(\021)1714 4035 y FA(for)39 b(the)g(stopping)f(times)h(whic)m(h)f(are)h(de\014ned)f(lik)m(e)h Fz(\034)3624 4050 y Fw(\030)118 4148 y FA(and)c Fz(\034)340 4162 y Fw(\021)417 4148 y FA(in)e(\(3.9\))k(and)e(\(3.6\))r(,)h(but)f (with)f Fz(h)h FA(and)g Fz(h)1884 4162 y Fs(1)1959 4148 y FA(replaced)f(b)m(y)i(2)p Fz(h)f FA(and)g(2)p Fz(h)2865 4162 y Fs(1)2905 4148 y FA(,)i(resp)s(ectiv)m(ely)-8 b(.)55 b(The)118 4261 y(probabilit)m(y)28 b(in)h(\(3.60\))k(is)c(b)s (ounded)f(b)m(y)312 4459 y Fr(P)373 4422 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)486 4431 y Ft(0)521 4422 y Fw(;)p Fs(0\))607 4358 y Fq(n)668 4459 y Fx(h)p Fz(\030)743 4473 y Fw(T)798 4459 y Fz(;)p 862 4386 55 4 v 15 w(X)7 b FA(\()p Fz(y)1003 4422 y Fs(det)1000 4482 y Fw(T)1106 4459 y FA(\))1141 4422 y Fv(\000)p Fs(1)1236 4459 y Fz(\030)1276 4473 y Fw(T)1331 4459 y Fx(i)25 b Fn(>)g Fz(h)1539 4422 y Fs(2)1579 4459 y FA(\(1)c Fx(\000)f(O)s FA(\()p Fz(h)1933 4473 y Fs(1)1973 4459 y FA(\)\))p Fz(;)h FA(~)-51 b Fz(\034)2123 4473 y Fw(\021)2191 4459 y Fz(>)25 b(T)2353 4358 y Fq(o)2434 4459 y FA(+)20 b Fr(P)2586 4422 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)2699 4431 y Ft(0)2734 4422 y Fw(;)p Fs(0\))2821 4358 y Fq(n)2886 4459 y FA(~)-50 b Fz(\034)2921 4473 y Fw(\021)2988 4459 y Fn(6)25 b Fz(T)3150 4358 y Fq(o)3210 4459 y Fz(:)194 b FA(\(3.64\))118 4666 y(Let)22 b Fz(H)355 4633 y Fs(2)420 4666 y FA(=)j Fz(h)568 4633 y Fs(2)608 4666 y FA(\(1)s Fx(\000)s(O)s FA(\()p Fz(h)927 4680 y Fs(1)968 4666 y FA(\)\).)38 b(As)22 b(in)f(the)h(pro)s(of)f(of)h(Prop)s (osition)e(3.1,)25 b(the)d(\014rst)f(term)h(on)f(the)h(righ)m(t-hand) 118 4779 y(side)29 b(can)i(b)s(e)f(further)f(decomp)s(osed)h(as)531 4973 y Fr(P)592 4935 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)705 4944 y Ft(0)741 4935 y Fw(;)p Fs(0\))827 4872 y Fq(n)888 4973 y Fx(h)p Fz(\030)963 4987 y Fw(T)1018 4973 y Fz(;)p 1082 4899 V 15 w(X)7 b FA(\()p Fz(y)1223 4935 y Fs(det)1220 4995 y Fw(T)1326 4973 y FA(\))1361 4935 y Fv(\000)p Fs(1)1456 4973 y Fz(\030)1496 4987 y Fw(T)1551 4973 y Fx(i)25 b Fn(>)g Fz(H)1790 4935 y Fs(2)1830 4973 y Fz(;)20 b FA(~)-50 b Fz(\034)1910 4987 y Fw(\021)1977 4973 y Fz(>)25 b(T)2139 4872 y Fq(o)738 5169 y Fn(6)g Fr(P)895 5132 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1008 5141 y Ft(0)1044 5132 y Fw(;)p Fs(0\))1130 5068 y Fq(n)1191 5092 y(\015)1191 5147 y(\015)p 1265 5096 V 1241 5169 a Fz(X)7 b FA(\()p Fz(y)1406 5132 y Fs(det)1403 5192 y Fw(T)1509 5169 y FA(\))1544 5132 y Fv(\000)p Fs(1)p Fw(=)p Fs(2)1709 5169 y Fz(\030)1753 5132 y Fs(0)1749 5192 y Fw(T)1804 5092 y Fq(\015)1804 5147 y(\015)1880 5169 y Fn(>)25 b Fz(H)2052 5183 y Fs(0)2091 5068 y Fq(o)2172 5169 y FA(+)20 b Fr(P)2324 5132 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)2437 5141 y Ft(0)2472 5132 y Fw(;)p Fs(0\))2559 5068 y Fq(n)2624 5169 y FA(~)-50 b Fz(\034)2659 5183 y Fw(\021)2726 5169 y Fz(>)25 b(T)8 b(;)20 b FA(~)-50 b Fz(\034)2963 5184 y Fw(\030)3026 5169 y Fn(6)25 b Fz(T)3188 5068 y Fq(o)829 5366 y FA(+)20 b Fr(P)981 5329 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1094 5338 y Ft(0)1130 5329 y Fw(;)p Fs(0\))1216 5265 y Fq(n)1277 5289 y(\015)1277 5343 y(\015)p 1351 5293 V 1327 5366 a Fz(X)7 b FA(\()p Fz(y)1492 5329 y Fs(det)1489 5389 y Fw(T)1595 5366 y FA(\))1630 5329 y Fv(\000)p Fs(1)p Fw(=)p Fs(2)1795 5366 y Fz(\030)1839 5329 y Fs(1)1835 5389 y Fw(T)1890 5289 y Fq(\015)1890 5343 y(\015)1966 5366 y Fn(>)25 b Fz(H)2138 5380 y Fs(1)2177 5366 y Fz(;)20 b FA(~)-50 b Fz(\034)2257 5380 y Fw(\021)2324 5366 y Fz(>)25 b(T)8 b(;)20 b FA(~)-50 b Fz(\034)2561 5381 y Fw(\030)2624 5366 y Fz(>)25 b(T)2786 5265 y Fq(o)829 5563 y FA(+)20 b Fr(P)981 5526 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1094 5535 y Ft(0)1130 5526 y Fw(;)p Fs(0\))1216 5462 y Fq(n)1277 5486 y(\015)1277 5540 y(\015)p 1351 5490 V 1327 5563 a Fz(X)7 b FA(\()p Fz(y)1492 5526 y Fs(det)1489 5586 y Fw(T)1595 5563 y FA(\))1630 5526 y Fv(\000)p Fs(1)p Fw(=)p Fs(2)1795 5563 y Fz(\030)1839 5526 y Fs(2)1835 5586 y Fw(T)1890 5486 y Fq(\015)1890 5540 y(\015)1966 5563 y Fn(>)25 b Fz(H)2138 5577 y Fs(2)2177 5563 y Fz(;)20 b FA(~)-50 b Fz(\034)2257 5577 y Fw(\021)2324 5563 y Fz(>)25 b(T)8 b(;)20 b FA(~)-50 b Fz(\034)2561 5578 y Fw(\030)2624 5563 y Fz(>)25 b(T)2786 5462 y Fq(o)2846 5563 y Fz(;)558 b FA(\(3.65\))1844 5871 y(28)p eop %%Page: 29 29 29 28 bop 118 280 a FA(where)38 b(w)m(e)h(c)m(ho)s(ose)h Fz(H)904 294 y Fs(1)943 280 y FA(,)h Fz(H)1085 294 y Fs(2)1162 280 y FA(t)m(wice)e(as)g(large)f(as)h(in)f(the)g(pro)s(of)g (of)h(Prop)s(osition)d(3.1,)42 b(while)37 b Fz(H)3513 294 y Fs(0)3591 280 y FA(=)118 393 y Fz(H)27 b Fx(\000)20 b Fz(H)388 407 y Fs(1)447 393 y Fx(\000)g Fz(H)614 407 y Fs(2)653 393 y FA(.)259 506 y(The)40 b(\014rst)f(term)h(on)g(the)g (righ)m(t-hand)f(side)f(can)j(b)s(e)e(estimated)h(as)g(in)f(Lemma)h (3.2,)k(with)38 b(the)118 619 y(di\013erence)e(that,)i(the)f(exp)s (ectation)f(of)h Fz(\030)1572 586 y Fs(0)1568 646 y Fw(T)1658 619 y FA(b)s(eing)e(exp)s(onen)m(tially)g(small)f(in)h Fz(T)10 b(=")p FA(,)39 b(it)d(leads)f(only)h(to)118 732 y(a)c(correction)g(of)g(order)f(e)1006 699 y Fv(\000)p Fs(2)p Fw(K)1156 708 y Ft(0)1190 699 y Fw(T)8 b(=")1326 732 y Fz(=)p FA(\(1)23 b Fx(\000)d FA(2)p Fz(\015)5 b FA(\))33 b(in)d(the)i(exp)s(onen)m(t.)45 b(The)32 b(second)f(and)g(the) h(third)e(term)118 845 y(can)37 b(b)s(e)f(estimated)h(b)m(y)f (Corollary)f(3.5)j(and)e(Lemma)g(3.3,)j(the)e(only)f(di\013erence)g (lying)f(in)g(a)i(larger)118 958 y(absolute)30 b(v)-5 b(alue)29 b(of)h(the)g(exp)s(onen)m(t,)g(b)s(ecause)g(w)m(e)g(enlarged) g Fz(h)g FA(and)f Fz(h)2529 972 y Fs(1)2569 958 y FA(.)40 b(The)29 b(last)h(term)g(v)-5 b(anishes)28 b(b)m(y)118 1071 y(our)22 b(c)m(hoice)g(of)g Fz(H)704 1085 y Fs(2)743 1071 y FA(.)38 b(Finally)-8 b(,)23 b(the)f(second)g(term)g(on)f(the)i (righ)m(t-hand)d(side)h(of)29 b(\(3.64\))c(can)d(b)s(e)f(estimated)118 1184 y(b)m(y)h(splitting)f(according)h(to)h(the)g(v)-5 b(alue)22 b(of)g Fz(\034)1603 1202 y Fv(B)r Fs(\(2)p Fw(h)p Fs(\))1808 1184 y FA(and)g(applying)e(Lemma)j(3.4)g(and)f (Corollary)f(3.5.)p 3595 1184 4 62 v 3599 1126 55 4 v 3599 1184 V 3653 1184 4 62 v 259 1366 a(W)-8 b(e)36 b(are)f(no)m(w)g (prepared)f(to)h(establish)f(an)g(impro)m(v)m(ed)g(estimate)i(on)f(the) g(distribution)c(of)k Fz(\034)3489 1385 y Fv(B)r Fs(\()p Fw(h)p Fs(\))3636 1366 y FA(.)118 1489 y(As)28 b(w)m(e)g(will)d (restart)k(the)f(pro)s(cess)f Fz(y)1358 1456 y Fs(det)1355 1511 y Fw(t)1488 1489 y FA(whenev)m(er)g Fz(t)h FA(is)f(a)h(m)m (ultiple)d(of)j Fz(T)13 b FA(,)28 b(w)m(e)h(need)e(the)h(assumptions) 118 1602 y(made)g(in)f(the)h(previous)e(section)i(to)h(hold)d (uniformly)f(in)h(the)j(initial)c(condition)h Fz(y)2955 1616 y Fs(0)3019 1602 y Fx(2)f(D)3175 1616 y Fs(0)3215 1602 y FA(.)40 b(Therefore)118 1715 y(w)m(e)f(will)d(in)m(tro)s(duce)h (replacemen)m(ts)i(for)f(some)h(of)g(the)g(notations)f(in)m(tro)s (duced)f(b)s(efore.)65 b(Note)40 b(that)118 1838 y Fz(\037)175 1805 y Fs(\(1\))269 1838 y FA(\()p Fz(t)p FA(\))e(=)g Fz(\037)576 1790 y Fs(\(1\))576 1850 y Fw(y)611 1859 y Ft(0)670 1838 y FA(\()p Fz(t)p FA(\))g(and)g Fz(\037)1053 1805 y Fs(\(2\))1147 1838 y FA(\()p Fz(t)p FA(\))g(=)f Fz(\037)1453 1790 y Fs(\(2\))1453 1850 y Fw(y)1488 1859 y Ft(0)1547 1838 y FA(\()p Fz(t)p FA(\))i(dep)s(end)d(on)h Fz(y)2190 1852 y Fs(0)2267 1838 y FA(via)h(the)g(principal)c(solution)i Fz(V)21 b FA(.)63 b(Also)118 1951 y Fz(\034)168 1918 y Fs(det)158 1978 y Fv(D)213 1987 y Ft(0)296 1951 y FA(=)25 b Fz(\034)442 1918 y Fs(det)432 1978 y Fv(D)487 1987 y Ft(0)544 1951 y FA(\()p Fz(y)624 1965 y Fs(0)663 1951 y FA(\))31 b(naturally)d(dep)s(ends)h(on)h Fz(y)1643 1965 y Fs(0)1682 1951 y FA(.)41 b(W)-8 b(e)31 b(de\014ne)1249 2144 y Fq(b)-59 b Fz(\037)1298 2106 y Fs(\(1\))1392 2144 y FA(\()p Fz(t)p FA(\))26 b(=)59 b(sup)1617 2222 y Fw(y)1652 2231 y Ft(0)1687 2222 y Fv(2D)1789 2231 y Ft(0)1838 2144 y Fz(\037)1895 2106 y Fs(\(1\))1895 2166 y Fw(y)1930 2175 y Ft(0)1989 2070 y Fq(\000)2031 2144 y Fz(t)20 b Fx(^)g Fz(\034)2215 2106 y Fs(det)2205 2166 y Fv(D)2260 2175 y Ft(0)2317 2144 y FA(\()p Fz(y)2397 2158 y Fs(0)2436 2144 y FA(\))2471 2070 y Fq(\001)2513 2144 y Fz(;)891 b FA(\(3.66\))1249 2364 y Fq(b)-59 b Fz(\037)1298 2327 y Fs(\(2\))1392 2364 y FA(\()p Fz(t)p FA(\))26 b(=)59 b(sup)1617 2443 y Fw(y)1652 2452 y Ft(0)1687 2443 y Fv(2D)1789 2452 y Ft(0)1838 2364 y Fz(\037)1895 2327 y Fs(\(2\))1895 2387 y Fw(y)1930 2396 y Ft(0)1989 2291 y Fq(\000)2031 2364 y Fz(t)20 b Fx(^)g Fz(\034)2215 2327 y Fs(det)2205 2387 y Fv(D)2260 2396 y Ft(0)2317 2364 y FA(\()p Fz(y)2397 2378 y Fs(0)2436 2364 y FA(\))2471 2291 y Fq(\001)2513 2364 y Fz(:)891 b FA(\(3.67\))118 2616 y(In)35 b(the)h(same)g(spirit,)f (the)g Fz(\037)1120 2583 y Fs(\()p Fw(i)p Fs(\))1203 2616 y FA(\()p Fz(T)13 b FA(\)-dep)s(enden)m(t)36 b Fx(O)s FA(\()p Fx(\001)p FA(\)-terms)g(in)f(the)g(de\014nitions)e(of)j Fz(\024)3153 2583 y Fs(+)3213 2616 y FA(\()p Fz(\013)p FA(\),)i Fz(\024)3456 2583 y Fv(0)3515 2616 y FA(and)118 2729 y(the)31 b(prefactors)f(lik)m(e)g Fz(C)936 2696 y Fs(+)929 2752 y Fw(n;m;\015)1118 2729 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))31 b(are)g(mo)s(di\014ed.)259 2856 y(W)-8 b(e)33 b(\014x)d(a)i(time)e Fz(T)44 b FA(of)31 b(order)g(1)h(satisfying)38 b Fq(b)-60 b Fz(\037)1809 2823 y Fs(\(2\))1904 2856 y FA(\()p Fz(T)13 b FA(\))27 b Fn(6)f FA(\()p Fz(\032)2246 2823 y Fs(2)2306 2856 y FA(+)21 b Fz(")p FA(\))2475 2823 y Fv(\000)p Fs(1)2570 2856 y FA(.)43 b Fz(T)g FA(is)31 b(c)m(hosen)g(in)f(suc)m(h)h(a)g(w)m (a)m(y)118 2969 y(that)k(whenev)m(er)g Fz(h)d Fn(6)g Fz(c)949 2983 y Fs(1)997 2969 y Fq(b)-59 b Fz(\037)1046 2936 y Fs(\(1\))1140 2969 y FA(\()p Fz(T)13 b FA(\))1276 2936 y Fv(\000)p Fs(1)1371 2969 y FA(,)35 b(Corollary)e(3.5)j(\(and)e (Lemma)h(3.7\))h(apply)-8 b(.)52 b(Note)36 b(that)f(larger)118 3082 y Fz(T)46 b FA(w)m(ould)32 b(b)s(e)h(p)s(ossible)d(unless)i Fz(\032)h FA(is)f(of)i(order)f(1,)h(but)f(for)g(larger)g Fz(T)46 b FA(the)33 b(constrain)m(t)g(on)h Fz(h)f FA(b)s(ecomes)118 3195 y(more)e(restrictiv)m(e)f(whic)m(h)f(is)g(not)i(desirable.)39 b(Ha)m(ving)31 b(c)m(hosen)f Fz(T)13 b FA(,)31 b(w)m(e)g(de\014ne)e (the)i(probabilities)832 3368 y Fz(P)890 3383 y Fw(k)933 3368 y FA(\()p Fz(h)p FA(\))26 b(=)f Fr(P)1238 3331 y Fs(0)p Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))1443 3294 y Fq(\010)1496 3368 y Fz(\034)1536 3387 y Fv(B)r Fs(\()p Fw(h)p Fs(\))1708 3368 y Fz(<)g(k)s(T)34 b Fx(^)19 b Fz(\034)2061 3382 y Fv(D)2116 3391 y Ft(0)2154 3294 y Fq(\011)2207 3368 y Fz(;)1197 b FA(\(3.68\))818 3524 y Fz(Q)890 3539 y Fw(k)933 3524 y FA(\()p Fz(h)p FA(\))26 b(=)f Fr(P)1238 3486 y Fs(0)p Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))1443 3450 y Fq(\010)1496 3524 y Fx(h)p Fz(\030)1571 3539 y Fw(k)r(T)1664 3524 y Fz(;)p 1728 3451 55 4 v 15 w(X)7 b FA(\()p Fz(y)1866 3539 y Fw(k)r(T)1961 3524 y FA(\))1996 3486 y Fv(\000)p Fs(1)2091 3524 y Fz(\030)2131 3539 y Fw(k)r(T)2224 3524 y Fx(i)25 b Fn(>)g Fz(h)2432 3486 y Fs(2)2472 3524 y Fz(;)15 b(\034)2552 3538 y Fv(D)2607 3547 y Ft(0)2671 3524 y Fn(>)25 b Fz(k)s(T)2883 3450 y Fq(\011)2936 3524 y Fz(:)468 b FA(\(3.69\))118 3697 y(Corollary)36 b(3.5)j(pro)m(vides)d (a)i(b)s(ound)e(for)h Fz(P)1622 3711 y Fs(1)1662 3697 y FA(\()p Fz(h)p FA(\),)j(and)d(Lemma)h(3.7)h(pro)m(vides)d(a)i(b)s (ound)d(for)j Fz(Q)3474 3711 y Fs(1)3513 3697 y FA(\()p Fz(h)p FA(\).)118 3810 y(Subsequen)m(t)e(b)s(ounds)f(are)i(computed)g (b)m(y)g(induction,)f(and)h(the)g(follo)m(wing)e(prop)s(osition)g (describ)s(es)118 3923 y(one)c(induction)d(step.)118 4085 y Fm(Prop)s(osition)36 b(3.8.)42 b Fy(L)-5 b(et)37 b FA(^)-49 b Fz(\024)26 b Fn(6)f Fz(\024)1270 4052 y Fs(+)1329 4085 y FA(\(0\))d Fx(^)e Fz(\024)1599 4052 y Fv(0)1622 4085 y Fy(.)42 b(Assume)32 b(that)i(for)f(some)h Fz(k)28 b Fx(2)d Fr(N)d Fy(,)1473 4270 y Fz(P)1531 4285 y Fw(k)1574 4270 y FA(\()p Fz(h)p FA(\))k Fn(6)f Fz(D)1893 4285 y Fw(k)1952 4270 y FA(e)1992 4233 y Fv(\000)s Fs(^)-38 b Fw(\024)o(h)2128 4209 y Ft(2)2163 4233 y Fw(=\033)2240 4209 y Ft(2)2279 4270 y Fz(;)1125 b FA(\(3.70\))1460 4430 y Fz(Q)1532 4445 y Fw(k)1574 4430 y FA(\()p Fz(h)p FA(\))26 b Fn(6)1837 4407 y Fq(b)1818 4430 y Fz(D)1893 4445 y Fw(k)1952 4430 y FA(e)1992 4393 y Fv(\000)s Fs(^)-38 b Fw(\024)o(h)2128 4369 y Ft(2)2163 4393 y Fw(=\033)2240 4369 y Ft(2)2295 4430 y Fz(:)1109 b FA(\(3.71\))118 4604 y Fy(Then)33 b(the)g(same)g(b)-5 b(ounds)34 b(hold)g(for)f Fz(k)j Fy(r)-5 b(eplac)g(e)g(d)35 b(by)e Fz(k)23 b FA(+)d(1)p Fy(,)33 b(pr)-5 b(ovide)g(d)981 4791 y Fz(D)1056 4806 y Fw(k)r Fs(+1)1214 4791 y Fn(>)25 b Fz(D)1385 4806 y Fw(k)1448 4791 y FA(+)20 b Fz(C)1611 4754 y Fs(+)1604 4814 y Fw(n;m;\015)1793 4791 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))2026 4768 y Fq(b)2006 4791 y Fz(D)2081 4806 y Fw(k)2217 4730 y Fz(\015)p 2135 4770 216 4 v 2135 4853 a(\015)25 b Fx(\000)f FA(^)-49 b Fz(\024)2376 4791 y FA(e)2416 4754 y Fs(\()p Fw(\015)t Fv(\000)s Fs(^)-38 b Fw(\024)p Fs(\))p Fw(h)2647 4730 y Ft(2)2682 4754 y Fw(=\033)2759 4730 y Ft(2)3429 4791 y FA(\(3.72\))999 4970 y Fq(b)981 4993 y Fz(D)1056 5008 y Fw(k)r Fs(+1)1214 4993 y Fn(>)1329 4970 y Fq(b)1310 4993 y Fz(D)1385 5008 y Fw(k)1448 4993 y FA(+)1557 4970 y Fq(b)1539 4993 y Fz(C)1604 5007 y Fw(n;m;\015)1793 4993 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))p Fz(:)1398 b FA(\(3.73\))118 5179 y Fm(Remark)27 b(3.9.)36 b FA(Note)25 b(that)f Fz(\015)12 b Fx(\000)e FA(^)-49 b Fz(\024)25 b FA(=)g Fx(O)s FA(\()p Fz(h)p FA(\))6 b(+)g Fx(O)s FA(\(\001\))g(+)g Fx(O)s FA(\(\(1)g(+)15 b Fq(b)-60 b Fz(\037)2332 5146 y Fs(\(1\))2431 5179 y FA(\()p Fz(T)13 b FA(\)\))p Fz(h)p FA(\))6 b(+)g Fx(O)2847 5105 y Fq(\000)2891 5179 y FA(e)2931 5146 y Fv(\000)p Fs(2)p Fw(K)3081 5155 y Ft(0)3116 5146 y Fw(T)i(=")3252 5179 y Fz(=)p FA(\(1)e Fx(\000)g FA(2)p Fz(\015)f FA(\))3592 5105 y Fq(\001)3636 5179 y FA(.)118 5292 y(In)26 b(the)g(case)i Fz(\024)621 5259 y Fs(+)680 5292 y FA(\(0\))f(=)e Fz(\024)970 5259 y Fv(0)1019 5292 y FA(=)f Fz(\015)5 b FA(,)28 b(w)m(e)f(ma)m(y)g (either)f(c)m(ho)s(ose)31 b(^)-49 b Fz(\024)26 b(<)f(\024)2306 5259 y Fs(+)2365 5292 y FA(\(0\))12 b Fx(^)g Fz(\024)2617 5259 y Fv(0)2642 5292 y FA(,)27 b(or)g(w)m(e)g(ma)m(y)g(replace)f (\(3.72\))118 5404 y(b)m(y)665 5574 y Fz(D)740 5589 y Fw(k)r Fs(+1)899 5574 y Fn(>)e Fz(D)1069 5589 y Fw(k)1133 5574 y FA(+)c Fz(C)1296 5537 y Fs(+)1289 5597 y Fw(n;m;\015)1477 5574 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))1710 5551 y Fq(b)1690 5574 y Fz(D)1765 5589 y Fw(k)1810 5446 y Fq(\024)1857 5574 y FA(1)21 b(+)f(log)2131 5446 y Fq(\022)2208 5504 y Fz(C)2280 5471 y Fs(+)2273 5526 y Fw(n;m;\015)2462 5504 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))p 2208 5553 469 4 v 2402 5630 a Fq(b)2383 5653 y Fz(D)2458 5668 y Fw(k)2702 5574 y FA(e)2742 5537 y Fw(\015)t(h)2823 5513 y Ft(2)2857 5537 y Fw(=\033)2934 5513 y Ft(2)2974 5446 y Fq(\023\025)3089 5574 y Fz(:)315 b FA(\(3.74\))1844 5871 y(29)p eop %%Page: 30 30 30 29 bop 118 280 a FA(Belo)m(w)31 b(w)m(e)g(will)d(optimize)h(with)g (resp)s(ect)i(to)j(^)-48 b Fz(\024)p FA(.)118 468 y Fg(Pr)n(oof)34 b(of)g(Pr)n(oposition)f(3.8.)46 b FA(W)-8 b(e)33 b(start)e(b)m(y)h (establishing)e(\(3.73\))r(.)43 b(The)31 b(Mark)m(o)m(v)i(prop)s(ert)m (y)d(al-)118 581 y(lo)m(ws)g(for)g(the)h(decomp)s(osition)151 785 y Fz(Q)223 800 y Fw(k)r Fs(+1)355 785 y FA(\()p Fz(h)p FA(\))176 940 y Fn(6)25 b Fr(P)333 902 y Fs(0)p Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))537 866 y Fq(\010)590 940 y Fz(\034)630 958 y Fv(B)r Fs(\()p Fw(h)p Fs(\))803 940 y Fz(<)g(k)s(T)8 b(;)15 b(\034)1090 954 y Fv(D)1145 963 y Ft(0)1209 940 y Fn(>)25 b Fz(k)s(T)1421 866 y Fq(\011)171 1112 y FA(+)20 b Fr(E)322 1075 y Fs(0)q Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))532 1012 y Fq(n)593 1112 y FA(1)638 1131 y Fv(f)p Fw(\034)704 1145 y Fk(B)q Ft(\()p Fl(h)p Ft(\))835 1131 y Fo(>)p Fw(k)r(T)10 b Fv(g)1019 1112 y Fr(P)1080 1075 y Fw(k)r(T)c(;)p Fs(\()p Fw(\030)1244 1087 y Fl(k)q(T)1326 1075 y Fw(;)p Fs(0\))1413 1039 y Fq(\010)1466 1112 y Fx(h)p Fz(\030)1541 1131 y Fs(\()p Fw(k)r Fs(+1\))p Fw(T)1779 1112 y Fz(;)p 1843 1039 55 4 v 15 w(X)h FA(\()p Fz(y)1981 1131 y Fs(\()p Fw(k)r Fs(+1\))p Fw(T)2221 1112 y FA(\))2256 1075 y Fv(\000)p Fs(1)2351 1112 y Fz(\030)2391 1131 y Fs(\()p Fw(k)r Fs(+1\))p Fw(T)2629 1112 y Fx(i)26 b Fn(>)e Fz(h)2837 1075 y Fs(2)2877 1112 y Fz(;)15 b(\034)2957 1126 y Fv(D)3012 1135 y Ft(0)3076 1112 y Fn(>)25 b FA(\()p Fz(k)f FA(+)c(1\))p Fz(T)3515 1039 y Fq(\011)3568 1012 y(o)176 1298 y Fn(6)25 b Fz(Q)344 1313 y Fw(k)386 1298 y FA(\()p Fz(h)p FA(\))d(+)639 1275 y Fq(b)621 1298 y Fz(C)686 1312 y Fw(n;m;\015)874 1298 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))g(e)1144 1260 y Fv(\000)s Fs(^)-38 b Fw(\024h)1281 1237 y Ft(2)1315 1260 y Fw(=\033)1392 1237 y Ft(2)1432 1298 y Fz(;)1972 b FA(\(3.75\))118 1502 y(where)32 b(the)g(initial)d(condition)i(\()p Fz(\030)1281 1517 y Fw(k)r(T)1374 1502 y Fz(;)15 b FA(0\))33 b(indicates)e(that)i (at)g(time)e Fz(k)s(T)13 b FA(,)33 b(w)m(e)f(also)g(restart)h(the)f (pro)s(cess)118 1615 y(of)d(the)h(deterministic)d(slo)m(w)i(v)-5 b(ariables)28 b Fz(y)1544 1582 y Fs(det)1541 1637 y Fw(t)1675 1615 y FA(in)g(the)h(p)s(oin)m(t)f Fz(y)2216 1630 y Fw(k)r(T)2335 1615 y Fx(2)c(D)2490 1629 y Fs(0)2530 1615 y FA(.)40 b(In)29 b(the)g(second)g(line,)f(w)m(e)i(used)118 1728 y(Lemma)h(3.7.)41 b(This)29 b(sho)m(ws)h(\(3.73\))r(.)259 1841 y(As)h(for)f(\(3.72\))r(,)h(w)m(e)f(again)h(start)g(from)f(a)g (decomp)s(osition,)g(similar)d(to)32 b(\(3.75\))r(:)386 2045 y Fz(P)444 2060 y Fw(k)r Fs(+1)577 2045 y FA(\()p Fz(h)p FA(\))27 b(=)d Fr(P)882 2008 y Fs(0)p Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))1087 1971 y Fq(\010)1140 2045 y Fz(\034)1180 2064 y Fv(B)r Fs(\()p Fw(h)p Fs(\))1353 2045 y Fz(<)h(k)s(T)33 b Fx(^)20 b Fz(\034)1706 2059 y Fv(D)1761 2068 y Ft(0)1799 1971 y Fq(\011)842 2218 y FA(+)g Fr(E)993 2180 y Fs(0)p Fw(;)p Fs(\(0)q Fw(;)p Fs(0\))1203 2117 y Fq(n)1264 2218 y FA(1)1309 2236 y Fv(f)p Fw(\034)1375 2250 y Fk(B)q Ft(\()p Fl(h)p Ft(\))1506 2236 y Fo(>)p Fw(k)r(T)10 b Fv(g)1689 2218 y Fr(P)1750 2180 y Fw(k)r(T)c(;)p Fs(\()p Fw(\030)1914 2192 y Fl(k)q(T)1997 2180 y Fw(;)p Fs(0\))2083 2144 y Fq(\010)2136 2218 y Fz(\034)2176 2236 y Fv(B)r Fs(\()p Fw(h)p Fs(\))2349 2218 y Fz(<)25 b FA(\()p Fz(k)f FA(+)c(1\))p Fz(T)33 b Fx(^)20 b Fz(\034)2929 2232 y Fv(D)2984 2241 y Ft(0)3022 2144 y Fq(\011)3075 2117 y(o)3136 2218 y Fz(:)268 b FA(\(3.76\))118 2440 y(Corollary)29 b(3.5)j(allo)m(ws)d(us)h(to)h(estimate)195 2644 y Fz(P)253 2659 y Fw(k)r Fs(+1)387 2644 y FA(\()p Fz(h)p FA(\))312 2816 y Fn(6)25 b Fz(P)466 2831 y Fw(k)509 2816 y FA(\()p Fz(h)p FA(\))c(+)f Fr(E)803 2778 y Fs(0)q Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))1013 2715 y Fq(n)1074 2816 y FA(1)1119 2840 y Fv(fh)p Fw(\030)1212 2852 y Fl(k)q(T)1296 2840 y Fw(;)p 1332 2788 43 3 v(X)5 b Fs(\()p Fw(y)1441 2852 y Fl(k)q(T)1523 2840 y Fs(\))1550 2821 y Fk(\000)p Ft(1)1633 2840 y Fw(\030)1664 2852 y Fl(k)q(T)1747 2840 y Fv(i)p Fo(6)p Fw(h)1870 2821 y Ft(2)1905 2840 y Fv(g)1944 2742 y Fq(\002)1982 2816 y Fz(')2041 2742 y Fq(\000)2083 2816 y Fx(h)p Fz(\030)2158 2831 y Fw(k)r(T)2252 2816 y Fz(;)p 2316 2743 55 4 v 15 w(X)i FA(\()p Fz(y)2454 2831 y Fw(k)r(T)2548 2816 y FA(\))2583 2778 y Fv(\000)p Fs(1)2678 2816 y Fz(\030)2718 2831 y Fw(k)r(T)2811 2816 y Fx(i)2846 2742 y Fq(\001)2908 2816 y Fx(^)20 b FA(1)3034 2742 y Fq(\003)3098 2711 y(\014)3098 2766 y(\014)3098 2820 y(\014)3153 2816 y Fz(\034)3193 2830 y Fv(D)3248 2839 y Ft(0)3312 2816 y Fn(>)25 b Fz(k)s(T)3524 2715 y Fq(o)763 2995 y Fx(\002)20 b Fr(P)915 2958 y Fs(0)p Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))1119 2922 y Fq(\010)1172 2995 y Fz(\034)1212 3009 y Fv(D)1267 3018 y Ft(0)1331 2995 y Fn(>)25 b Fz(k)s(T)1543 2922 y Fq(\011)1596 2995 y Fz(;)1808 b FA(\(3.77\))118 3199 y(with)1027 3312 y Fz(')p FA(\()p Fz(s)p FA(\))26 b(=)f Fz(C)1393 3275 y Fs(+)1386 3335 y Fw(n;m;\015)1575 3312 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))g(e)1844 3275 y Fs(\()p Fw(\015)t Fv(\000)s Fs(^)-38 b Fw(\024)q Fs(\))p Fw(h)2076 3251 y Ft(2)2110 3275 y Fw(=\033)2187 3251 y Ft(2)2242 3312 y FA(e)2283 3275 y Fv(\000)p Fw(\015)t Fs(\()p Fw(h)2446 3251 y Ft(2)2481 3275 y Fv(\000)p Fw(s)p Fs(\))p Fw(=\033)2673 3251 y Ft(2)2727 3312 y Fz(:)677 b FA(\(3.78\))118 3479 y(\(3.71\))33 b(sho)m(ws)d(that)834 3684 y Fr(P)895 3646 y Fs(0)p Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))1099 3610 y Fq(\010)1152 3684 y Fx(h)p Fz(\030)1227 3699 y Fw(k)r(T)1321 3684 y Fz(;)p 1385 3610 V 15 w(X)7 b FA(\()p Fz(y)1523 3699 y Fw(k)r(T)1617 3684 y FA(\))1652 3646 y Fv(\000)p Fs(1)1747 3684 y Fz(\030)1787 3699 y Fw(k)r(T)1880 3684 y Fx(i)26 b Fz(<)f(s)2105 3606 y Fq(\014)2105 3661 y(\014)2160 3684 y Fz(\034)2200 3698 y Fv(D)2255 3707 y Ft(0)2319 3684 y Fn(>)g Fz(k)s(T)2531 3610 y Fq(\011)2609 3684 y Fn(>)g Fz(')2764 3699 y Fw(k)2807 3684 y FA(\()p Fz(s)p FA(\))p Fz(;)484 b FA(\(3.79\))118 3888 y(where)990 4001 y Fz(')1049 4016 y Fw(k)1092 4001 y FA(\()p Fz(s)p FA(\))1220 3999 y(:)1241 4001 y(=)1312 3927 y Fq(\000)1353 4001 y FA(1)21 b Fx(\000)1528 3978 y Fq(b)1510 4001 y Fz(D)1585 4016 y Fw(k)1643 4001 y FA(e)1683 3963 y Fv(\000)s Fs(^)-38 b Fw(\024s=\033)1889 3940 y Ft(2)1928 3927 y Fq(\001\016)2022 4001 y Fr(P)2083 3963 y Fs(0)p Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))2288 3927 y Fq(\010)2341 4001 y Fz(\034)2381 4015 y Fv(D)2436 4024 y Ft(0)2499 4001 y Fn(>)25 b Fz(k)s(T)2711 3927 y Fq(\011)2764 4001 y Fz(:)640 b FA(\(3.80\))118 4168 y(The)30 b(functions)f Fz(')i FA(and)e Fz(')1022 4183 y Fw(k)1096 4168 y FA(ful\014l)e(the)k(assumptions)e(of)h(Lemma)h (3.6)g(with)737 4372 y(e)777 4334 y Fw(\015)t(s)850 4343 y Ft(2)885 4334 y Fw(=\033)962 4311 y Ft(2)1027 4372 y FA(=)24 b Fz(C)1194 4334 y Fs(+)1187 4394 y Fw(n;m;\015)1376 4372 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))1589 4334 y Fv(\000)p Fs(1)1700 4372 y FA(e)1743 4334 y Fs(^)-38 b Fw(\024h)1822 4311 y Ft(2)1856 4334 y Fw(=\033)1933 4311 y Ft(2)2170 4372 y FA(and)196 b(e)2557 4334 y Fs(^)-38 b Fw(\024s)2628 4343 y Ft(1)2662 4334 y Fw(=\033)2739 4311 y Ft(2)2804 4372 y FA(=)2918 4349 y Fq(b)2900 4372 y Fz(D)2975 4387 y Fw(k)3018 4372 y Fz(:)386 b FA(\(3.81\))118 4576 y(F)-8 b(or)31 b Fz(h)333 4543 y Fs(2)398 4576 y Fn(6)25 b Fz(s)537 4590 y Fs(1)576 4576 y FA(,)31 b(\(3.70\))h(b)s (ecomes)f(trivial,)e(while)f(for)i Fz(h)1980 4543 y Fs(2)2045 4576 y Fz(>)25 b(s)2184 4590 y Fs(1)2223 4576 y FA(,)31 b(Lemma)f(3.6)i(sho)m(ws)306 4780 y Fz(P)364 4795 y Fw(k)r Fs(+1)498 4780 y FA(\()p Fz(h)p FA(\))26 b Fn(6)f Fz(P)800 4795 y Fw(k)843 4780 y FA(\()p Fz(h)p FA(\))c Fx(\000)f Fz(')p FA(\()p Fz(h)1223 4743 y Fs(2)1284 4780 y Fx(^)g Fz(s)1408 4794 y Fs(2)1447 4780 y FA(\))1482 4707 y Fq(\002)1520 4780 y FA(1)h Fx(\000)f Fr(P)1738 4743 y Fs(0)p Fw(;)p Fs(\(0)p Fw(;)p Fs(0\))1942 4707 y Fq(\010)1995 4780 y Fx(h)p Fz(\030)2070 4795 y Fw(k)r(T)2164 4780 y Fz(;)p 2228 4707 V 15 w(X)7 b FA(\()p Fz(y)2366 4795 y Fw(k)r(T)2460 4780 y FA(\))2495 4743 y Fv(\000)p Fs(1)2590 4780 y Fz(\030)2630 4795 y Fw(k)r(T)2723 4780 y Fx(i)26 b Fz(<)f(h)2932 4743 y Fs(2)2972 4780 y Fz(;)15 b(\034)3052 4794 y Fv(D)3107 4803 y Ft(0)3170 4780 y Fn(>)25 b Fz(k)s(T)3382 4707 y Fq(\011\003)762 5000 y FA(+)20 b Fz(')p FA(\()p Fz(s)990 5014 y Fs(1)1030 5000 y FA(\))g(+)1176 4877 y Fq(Z)1267 4903 y Fw(s)1300 4912 y Ft(2)1334 4903 y Fv(^)p Fw(h)1422 4879 y Ft(2)1227 5083 y Fw(s)1260 5092 y Ft(1)1476 5000 y Fz(')1535 4963 y Fv(0)1559 5000 y FA(\()p Fz(s)p FA(\))1691 4977 y Fq(b)1672 5000 y Fz(D)1747 5015 y Fw(k)1805 5000 y FA(e)1846 4963 y Fv(\000)s Fs(^)-38 b Fw(\024)o(s=\033)2051 4939 y Ft(2)2106 5000 y FA(d)o Fz(s)646 5226 y Fn(6)25 b Fz(P)800 5241 y Fw(k)843 5226 y FA(\()p Fz(h)p FA(\))c(+)f Fz(C)1149 5188 y Fs(+)1142 5248 y Fw(n;m;\015)1331 5226 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))1564 5203 y Fq(b)1544 5226 y Fz(D)1619 5241 y Fw(k)1755 5164 y Fz(\015)p 1673 5205 216 4 v 1673 5288 a(\015)25 b Fx(\000)f FA(^)-49 b Fz(\024)1914 5226 y FA(e)1954 5188 y Fs(\()p Fw(\015)t Fv(\000)s Fs(^)-38 b Fw(\024)p Fs(\))p Fw(h)2185 5165 y Ft(2)2220 5188 y Fw(=\033)2297 5165 y Ft(2)2352 5226 y FA(e)2392 5188 y Fv(\000)s Fs(^)g Fw(\024h)2529 5165 y Ft(2)2563 5188 y Fw(=\033)2640 5165 y Ft(2)2695 5226 y Fz(:)709 b FA(\(3.82\))118 5474 y(No)m(w,)31 b(\(3.72\))i(is)c (immediate.)p 3595 5474 4 62 v 3599 5416 55 4 v 3599 5474 V 3653 5474 4 62 v 1844 5871 a(30)p eop %%Page: 31 31 31 30 bop 259 280 a FA(Rep)s(eated)44 b(application)d(of)j(the)f (previous)f(result)g(\014nally)f(leads)h(to)i(the)g(follo)m(wing)d (estimate,)118 393 y(whic)m(h)36 b(is)g(equiv)-5 b(alen)m(t)36 b(to)i(the)f(upp)s(er)e(b)s(ound)g(in)h(Theorem)h(2.4.)61 b(Note)39 b(that)e(the)g(lo)m(w)m(er)h(b)s(ound)d(in)118 506 y(Theorem)42 b(2.4)i(is)d(a)i(direct)e(consequence)i(of)g(the)f(lo) m(w)m(er)h(b)s(ound)d(in)h(Corollary)g(3.5,)47 b(so)42 b(that)h(the)118 619 y(corollary)30 b(b)s(elo)m(w)f(completes)i(the)g (pro)s(of)e(of)i(the)f(theorem.)118 807 y Fm(Corollary)37 b(3.10.)42 b Fy(Assume)32 b(that)i Fz(y)1411 821 y Fs(0)1476 807 y Fx(2)24 b(D)1631 821 y Fs(0)1671 807 y Fy(,)32 b Fz(x)1783 821 y Fs(0)1848 807 y FA(=)f(\026)-51 b Fz(x)p FA(\()p Fz(y)2076 821 y Fs(0)2115 807 y Fz(;)15 b(")p FA(\))p Fy(.)43 b(Then,)33 b(for)g(every)g Fz(t)25 b(>)g FA(0)p Fy(,)33 b(we)f(have)348 1011 y Fr(P)409 973 y Fs(0)p Fw(;)p Fs(\()p Fw(x)531 982 y Ft(0)566 973 y Fw(;y)621 982 y Ft(0)655 973 y Fs(\))687 937 y Fq(\010)740 1011 y Fz(\034)780 1029 y Fv(B)r Fs(\()p Fw(h)p Fs(\))952 1011 y Fz(<)25 b(t)20 b Fx(^)g Fz(\034)1222 1025 y Fv(D)1277 1034 y Ft(0)1315 937 y Fq(\011)559 1228 y Fn(6)25 b Fz(C)727 1190 y Fs(+)720 1250 y Fw(n;m;\015)909 1228 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))1122 1099 y Fq(\024)1171 1228 y FA(1)21 b(+)1345 1205 y Fq(b)1327 1228 y Fz(C)1392 1242 y Fw(n;m;\015)1581 1228 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))1794 1099 y Fq(\022)1872 1166 y FA(1)p 1872 1207 46 4 v 1872 1290 a(2)1948 1228 y(+)2065 1166 y Fz(t)p 2049 1207 66 4 v 2049 1290 a(T)2125 1099 y Fq(\023)2191 1122 y Fs(2)2381 1166 y Fz(\015)p 2241 1207 332 4 v 2241 1290 a FA(2\()p Fz(\015)26 b Fx(\000)d FA(^)-48 b Fz(\024)p FA(\))2583 1099 y Fq(\025)2646 1228 y FA(e)2686 1190 y Fv(\000)p Fs(\(2)s(^)-38 b Fw(\024)p Fv(\000)p Fw(\015)t Fs(\))p Fw(h)3007 1167 y Ft(2)3042 1190 y Fw(=\033)3119 1167 y Ft(2)3174 1228 y Fz(:)230 b FA(\(3.83\))118 1477 y Fy(In)33 b(addition,)h(the)f(distribution)h(of)f(the)g(endp)-5 b(oint)34 b Fz(\030)1925 1491 y Fw(t)1987 1477 y Fy(satis\014es)413 1726 y Fr(P)474 1688 y Fs(0)p Fw(;)p Fs(\()p Fw(x)596 1697 y Ft(0)631 1688 y Fw(;y)686 1697 y Ft(0)720 1688 y Fs(\))752 1652 y Fq(\010)805 1726 y Fx(h)p Fz(\030)880 1740 y Fw(t)909 1726 y Fz(;)p 973 1653 55 4 v 15 w(X)7 b FA(\()p Fz(y)1111 1740 y Fw(t)1142 1726 y FA(\))1177 1688 y Fv(\000)p Fs(1)1272 1726 y Fz(\030)1312 1740 y Fw(t)1341 1726 y Fx(i)26 b Fn(>)e Fz(h)1549 1688 y Fs(2)1589 1726 y Fz(;)15 b(\034)1669 1740 y Fv(D)1724 1749 y Ft(0)1788 1726 y Fn(>)25 b Fz(t)1917 1652 y Fq(\011)1995 1726 y Fn(6)2109 1703 y Fq(b)2091 1726 y Fz(C)2156 1740 y Fw(n;m;\015)2345 1726 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))2558 1598 y Fq(\030)2639 1664 y Fz(t)p 2622 1705 66 4 v 2622 1788 a(T)2698 1598 y Fq(\031)2766 1726 y FA(e)2806 1688 y Fv(\000)s Fs(^)-38 b Fw(\024h)2943 1665 y Ft(2)2977 1688 y Fw(=\033)3054 1665 y Ft(2)3109 1726 y Fz(:)295 b FA(\(3.84\))118 1980 y Fg(Pr)n(oof:)45 b FA(W)-8 b(e)40 b(already)e(kno)m(w)h(the)g(b)s (ounds)d(\(3.70\))41 b(and)d(\(3.71\))j(to)f(hold)d(for)i Fz(k)j FA(=)d(1,)i(with)d Fz(D)3512 1994 y Fs(1)3591 1980 y FA(=)118 2093 y Fz(C)190 2060 y Fs(+)183 2115 y Fw(n;m;\015)372 2093 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))27 b(and)802 2070 y Fq(b)784 2093 y Fz(D)859 2107 y Fs(1)924 2093 y FA(=)1038 2070 y Fq(b)1020 2093 y Fz(C)1085 2107 y Fw(n;m;\015)1273 2093 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\).)41 b(No)m(w)26 b(the)g(inductiv)m(e)e(relations)h (\(3.72\))i(and)e(\(3.73\))j(are)e(seen)118 2206 y(to)31 b(b)s(e)f(satis\014ed)f(b)m(y)743 2387 y Fq(b)724 2410 y Fz(D)799 2425 y Fw(k)867 2410 y FA(=)c Fz(k)1032 2387 y Fq(b)1013 2410 y Fz(C)1078 2424 y Fw(n;m;\015)1267 2410 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))p Fz(;)724 2646 y(D)799 2661 y Fw(k)867 2646 y FA(=)25 b Fz(C)1035 2609 y Fs(+)1028 2669 y Fw(n;m;\015)1217 2646 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))1430 2518 y Fq(\024)1479 2646 y FA(1)21 b(+)1654 2623 y Fq(b)1636 2646 y Fz(C)1701 2660 y Fw(n;m;\015)1890 2646 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))2196 2585 y Fz(\015)p 2113 2625 216 4 v 2113 2708 a(\015)26 b Fx(\000)e FA(^)-49 b Fz(\024)2355 2646 y FA(e)2395 2609 y Fs(\()p Fw(\015)t Fv(\000)s Fs(^)-38 b Fw(\024)p Fs(\))p Fw(h)2626 2585 y Ft(2)2661 2609 y Fw(=\033)2738 2585 y Ft(2)2794 2532 y Fw(k)r Fv(\000)p Fs(1)2793 2560 y Fq(X)2797 2755 y Fw(j)t Fs(=1)2939 2646 y Fz(j)2981 2518 y Fq(\025)3030 2646 y Fz(:)374 b FA(\(3.85\))118 2955 y(The)30 b(conclusion)e(follo)m (ws)h(b)m(y)h(taking)g Fz(k)e FA(=)d Fx(d)p Fz(t=T)13 b Fx(e)31 b FA(and)f(b)s(ounding)d(the)j(sum)f(b)m(y)2936 2920 y Fs(1)p 2936 2935 36 4 v 2936 2987 a(2)2981 2955 y FA(\()p Fz(t=T)13 b FA(\)\()p Fz(t=T)34 b FA(+)19 b(1\))26 b Fn(6)128 3032 y Fs(1)p 128 3047 V 128 3100 a(2)173 3068 y FA(\()p Fz(t=T)34 b FA(+)20 b(1)p Fz(=)p FA(2\))634 3035 y Fs(2)675 3068 y FA(.)p 3595 3068 4 62 v 3599 3010 55 4 v 3599 3068 V 3653 3068 4 62 v 259 3256 a(T)-8 b(o)35 b(complete)g(the)g(pro)s(of)e(of)i(Theorem)f(2.4,)j(w)m(e)e(\014rst)e (optimize)h(our)g(c)m(hoice)h(of)j(^)-48 b Fz(\024)p FA(,)36 b(taking)e(in)m(to)118 3369 y(accoun)m(t)e(the)f(constrain)m(t) j(^)-49 b Fz(\024)26 b Fn(6)f Fz(\024)1264 3336 y Fs(+)1323 3369 y FA(\(0\))c Fx(^)f Fz(\024)1592 3336 y Fv(0)1616 3369 y FA(.)41 b(By)30 b(doing)g(so,)h(w)m(e)f(\014nd)f(that)1276 3567 y Fz(\015)p 1136 3608 332 4 v 1136 3691 a FA(2\()p Fz(\015)d Fx(\000)e FA(^)-49 b Fz(\024)q FA(\))1493 3628 y(e)1534 3591 y Fv(\000)p Fs(\(2)s(^)-38 b Fw(\024)p Fv(\000)p Fw(\015)t Fs(\))p Fw(h)1855 3567 y Ft(2)1890 3591 y Fw(=\033)1967 3567 y Ft(2)2032 3628 y Fn(6)2138 3567 y FA(2)p Fz(h)2235 3534 y Fs(2)p 2138 3608 138 4 v 2159 3691 a Fz(\033)2214 3665 y Fs(2)2300 3628 y FA(e)2340 3591 y Fv(\000)p Fw(\024h)2477 3567 y Ft(2)2511 3591 y Fw(=\033)2588 3567 y Ft(2)2628 3628 y Fz(;)776 b FA(\(3.86\))118 3876 y(where)30 b(w)m(e)h(ha)m(v)m(e)g(set)893 4081 y Fz(\024)25 b FA(=)g Fz(\015)1118 4007 y Fq(\002)1156 4081 y FA(1)c Fx(\000)f(O)s FA(\()p Fz(h)p FA(\))h Fx(\000)f(O)s FA(\(\001\))g Fx(\000)g(O)2029 4007 y Fq(\000)2071 4081 y FA(e)2111 4043 y Fv(\000)p Fu(c)l(onst)q Fw(=")2412 4081 y Fz(=)p FA(\(1)h Fx(\000)f FA(2)p Fz(\015)5 b FA(\))2781 4007 y Fq(\001)q(\003)2862 4081 y Fz(:)542 b FA(\(3.87\))118 4285 y(Simplifying)26 b(the)k(prefactor)h(in)e(\(3.83\))k(\014nally)28 b(yields)264 4544 y Fr(P)325 4507 y Fs(0)p Fw(;)p Fs(\()p Fw(x)447 4516 y Ft(0)482 4507 y Fw(;y)537 4516 y Ft(0)571 4507 y Fs(\))603 4471 y Fq(\010)656 4544 y Fz(\034)696 4563 y Fv(B)r Fs(\()p Fw(h)p Fs(\))868 4544 y Fz(<)d(t)20 b Fx(^)g Fz(\034)1138 4558 y Fv(D)1193 4567 y Ft(0)1231 4471 y Fq(\011)1310 4544 y Fn(6)1415 4483 y FA(\(1)h(+)f Fz(t)p FA(\))1675 4450 y Fs(2)p 1415 4523 300 4 v 1506 4607 a FA(\001)p Fz(")1725 4443 y Fq(h)1768 4544 y FA(\(1)h Fx(\000)f FA(2)p Fz(\015)5 b FA(\))2092 4507 y Fv(\000)p Fw(n)2215 4544 y FA(+)20 b(e)2346 4507 y Fw(n=)p Fs(4)2479 4544 y FA(+)15 b(e)2605 4507 y Fw(m=)p Fs(4)2742 4443 y Fq(i)2797 4483 y Fz(h)2849 4450 y Fs(2)p 2795 4523 95 4 v 2795 4607 a Fz(\033)2850 4580 y Fs(2)2915 4544 y FA(e)2955 4507 y Fv(\000)p Fw(\024h)3092 4483 y Ft(2)3126 4507 y Fw(=\033)3203 4483 y Ft(2)3258 4544 y Fz(:)146 b FA(\(3.88\))118 4811 y Fp(3.3)112 b(Approac)m(hing)38 b(the)f(adiabatic)g(manifold)118 4983 y FA(The)f(follo)m(wing)f(result) h(giv)m(es)h(a)g(rather)g(rough)f(description)e(of)j(the)g(b)s(eha)m (viour)e(of)i(paths)g(starting)118 5096 y(at)f(a)h(\(su\016cien)m(tly)d (small\))h(distance)g(of)h(order)f(1)h(from)f(the)h(adiabatic)f (manifold.)55 b(It)35 b(is,)i(ho)m(w)m(ev)m(er,)118 5209 y(su\016cien)m(t)26 b(to)i(sho)m(w)e(that)i(with)d(large)i(probabilit)m (y)-8 b(,)26 b(these)h(paths)f(will)e(reac)m(h)k(the)f(set)g Fx(B)s FA(\()p Fz(h)p FA(\),)h(for)f(some)118 5322 y Fz(h)f(>)f(\033)s FA(,)30 b(in)g(a)g(time)g(of)h(order)f Fz(")p Fx(j)p FA(log)17 b Fz(h)p Fx(j)p FA(.)1844 5871 y(31)p eop %%Page: 32 32 32 31 bop 118 280 a Fm(Prop)s(osition)43 b(3.11.)i Fy(L)-5 b(et)38 b Fz(t)f Fy(satisfy)i(the)f(hyp)-5 b(otheses)40 b(of)e(Cor)-5 b(ol)5 b(lary)40 b(3.5.)58 b(Then)38 b(ther)-5 b(e)38 b(exist)g(c)-5 b(on-)118 393 y(stants)40 b Fz(h)441 407 y Fs(0)481 393 y Fy(,)f Fz(\016)588 407 y Fs(0)628 393 y Fy(,)h Fz(c)735 407 y Fs(0)813 393 y Fy(and)f Fz(K)1072 407 y Fs(0)1150 393 y Fy(such)g(that,)i(for)e Fz(h)d Fn(6)g Fz(h)1982 407 y Fs(0)2022 393 y Fy(,)j Fz(\016)h Fn(6)35 b Fz(\016)2315 407 y Fs(0)2355 393 y Fy(,)40 b Fz(\015)h Fx(2)35 b FA(\(0)p Fz(;)15 b FA(1)p Fz(=)p FA(2\))42 b Fy(and)d FA(\001)f Fy(su\016ciently)118 506 y(smal)5 b(l,)666 771 y FA(sup)364 860 y Fw(\030)395 869 y Ft(0)450 860 y Fs(:)35 b Fv(h)p Fw(\030)563 869 y Ft(0)598 860 y Fw(;)p 634 808 43 3 v(X)5 b Fs(\()p Fw(y)743 869 y Ft(0)777 860 y Fs(\))804 841 y Fk(\000)p Ft(1)887 860 y Fw(\030)918 869 y Ft(0)953 860 y Fv(i)p Fo(6)p Fw(\016)1068 841 y Ft(2)1104 771 y Fr(P)1165 734 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1278 743 y Ft(0)1313 734 y Fw(;)p Fs(0\))1399 643 y Fq(\032)1583 771 y FA(sup)1468 848 y Fs(0)p Fo(6)p Fw(s)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)1749 859 y Fk(D)1797 874 y Ft(0)1910 630 y Fq(\012)1952 704 y Fz(\030)1992 718 y Fw(s)2029 704 y Fz(;)p 2093 631 55 4 v 15 w(X)i FA(\()p Fz(y)2231 718 y Fw(s)2269 704 y FA(\))2304 671 y Fv(\000)p Fs(1)2398 704 y Fz(\030)2438 718 y Fw(s)2475 630 y Fq(\013)p 1861 750 706 4 v 1861 837 a FA(\()p Fz(h)21 b FA(+)f Fz(c)2099 851 y Fs(0)2139 837 y Fz(\016)f FA(e)2238 811 y Fv(\000)p Fw(K)2353 820 y Ft(0)2387 811 y Fw(s=")2492 837 y FA(\))2527 811 y Fs(2)2602 771 y Fn(>)25 b FA(1)2743 643 y Fq(\033)3429 771 y FA(\(3.89\))1129 1050 y Fn(6)1225 922 y Fq(\030)1330 989 y Fz(t)p 1288 1030 119 4 v 1288 1113 a FA(\001)p Fz(")1416 922 y Fq(\031\024)1725 989 y FA(1)p 1527 1030 443 4 v 1527 1117 a(\(1)c Fx(\000)f FA(2)p Fz(\015)5 b FA(\))1851 1090 y Fw(n=)p Fs(2)1999 1050 y FA(+)2090 977 y Fq(\000)2132 1050 y FA(e)2172 1013 y Fw(n=)p Fs(4)2305 1050 y FA(+2)15 b(e)2477 1013 y Fw(m=)p Fs(4)2614 977 y Fq(\001)2671 1050 y FA(e)2711 1013 y Fv(\000)p Fw(\024h)2848 990 y Ft(2)2882 1013 y Fw(=\033)2959 990 y Ft(2)2999 922 y Fq(\025)3062 1050 y FA(e)3102 1013 y Fv(\000)p Fw(\024h)3239 990 y Ft(2)3273 1013 y Fw(=\033)3350 990 y Ft(2)3390 1050 y Fz(;)118 1307 y Fy(wher)-5 b(e)34 b Fz(\024)25 b FA(=)g Fz(\015)5 b FA([1)21 b Fx(\000)f(O)s FA(\()p Fz(h)p FA(\))i Fx(\000)d(O)s FA(\(\001\))i Fx(\000)f(O)s FA(\()p Fz(\016)s FA(\)])p Fy(.)118 1495 y Fg(Pr)n(oof:)45 b FA(W)-8 b(e)32 b(start)g(again)e(b)m(y)h(considering)e(an)i(in)m (terv)-5 b(al)30 b([)p Fz(s;)15 b(t)p FA(])31 b(with)f Fz(t)20 b Fx(\000)g Fz(s)26 b FA(=)g(\001)p Fz(")p FA(.)42 b(Let)32 b Fz(y)3266 1462 y Fs(det)3263 1519 y(0)3394 1495 y FA(=)26 b Fz(y)3536 1509 y Fs(0)3601 1495 y Fx(2)118 1608 y(D)188 1622 y Fs(0)228 1608 y FA(.)40 b(Then)875 1869 y Fz(P)e FA(=)25 b Fr(P)1128 1831 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1241 1840 y Ft(0)1276 1831 y Fw(;)p Fs(0\))1363 1740 y Fq(\032)1608 1869 y FA(sup)1431 1945 y Fw(s)p Fo(6)p Fw(u)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)1718 1956 y Fk(D)1766 1971 y Ft(0)1805 1945 y Fv(^)p Fw(\034)1883 1953 y Fl(\021)1987 1728 y Fq(\012)2030 1801 y Fz(\030)2070 1815 y Fw(u)2114 1801 y Fz(;)p 2178 1728 55 4 v 15 w(X)7 b FA(\()p Fz(y)2316 1815 y Fw(u)2362 1801 y FA(\))2397 1768 y Fv(\000)p Fs(1)2492 1801 y Fz(\030)2532 1815 y Fw(u)2577 1728 y Fq(\013)p 1946 1848 714 4 v 1946 1935 a FA(\()p Fz(h)21 b FA(+)f Fz(c)2184 1949 y Fs(0)2224 1935 y Fz(\016)f FA(e)2323 1908 y Fv(\000)p Fw(K)2438 1917 y Ft(0)2472 1908 y Fw(u=")2585 1935 y FA(\))2620 1909 y Fs(2)2695 1869 y Fn(>)25 b FA(1)2836 1740 y Fq(\033)971 2146 y Fn(6)g Fr(P)1128 2108 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)1241 2117 y Ft(0)1276 2108 y Fw(;)p Fs(0\))1363 2018 y Fq(\032)1510 2146 y FA(sup)1431 2223 y Fw(s)p Fo(6)p Fw(u)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)1741 2072 y Fq(\012)1784 2146 y Fz(\030)1824 2160 y Fw(u)1869 2146 y Fz(;)p 1932 2073 55 4 v 15 w(X)7 b FA(\()p Fz(y)2074 2108 y Fs(det)2071 2168 y Fw(u)2177 2146 y FA(\))2212 2108 y Fv(\000)p Fs(1)2307 2146 y Fz(\030)2347 2160 y Fw(u)2391 2072 y Fq(\013)2460 2146 y Fn(>)24 b Fz(H)2638 2108 y Fs(2)2678 2018 y Fq(\033)2746 2146 y Fz(;)658 b FA(\(3.90\))118 2405 y(where)30 b Fz(\034)40 b FA(is)30 b(a)g(stopping)g(time)g(de\014ned)f(b)m(y)236 2609 y Fz(\034)35 b FA(=)25 b Fz(\034)447 2623 y Fv(D)502 2632 y Ft(0)560 2609 y Fx(^)20 b Fz(\034)681 2623 y Fw(\021)743 2609 y Fx(^)g FA(inf)934 2536 y Fq(\010)987 2609 y Fz(u)26 b Fn(>)f FA(0)10 b(:)31 b Fx(h)p Fz(\030)1347 2623 y Fw(u)1392 2609 y Fz(;)p 1455 2536 V 15 w(X)7 b FA(\()p Fz(y)1597 2572 y Fs(det)1594 2632 y Fw(u)1700 2609 y FA(\))1735 2572 y Fv(\000)p Fs(1)1830 2609 y Fz(\030)1870 2623 y Fw(u)1914 2609 y Fx(i)26 b Fn(>)f FA(\()p Fz(h)c FA(+)f Fz(c)2309 2623 y 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Fw(t=")p 1788 3403 1137 4 v 2309 3486 a Fz(\033)2364 3460 y Fs(2)2935 3323 y Fq(i)2977 3424 y Fz(:)427 b FA(\(3.92\))118 3676 y(F)-8 b(ollo)m(wing)37 b(the)g(pro)s(of)f(of)i(Lemma)f(3.3,)j(and)c(taking)h(in)m(to)g(accoun) m(t)i(the)e(new)g(de\014nition)e(of)i Fz(\034)10 b FA(,)39 b(w)m(e)118 3789 y(further)29 b(obtain)h(that)436 4026 y Fz(P)494 4040 y Fs(1)559 4026 y Fn(6)25 b FA(e)696 3988 y Fw(n=)p Fs(4)828 4026 y FA(exp)967 3925 y Fq(n)1028 4026 y Fx(\000)1109 3964 y Fz(H)1192 3931 y Fs(2)1185 3988 y(1)p 1109 4005 123 4 v 1122 4088 a Fz(\033)1177 4062 y Fs(2)2110 3964 y FA(1)p 1251 4005 1765 4 v 1251 4092 a Fz(M)1349 4060 y Fs(2)1339 4118 y(1)1388 4092 y Fy(c)-5 b(onst)34 b FA([\()p Fz(h)21 b FA(+)f Fz(h)1899 4106 y Fs(1)1939 4092 y FA(\))1974 4065 y Fs(2)2014 4092 y FA(\011\()p Fz(t)p FA(\))g(+)g Fz(c)2338 4060 y Fs(2)2338 4118 y(0)2378 4092 y Fz(\016)2421 4065 y Fs(2)2461 4092 y FA(\()p Fz(t=")p FA(\))15 b(e)2708 4065 y Fv(\000)p Fs(2)p Fw(K)2858 4074 y Ft(0)2892 4065 y Fw(t=")2990 4092 y FA(])3025 3925 y Fq(o)3086 4026 y Fz(:)318 b FA(\(3.93\))118 4281 y(As)30 b(for)h Fz(P)450 4295 y Fs(2)489 4281 y FA(,)g(it)f(can)h(b)s(e)e(estimated)i(trivially)-8 b(,)28 b(pro)m(vided)841 4521 y Fz(H)917 4535 y Fs(2)981 4521 y Fn(>)d Fy(c)-5 b(onst)1331 4460 y Fz(M)p 1321 4501 117 4 v 1321 4584 a(K)1398 4598 y Fs(0)1448 4421 y Fq(h)1491 4521 y FA(\()p Fz(h)1578 4484 y Fs(2)1638 4521 y FA(+)20 b Fz(hh)1833 4535 y Fs(1)1873 4521 y FA(\)\002\()p Fz(t)p FA(\))h(+)f Fz(c)2233 4484 y Fs(2)2233 4544 y(0)2273 4521 y Fz(\016)2316 4484 y Fs(2)2371 4521 y FA(e)2412 4484 y Fw(K)2472 4493 y Ft(0)2506 4484 y Fs(\001)2584 4521 y FA(e)2624 4484 y Fv(\000)p Fw(K)2739 4493 y Ft(0)2773 4484 y Fw(t=")2871 4421 y Fq(i)2914 4521 y Fz(:)490 b FA(\(3.94\))118 4773 y(Cho)s(osing)29 b Fz(H)591 4787 y Fs(1)660 4773 y FA(in)g(suc)m(h)h(a)h(w)m(a)m(y)h(that)f(the)f(exp)s (onen)m(t)h(in)e(\(3.93\))j(equals)e Fz(H)2704 4740 y Fs(2)2743 4773 y Fz(=\033)2843 4740 y Fs(2)2883 4773 y FA(,)h(w)m(e)g(obtain)164 5033 y Fz(P)38 b Fn(6)356 4904 y Fq(\022)631 4971 y FA(1)p 433 5012 443 4 v 433 5099 a(\(1)21 b Fx(\000)f FA(2)p Fz(\015)5 b FA(\))757 5072 y Fw(n=)p Fs(2)903 5033 y FA(+)19 b(e)1033 4995 y Fw(n=)p Fs(4)1166 5033 y FA(e)1206 4995 y Fv(\000)p Fw(H)1324 4972 y Ft(2)1359 4995 y Fw(=)p Fs(\(2)p Fw(\033)1498 4972 y Ft(2)1534 4995 y Fs(\))1566 4904 y Fq(\023)1648 5033 y FA(exp)1787 4932 y Fq(n)1847 5033 y Fx(\000)p Fz(\015)1980 4971 y(H)2063 4938 y Fs(2)p 1980 5012 123 4 v 1994 5095 a Fz(\033)2049 5069 y Fs(2)2112 4959 y Fq(\002)2150 5033 y FA(1)g Fx(\000)f(O)s FA(\(\001\))h Fx(\000)f(O)s FA(\()p Fz(h)h FA(+)f Fz(h)2954 5047 y Fs(1)3013 5033 y FA(+)g Fz(c)3141 5047 y Fs(0)3181 5033 y Fz(\016)s FA(\))3259 4959 y Fq(\003)3298 4932 y(o)3358 5033 y Fz(;)46 b FA(\(3.95\))118 5304 y(where)34 b(w)m(e)h(c)m(ho)s (ose)g Fz(h)867 5318 y Fs(1)941 5304 y FA(prop)s(ortional)d(to)j Fz(h)23 b FA(+)g Fz(c)1791 5318 y Fs(0)1831 5304 y Fz(\016)c FA(e)1930 5271 y Fw(K)1990 5280 y Ft(0)2024 5271 y Fs(\001)2102 5304 y FA(e)2143 5271 y Fv(\000)p Fw(K)2258 5280 y Ft(0)2292 5271 y Fw(t=")2389 5304 y FA(.)52 b(The)34 b(remainder)f(of)h(the)h (pro)s(of)e(is)118 5417 y(similar)28 b(to)j(the)g(pro)s(ofs)e(of)i (Lemma)f(3.4)i(and)d(Corollary)g(3.5.)p 3595 5417 4 62 v 3599 5359 55 4 v 3599 5417 V 3653 5417 4 62 v 1844 5871 a(32)p eop %%Page: 33 33 33 32 bop 259 280 a FA(The)31 b(preceding)f(lemma)h(sho)m(ws)g(that)h (after)g(a)g(time)f Fz(t)2145 294 y Fs(1)2215 280 y FA(of)g(order)g Fz(")p Fx(j)p FA(log)17 b Fz(h)p Fx(j)p FA(,)33 b(the)e(paths)g(are)h (lik)m(ely)118 393 y(to)k(ha)m(v)m(e)g(reac)m(hed)g Fx(B)s FA(\()p Fz(h)p FA(\).)55 b(As)35 b(in)e(Lemma)i(3.7,)j(an)d(impro)m(v)m (ed)f(b)s(ound)f(for)i(the)g(distribution)c(of)k(the)118 506 y(endp)s(oin)m(t)c Fz(\030)539 520 y Fw(t)564 529 y Ft(1)636 506 y FA(can)i(b)s(e)f(obtained.)47 b(Rep)s(eating)32 b(the)h(argumen)m(ts)g(leading)f(to)h(Theorem)f(2.4,)j(namely)118 619 y(using)30 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4545 y Fq(\000)1962 4618 y FA(\001)g(+)g Fz(")h FA(+)f Fz(\026h)g FA(+)g Fz(h)p Fx(k)p 2642 4545 45 4 v(Z)7 b(k)2736 4637 y Fs([0)p Fw(;t)p Fs(])2861 4618 y Fx(k)p Fz(Y)2980 4581 y Fv(\000)p Fs(1)3074 4618 y Fx(k)3119 4570 y Fs(1)p Fw(=)p Fs(2)3119 4651 y([0)p Fw(;t)p Fs(])3243 4618 y Fz(\037)3300 4581 y Fs(\(1\))3394 4618 y FA(\()p Fz(t)p FA(\))3497 4545 y Fq(\001)3539 4517 y(i)3582 4490 y(\033)338 4878 y FA(+)20 b(e)475 4823 y Ff(b)469 4840 y Fs(\010\()p Fw(t)p Fs(\))p Fw(=)p Fs(4)677 4823 y Ff(b)669 4840 y Fs(\011)r(\()p Fw(t)p Fs(\))826 4878 y FA(exp)980 4750 y Fq(\032)1048 4878 y Fx(\000)1130 4816 y Fz(h)1182 4783 y Fs(2)p 1129 4857 95 4 v 1129 4940 a Fz(\033)1184 4914 y Fs(2)1317 4816 y Fz(\026)1372 4783 y Fs(2)1411 4816 y FA(\(1)h Fx(\000)f(O)s FA(\(\001\)\))p 1243 4857 691 4 v 1243 4956 a(16)p Fz(M)1431 4925 y Fs(2)1421 4982 y(1)1472 4956 y Fx(k)p 1541 4883 45 4 v(Z)7 b(k)1635 4975 y Fs([0)p Fw(;t)p Fs(])1770 4933 y Fq(b)1760 4956 y FA(\011)o(\()p Fz(t)p FA(\))1944 4750 y Fq(\033)3429 4878 y FA(\(4.11\))118 5143 y Fy(holds)34 b(for)g(al)5 b(l)33 b Fz(h)25 b(<)g FA(1)p Fz(=\026)p Fy(.)118 5318 y Fg(Pr)n(oof:)45 b FA(W)-8 b(riting)30 b Fz(\020)851 5332 y Fw(u)921 5318 y FA(=)25 b Fx(U)9 b FA(\()p Fz(u;)15 b(s)p FA(\)\007)1359 5332 y Fw(u)1404 5318 y FA(,)31 b(w)m(e)f(ha)m(v)m(e)264 5556 y Fr(P)325 5518 y Fs(0)p Fw(;\020)411 5527 y Ft(0)450 5428 y Fq(\032)637 5556 y FA(sup)518 5633 y Fw(s)p Fo(6)p Fw(u)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)805 5644 y Fk(D)853 5659 y Ft(0)907 5482 y Fq(\012)950 5556 y Fz(\020)990 5570 y Fw(u)1035 5556 y Fz(;)p 1099 5483 V 15 w Fx(Z)7 b FA(\()p Fz(u)p FA(\))1270 5518 y Fv(\000)p Fs(1)1366 5556 y Fz(\020)1406 5570 y Fw(u)1450 5482 y Fq(\013)1519 5556 y Fn(>)24 b Fz(h)1666 5518 y Fs(2)1706 5428 y Fq(\033)1800 5556 y FA(=)h Fr(P)1957 5518 y Fs(0)p Fw(;\020)2043 5527 y Ft(0)2082 5428 y Fq(\032)2325 5556 y FA(sup)2150 5633 y Fw(s)p Fo(6)p Fw(u)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)2437 5644 y Fk(D)2485 5659 y Ft(0)2524 5633 y Fv(^)p Fw(\034)2602 5645 y Fl(\020)2638 5556 y Fx(kQ)p FA(\()p Fz(u)p FA(\)\007)2950 5570 y Fw(u)2996 5556 y Fx(k)h Fn(>)f Fz(h)3215 5428 y Fq(\033)3429 5556 y FA(\(4.12\))1844 5871 y(34)p eop %%Page: 35 35 35 34 bop 118 280 a FA(where)30 b Fx(Q)p FA(\()p Fz(u)p FA(\))h(is)f(the)g(symmetric)g(matrix)g(de\014ned)f(b)m(y)1254 477 y Fx(Q)p FA(\()p Fz(u)p FA(\))1450 440 y Fs(2)1516 477 y FA(=)c Fx(U)9 b FA(\()p Fz(u;)15 b(s)p FA(\))1883 440 y Fw(T)p 1962 404 45 4 v 1938 477 a Fx(Z)7 b FA(\()p Fz(u)p FA(\))2133 440 y Fv(\000)p Fs(1)2229 477 y Fx(U)i FA(\()p Fz(u;)15 b(s)p FA(\))p Fz(:)904 b FA(\(4.13\))118 674 y(As)30 b(in)e(the)h(pro)s(of)g(of)g(Prop)s(osition)f(3.1,)j(w)m(e) f(w)m(an)m(t)g(to)g(eliminate)e(the)i Fz(u)p FA(-dep)s(endence)f(of)g Fx(Q)h FA(in)e(\(4.12\))r(.)118 787 y(It)36 b(turns)f(out)i(that)g(the) f(relation)f Fx(kQ)p FA(\()p Fz(u)p FA(\))p Fx(Q)p FA(\()p Fz(t)p FA(\))1756 754 y Fv(\000)p Fs(1)1853 787 y Fx(k)g FA(=)g(1)25 b(+)e Fx(O)s FA(\(\001\))37 b(still)d(holds)h(in)g(the)h (presen)m(t)g(situ-)118 900 y(ation,)f(although)f(the)g(pro)s(of)f(is)g (less)g(straigh)m(tforw)m(ard)h(than)g(b)s(efore.)51 b(W)-8 b(e)35 b(establish)e(this)g(result)g(in)118 1013 y(Lemma)e(4.2)g(b)s(elo)m(w.)259 1126 y(Splitting)j(\007)710 1140 y Fw(u)792 1126 y FA(in)m(to)i(the)h(sum)f(\007)1415 1140 y Fw(u)1495 1126 y FA(=)g(\007)1673 1093 y Fs(0)1673 1148 y Fw(u)1742 1126 y FA(+)24 b(\007)1908 1093 y Fs(1)1908 1148 y Fw(u)1977 1126 y FA(+)g(\007)2143 1093 y Fs(2)2143 1148 y Fw(u)2188 1126 y FA(,)38 b(where)e(the)h(\007)2754 1093 y Fw(i)2754 1148 y(u)2836 1126 y FA(are)g(de\014ned)e(in)g(a)i(w)m (a)m(y)118 1238 y(analogous)d(to)h(\(3.16\))r(,)g(w)m(e)f(can)g (estimate)h(the)e(probabilit)m(y)f(in)g(\(4.12\))k(b)m(y)e(the)g(sum)f Fz(P)3170 1252 y Fs(0)3232 1238 y FA(+)22 b Fz(P)3383 1252 y Fs(1)3445 1238 y FA(+)g Fz(P)3596 1252 y Fs(2)3636 1238 y FA(,)118 1351 y(where)1052 1575 y Fz(P)1110 1589 y Fs(0)1175 1575 y FA(=)j Fr(P)1332 1538 y Fs(0)p Fw(;\020)1418 1547 y Ft(0)1457 1447 y Fq(\032)1644 1575 y FA(sup)1526 1652 y Fw(s)p Fo(6)p Fw(u)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)1813 1663 y Fk(D)1861 1678 y Ft(0)1899 1575 y Fx(kQ)p FA(\()p Fz(t)p FA(\)\007)2192 1538 y Fs(0)2192 1598 y Fw(u)2238 1575 y Fx(k)h Fn(>)f Fz(H)2481 1589 y Fs(0)2520 1447 y Fq(\033)2588 1575 y Fz(;)1052 1852 y(P)1110 1866 y Fs(1)1175 1852 y FA(=)g Fr(P)1332 1815 y Fs(0)p Fw(;\020)1418 1824 y Ft(0)1457 1724 y Fq(\032)1701 1852 y FA(sup)1526 1929 y Fw(s)p Fo(6)p Fw(u)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)1813 1940 y Fk(D)1861 1955 y Ft(0)1899 1929 y Fv(^)p Fw(\034)1977 1941 y Fl(\020)2013 1852 y Fx(kQ)p FA(\()p Fz(t)p FA(\)\007)2306 1815 y Fs(1)2306 1875 y Fw(u)2352 1852 y Fx(k)h Fn(>)f Fz(H)2595 1866 y Fs(1)2634 1724 y Fq(\033)2702 1852 y Fz(;)702 b FA(\(4.14\))1052 2129 y Fz(P)1110 2143 y Fs(2)1175 2129 y FA(=)25 b Fr(P)1332 2092 y Fs(0)p Fw(;\020)1418 2101 y Ft(0)1457 2001 y Fq(\032)1701 2129 y FA(sup)1526 2206 y Fw(s)p Fo(6)p Fw(u)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)1813 2217 y Fk(D)1861 2232 y Ft(0)1899 2206 y Fv(^)p Fw(\034)1977 2218 y Fl(\020)2013 2129 y Fx(kQ)p FA(\()p Fz(t)p FA(\)\007)2306 2092 y Fs(2)2306 2152 y Fw(u)2352 2129 y Fx(k)h Fn(>)f Fz(H)2595 2143 y Fs(2)2634 2001 y Fq(\033)2702 2129 y Fz(;)118 2402 y FA(and)h Fz(H)367 2416 y Fs(0)418 2402 y FA(+)12 b Fz(H)577 2416 y Fs(1)629 2402 y FA(+)g Fz(H)788 2416 y Fs(2)852 2402 y FA(=)25 b Fz(h)p FA(\(1)12 b Fx(\000)g(O)s FA(\(\001\)\).)42 b(F)-8 b(ollo)m(wing)26 b(the)h(pro)s(of)e(of)i (Lemma)f(3.2,)j(it)d(is)g(straigh)m(tforw)m(ard)118 2515 y(to)31 b(sho)m(w)f(that)766 2727 y Fz(P)824 2741 y Fs(0)889 2727 y Fn(6)1279 2666 y FA(1)p 994 2706 615 4 v 994 2793 a(\(1)21 b Fx(\000)f FA(2)p Fz(\015)5 b FA(\))1318 2767 y Fs(\()p Fw(n)p Fs(+)p Fw(m)p Fs(\))p Fw(=)p Fs(2)1634 2727 y FA(exp)1773 2626 y Fq(n)1834 2727 y Fx(\000)1936 2666 y Fz(\015)p 1915 2706 95 4 v 1915 2789 a(\033)1970 2763 y Fs(2)2019 2727 y FA(\()p Fz(H)2137 2689 y Fs(2)2130 2749 y(0)2197 2727 y Fx(\000)20 b Fz(\013)2346 2689 y Fs(2)2386 2727 y Fz(h)2438 2689 y Fs(2)2478 2727 y FA(\)\(1)h Fx(\000)f(O)s FA(\()p Fz(")p FA(\)\))2927 2626 y Fq(o)2989 2727 y Fz(;)415 b FA(\(4.15\))118 2976 y(the)33 b(sole)g(di\013erence)f (b)s(eing)g(the)h(factor)h Fx(O)s FA(\()p Fz(")p FA(\))f(in)f(the)h (exp)s(onen)m(t)g(whic)m(h)e(stems)j(from)e(the)h(fact)h(that)118 3089 y Fx(h)p Fz(\020)193 3103 y Fs(0)233 3089 y Fz(;)p 296 3016 45 4 v 15 w Fx(Z)7 b FA(\(0\))461 3056 y Fv(\000)p Fs(1)557 3089 y Fz(\020)597 3103 y Fs(0)636 3089 y Fx(i)25 b FA(=)g Fx(h)p Fz(\030)867 3103 y Fs(0)907 3089 y Fz(;)p 971 3016 55 4 v 15 w(X)7 b FA(\(0\))1144 3056 y Fv(\000)p Fs(1)1240 3089 y Fz(\030)1280 3103 y Fs(0)1320 3089 y Fx(i)p FA(\(1)j(+)g Fx(O)s FA(\()p Fz(")p FA(\)\).)41 b(F)-8 b(urthermore,)27 b(similar)c(argumen)m(ts)j(as)f(in)f(the)i(pro) s(of)118 3202 y(of)31 b(Lemma)f(3.3)i(lead)e(to)h(the)f(b)s(ound)1000 3453 y Fz(P)1058 3467 y Fs(1)1123 3453 y Fn(6)25 b FA(exp)1358 3325 y Fq(\032)1426 3453 y Fx(\000)1507 3383 y FA(\()p Fz(H)1625 3350 y Fs(2)1618 3407 y(1)1684 3383 y Fx(\000)20 b FA(2)p Fz(\033)1875 3350 y Fs(2)1915 3383 y Fz(M)2013 3350 y Fs(2)2003 3407 y(1)2053 3383 y Fz(h)2105 3350 y Fs(2)2145 3383 y Fx(k)p 2213 3310 45 4 v(Z)7 b(k)2308 3401 y Fs([0)p Fw(;t)p Fs(])2440 3360 y Fq(b)2432 3383 y FA(\010\()p Fz(t)p FA(\)\))2636 3350 y Fs(2)p 1507 3433 1171 4 v 1592 3532 a FA(16)p Fz(\033)1737 3506 y Fs(2)1777 3532 y Fz(M)1875 3501 y Fs(2)1865 3558 y(1)1915 3532 y Fz(h)1967 3506 y Fs(2)2007 3532 y Fz(H)2090 3501 y Fs(2)2083 3558 y(1)2129 3532 y Fx(k)p 2198 3459 45 4 v(Z)g(k)2292 3551 y Fs([0)p Fw(;t)p Fs(])2427 3509 y Fq(b)2417 3532 y FA(\011\()p Fz(t)p FA(\))2686 3325 y Fq(\033)2755 3453 y Fz(:)649 b FA(\(4.16\))118 3722 y(Finally)-8 b(,)29 b(the)i(estimate)269 3967 y Fx(kQ)p FA(\()p Fz(t)p FA(\)\007)562 3929 y Fs(2)562 3989 y Fw(u)p Fv(^)p Fw(\034)681 4001 y Fl(\020)722 3967 y Fx(k)767 3929 y Fs(2)832 3967 y Fn(6)928 3843 y Fq(Z)1019 3869 y Fw(u)p Fv(^)p Fw(\034)1138 3881 y Fl(\020)979 4049 y Fs(0)1193 3843 y Fq(Z)1284 3869 y Fw(u)p Fv(^)p Fw(\034)1403 3881 y Fl(\020)1244 4049 y Fs(0)1443 3889 y Fq(\015)1443 3944 y(\015)1494 3967 y Fx(B)s FA(\()p Fz(\020)1632 3981 y Fw(v)1672 3967 y Fz(;)15 b(v)s(;)g(")p FA(\))1876 3929 y Fw(T)1933 3967 y Fx(U)9 b FA(\()p Fz(t;)15 b(v)s FA(\))2189 3929 y Fw(T)p 2268 3894 V 2245 3967 a Fx(Z)7 b FA(\()p Fz(t)p FA(\))2421 3929 y Fv(\000)p Fs(1)2516 3967 y Fx(U)i FA(\()p Fz(t;)15 b(w)r FA(\))p Fx(B)s FA(\()p Fz(\020)2930 3981 y Fw(w)2988 3967 y Fz(;)g(w)r(;)g(")p FA(\))3212 3889 y Fq(\015)3212 3944 y(\015)3279 3967 y FA(d)p Fz(v)j FA(d)p Fz(w)832 4170 y Fn(6)25 b Fy(c)-5 b(onst)34 b Fz(h)1215 4132 y Fs(4)1255 4170 y Fx(k)p 1323 4097 V(Z)7 b(k)1418 4132 y Fs(2)1418 4196 y([0)p Fw(;t)p Fs(])1542 4096 y Fq(\000)1584 4170 y FA(1)21 b(+)f Fx(k)p Fz(Y)1859 4132 y Fv(\000)p Fs(1)1953 4170 y Fx(k)1998 4188 y Fs([0)p Fw(;t)p Fs(])2122 4170 y Fz(\037)2179 4132 y Fs(\(1\))2274 4170 y FA(\()p Fz(u)p FA(\))2396 4132 y Fs(2)2436 4096 y Fq(\001)3429 4170 y FA(\(4.17\))118 4403 y(sho)m(ws)28 b(that)g Fz(P)629 4417 y Fs(2)694 4403 y FA(=)d(0)k(for)f Fz(H)1077 4417 y Fs(2)1141 4403 y Fn(>)d Fx(O)s FA(\()p Fz(h)1399 4370 y Fs(2)1439 4403 y Fx(k)p 1508 4330 V(Z)7 b(k)1602 4421 y Fs([0)p Fw(;t)p Fs(])1727 4403 y Fx(k)p Fz(Y)1845 4370 y Fv(\000)p Fs(1)1939 4403 y Fx(k)1984 4355 y Fs(1)p Fw(=)p Fs(2)1984 4436 y([0)p Fw(;t)p Fs(])2108 4403 y Fz(\037)2165 4370 y Fs(\(1\))2260 4403 y FA(\()p Fz(t)p FA(\)\).)40 b(Hence)29 b(\(4.11\))h(follo)m(ws)e(b)m(y)f(taking) 118 4537 y Fz(H)194 4551 y Fs(1)258 4537 y FA(=)e Fz(\026h)461 4504 y Fs(2)501 4537 y FA(\(1)c Fx(\000)f(O)s FA(\(\001\)\).)p 3595 4537 4 62 v 3599 4479 55 4 v 3599 4537 V 3653 4537 4 62 v 259 4723 a(In)30 b(the)g(pro)s(of)g(of)h(Prop)s(osition)d(4.1,)k (w)m(e)f(ha)m(v)m(e)g(used)f(the)g(follo)m(wing)f(estimate.)118 4904 y Fm(Lemma)k(4.2.)42 b Fy(F)-7 b(or)34 b FA(\001)25 b(=)g(\()p Fz(t)20 b Fx(\000)g Fz(s)p FA(\))p Fz(=")33 b Fy(su\016ciently)g(smal)5 b(l,)1279 5101 y FA(sup)1243 5178 y Fw(s)p Fo(6)p Fw(u)p Fo(6)p Fw(t)1452 5101 y Fx(kQ)p FA(\()p Fz(u)p FA(\))p Fx(Q)p FA(\()p Fz(t)p FA(\))1870 5064 y Fv(\000)p Fs(1)1966 5101 y Fx(k)26 b FA(=)f(1)c(+)f Fx(O)s FA(\(\001\))p Fz(:)893 b FA(\(4.18\))118 5366 y Fg(Pr)n(oof:)45 b FA(Using)30 b(the)g(fact)h(that)g Fx(Q)p FA(\()p Fz(v)s FA(\))1456 5333 y Fv(\000)p Fs(2)1582 5366 y FA(satis\014es)f(the)h(ODE)925 5544 y(d)p 894 5585 98 4 v 894 5668 a(d)p Fz(v)1002 5605 y Fx(Q)p FA(\()p Fz(v)s FA(\))1193 5568 y Fv(\000)p Fs(2)1314 5605 y FA(=)25 b Fx(U)9 b FA(\()p Fz(s;)15 b(v)s FA(\))p Fx(F)1741 5619 y Fs(0)1781 5605 y FA(\()p Fz(y)1864 5568 y Fs(det)1861 5628 y Fw(v)1967 5605 y Fz(;)g(")p FA(\))p Fx(F)2149 5619 y Fs(0)2190 5605 y FA(\()p Fz(y)2273 5568 y Fs(det)2270 5628 y Fw(v)2375 5605 y Fz(;)g(")p FA(\))2492 5568 y Fw(T)2548 5605 y Fx(U)9 b FA(\()p Fz(s;)15 b(v)s FA(\))2814 5568 y Fw(T)2870 5605 y Fz(;)534 b FA(\(4.19\))1844 5871 y(35)p eop %%Page: 36 36 36 35 bop 118 280 a FA(w)m(e)31 b(obtain)f(the)g(relation)385 518 y Fx(Q)p FA(\()p Fz(u)p FA(\))581 481 y Fs(2)621 518 y Fx(Q)p FA(\()p Fz(t)p FA(\))798 481 y Fv(\000)p Fs(2)919 518 y FA(=)25 b(1)-20 b(l)19 b(+)h Fx(Q)p FA(\()p Fz(u)p FA(\))1372 481 y Fs(2)1428 395 y Fq(Z)1519 421 y Fw(t)1478 601 y(u)1564 518 y Fx(U)9 b FA(\()p Fz(s;)15 b(v)s FA(\))p Fx(F)1895 532 y Fs(0)1935 518 y FA(\()p Fz(y)2018 481 y Fs(det)2015 541 y Fw(v)2121 518 y Fz(;)g(")p FA(\))p Fx(F)2303 532 y Fs(0)2344 518 y FA(\()p Fz(y)2427 481 y Fs(det)2424 541 y Fw(v)2529 518 y Fz(;)g(")p FA(\))2646 481 y Fw(T)2702 518 y Fx(U)9 b FA(\()p Fz(s;)15 b(v)s FA(\))2968 481 y Fw(T)3039 518 y FA(d)p Fz(v)s(:)267 b FA(\(4.20\))118 769 y(The)30 b(de\014nition)e(of)j Fx(F)878 783 y Fs(0)948 769 y FA(and)e(the)i(b)s(ound)d(\(4.4\))k(on)f Fx(k)p Fz(S)5 b Fx(k)31 b FA(allo)m(w)f(us)f(to)j(write)306 1053 y Fx(U)9 b FA(\()p Fz(s;)15 b(v)s FA(\))p Fx(F)637 1067 y Fs(0)678 1053 y FA(\()p Fz(y)761 1016 y Fs(det)758 1076 y Fw(v)863 1053 y Fz(;)g(")p FA(\))p Fx(F)1045 1067 y Fs(0)1086 1053 y FA(\()p Fz(y)1169 1016 y Fs(det)1166 1076 y Fw(v)1271 1053 y Fz(;)g(")p FA(\))1388 1016 y Fw(T)1445 1053 y Fx(U)9 b FA(\()p Fz(s;)15 b(v)s FA(\))1711 1016 y Fw(T)1792 1053 y FA(=)1888 898 y Fq( )2092 970 y Fx(O)s FA(\(1)p Fz(=")p FA(\))217 b Fx(O)s FA(\(\001)20 b(+)g Fz(\032=)2975 904 y Fx(p)p 3051 904 43 4 v 66 x Fz(")q FA(\))1960 1136 y Fx(O)s FA(\(\001)g(+)g Fz(\032=)2349 1071 y Fx(p)p 2425 1071 V 65 x Fz(")p FA(\))105 b Fx(O)s FA(\(\001)2793 1103 y Fs(2)2832 1136 y Fz(")21 b FA(+)f Fz(\032)3033 1103 y Fs(2)3072 1136 y FA(\))3129 898 y Fq(!)3216 1053 y Fz(:)188 b FA(\(4.21\))118 1350 y(Using)29 b(the)h(estimate)g(\(4.7\))h(for)p 1270 1277 45 4 v 30 w Fx(Z)1320 1317 y Fv(\000)p Fs(1)1444 1350 y FA(and)e(the)g(fact)i (that)f(w)m(e)g(in)m(tegrate)h(o)m(v)m(er)g(an)e(in)m(terv)-5 b(al)29 b(of)h(length)118 1463 y(\001)p Fz(")p FA(,)h(it)f(follo)m(ws)f (that)949 1742 y Fx(Q)p FA(\()p Fz(u)p FA(\))1145 1705 y Fs(2)1186 1742 y Fx(Q)p FA(\()p Fz(t)p FA(\))1363 1705 y Fv(\000)p Fs(2)1478 1742 y Fx(\000)20 b FA(1)-20 b(l)25 b(=)g(\001)1832 1587 y Fq( )1904 1659 y Fx(O)s FA(\(1\))84 b Fx(O)s FA(\(\001)p Fz(")20 b FA(+)g Fz(\032)2564 1594 y Fx(p)p 2640 1594 43 4 v 65 x Fz(")p FA(\))1904 1825 y Fx(O)s FA(\(1\))122 b Fx(O)s FA(\()p Fz(")20 b FA(+)g Fz(\032)2526 1760 y Fx(p)p 2602 1760 V 65 x Fz(")q FA(\))2718 1587 y Fq(!)2805 1742 y Fz(;)599 b FA(\(4.22\))118 2027 y(whic)m(h)29 b(implies)f(\(4.18\))r(.)p 3595 2027 4 62 v 3599 1969 55 4 v 3599 2027 V 3653 2027 4 62 v 259 2214 a(No)m(w,)34 b(Theorem)d(2.5)i(follo)m(ws)f(from)f(Prop)s(osition) f(4.1,)k(b)m(y)e(taking)g(a)g(regular)f(partition)g(of)h([0)p Fz(;)15 b(t)p FA(])118 2327 y(with)32 b(spacing)h(\001)p Fz(")h FA(and)e Fz(\026)f FA(=)f(4)p Fz(M)1306 2341 y Fs(1)1346 2327 y Fx(k)p 1414 2254 45 4 v(Z)7 b(k)1509 2279 y Fs(1)p Fw(=)p Fs(2)1509 2360 y([0)p Fw(;t)p Fs(])1643 2304 y Fq(b)1633 2327 y FA(\011\()p Fz(t)p FA(\))1807 2294 y Fs(1)p Fw(=)p Fs(2)1918 2327 y FA(.)49 b(W)-8 b(e)35 b(use)e(in)f(particular)g(the)i(fact)g(that)3396 2304 y Fq(b)3386 2327 y FA(\011\()p Fz(t)p FA(\))d(=)118 2440 y Fx(O)s FA(\(1)21 b(+)f Fz(\037)442 2407 y Fs(\(1\))536 2440 y FA(\()p Fz(t)p FA(\))h(+)f Fz(\037)808 2407 y Fs(\(2\))902 2440 y FA(\()p Fz(t)p FA(\)\).)118 2727 y FB(5)135 b(Pro)t(ofs)45 b({)g(Bifurcations)118 2930 y FA(W)-8 b(e)32 b(consider)e(in)f(this)h(section)h(the)g(b)s(eha)m (viour)f(of)h(the)g(SDE)f(\(2.1\))j(near)e(a)g(bifurcation)e(p)s(oin)m (t.)41 b(The)118 3043 y(system)31 b(can)f(b)s(e)g(written)f(in)h(the)g (form)838 3280 y(d)o Fz(\030)932 3242 y Fv(\000)928 3304 y Fw(t)1016 3280 y FA(=)1122 3218 y(1)p 1122 3259 46 4 v 1124 3342 a Fz(")1197 3256 y FA(^)1178 3280 y Fz(f)1233 3242 y Fv(\000)1291 3280 y FA(\()p Fz(\030)1370 3242 y Fv(\000)1366 3304 y Fw(t)1429 3280 y Fz(;)15 b(z)1511 3294 y Fw(t)1542 3280 y Fz(;)g(y)1627 3294 y Fw(t)1656 3280 y Fz(;)g(")p FA(\))g(d)q Fz(t)20 b FA(+)2025 3218 y Fz(\033)p 1994 3259 119 4 v 1994 3277 a Fx(p)p 2070 3277 43 4 v 66 x Fz(")2140 3257 y Fq(b)2122 3280 y Fz(F)2193 3242 y Fv(\000)2252 3280 y FA(\()p Fz(\030)2331 3242 y Fv(\000)2327 3304 y Fw(t)2391 3280 y Fz(;)15 b(z)2473 3294 y Fw(t)2503 3280 y Fz(;)g(y)2588 3294 y Fw(t)2617 3280 y Fz(;)g(")p FA(\))g(d)q Fz(W)2887 3294 y Fw(t)2917 3280 y Fz(;)869 3518 y FA(d)o Fz(z)961 3532 y Fw(t)1016 3518 y FA(=)1122 3456 y(1)p 1122 3497 46 4 v 1124 3580 a Fz(")1197 3494 y FA(^)1178 3518 y Fz(f)1233 3480 y Fs(0)1271 3518 y FA(\()p Fz(\030)1350 3480 y Fv(\000)1346 3542 y Fw(t)1410 3518 y Fz(;)g(z)1492 3532 y Fw(t)1522 3518 y Fz(;)g(y)1607 3532 y Fw(t)1637 3518 y Fz(;)g(")p FA(\))g(d)p Fz(t)20 b FA(+)2006 3456 y Fz(\033)p 1974 3497 119 4 v 1974 3515 a Fx(p)p 2050 3515 43 4 v 65 x Fz(")2120 3495 y Fq(b)2102 3518 y Fz(F)2173 3480 y Fs(0)2213 3518 y FA(\()p Fz(\030)2292 3480 y Fv(\000)2288 3542 y Fw(t)2351 3518 y Fz(;)15 b(z)2433 3532 y Fw(t)2464 3518 y Fz(;)g(y)2549 3532 y Fw(t)2578 3518 y Fz(;)g(")p FA(\))g(d)q Fz(W)2848 3532 y Fw(t)2878 3518 y Fz(;)866 3724 y FA(d)p Fz(y)962 3738 y Fw(t)1016 3724 y FA(=)28 b(^)-48 b Fz(g)t FA(\()p Fz(\030)1238 3686 y Fv(\000)1234 3748 y Fw(t)1297 3724 y Fz(;)15 b(z)1379 3738 y Fw(t)1410 3724 y Fz(;)g(y)1495 3738 y Fw(t)1524 3724 y Fz(;)g(")p FA(\))g(d)q Fz(t)20 b FA(+)g Fz(\033)1907 3687 y Fv(0)1948 3701 y Fq(b)1930 3724 y Fz(G)q FA(\()p Fz(\030)2081 3686 y Fv(\000)2077 3748 y Fw(t)2140 3724 y Fz(;)15 b(z)2222 3738 y Fw(t)2253 3724 y Fz(;)g(y)2338 3738 y Fw(t)2367 3724 y Fz(;)g(")p FA(\))g(d)q Fz(W)2637 3738 y Fw(t)2666 3724 y Fz(;)3475 3482 y FA(\(5.1\))118 3926 y(compare)26 b(\(2.63\))i(and)d(\(2.64\))r(.)40 b(W)-8 b(e)26 b(consider)f(the)h (dynamics)e(as)i(long)f(as)h(\()p Fz(z)2737 3940 y Fw(t)2767 3926 y Fz(;)15 b(y)2852 3940 y Fw(t)2882 3926 y FA(\))26 b(ev)m(olv)m(es)h(in)d(a)i(neigh-)118 4039 y(b)s(ourho)s(o)s(d)e Fx(N)39 b FA(of)27 b(the)f(bifurcation)e(p)s(oin)m(t,)j(whic)m(h)e(is)g (su\016cien)m(tly)g(small)g(for)h(the)h(adiabatic)e(manifold)118 4152 y(to)34 b(b)s(e)f(uniformly)d(asymptotically)i(stable,)i(that)g (is,)f(all)f(the)h(eigen)m(v)-5 b(alues)33 b(of)h Fz(@)2918 4166 y Fw(x)2981 4128 y FA(^)2962 4152 y Fz(f)3017 4119 y Fv(\000)3075 4152 y FA(\(0)p Fz(;)15 b(z)t(;)g(y)s(;)g(")p FA(\))36 b(ha)m(v)m(e)118 4265 y(negativ)m(e)c(real)e(parts,)g (uniformly)d(b)s(ounded)i(a)m(w)m(a)m(y)j(from)e(zero.)118 4508 y Fp(5.1)112 b(Exit)36 b(from)h Fd(B)953 4472 y Fv(\000)1013 4508 y FF(\()p Fc(h)p FF(\))118 4680 y FA(Let)31 b Fz(h)333 4694 y Fw(\021)375 4680 y Fz(;)15 b(h)467 4694 y Fw(z)533 4680 y Fn(>)25 b FA(0.)41 b(In)30 b(addition)e(to)k (the)e(stopping)f(time)1336 4884 y Fz(\034)1376 4898 y Fw(\021)1443 4884 y FA(=)c(inf)1649 4810 y Fq(\010)1702 4884 y Fz(s)g(>)g FA(0)10 b(:)31 b Fx(k)p Fz(\021)2067 4898 y Fw(s)2105 4884 y Fx(k)25 b Fn(>)g Fz(h)2323 4898 y Fw(\021)2366 4810 y Fq(\011)2419 4884 y Fz(;)1031 b FA(\(5.2\))118 5088 y(cf.)31 b(\(3.6\))r(,)f(w)m(e)h(in)m(tro)s(duce)e (the)i(corresp)s(onding)d(stopping)i(time)g(for)g Fz(z)2521 5102 y Fw(s)2578 5088 y Fx(\000)20 b Fz(z)2715 5055 y Fs(det)2711 5111 y Fw(s)2817 5088 y FA(,)31 b(namely)-8 b(,)1210 5292 y Fz(\034)1250 5306 y Fw(z)1314 5292 y FA(=)25 b(inf)1521 5219 y Fq(\010)1574 5292 y Fz(s)g(>)g FA(0)10 b(:)31 b Fx(k)p Fz(z)1936 5306 y Fw(s)1994 5292 y Fx(\000)19 b Fz(z)2130 5255 y Fs(det)2126 5315 y Fw(s)2233 5292 y Fx(k)26 b Fn(>)f Fz(h)2452 5306 y Fw(z)2492 5219 y Fq(\011)2545 5292 y Fz(:)905 b FA(\(5.3\))118 5497 y(The)30 b(follo)m(wing)f(result)g(is)h(obtained)f(using)g(almost)i (the)f(same)h(line)e(of)h(though)m(t)h(as)g(in)e(Section)h(3.1.)1844 5871 y(36)p eop %%Page: 37 37 37 36 bop 118 280 a Fm(Prop)s(osition)43 b(5.1.)i Fy(L)-5 b(et)39 b Fz(t)e Fy(b)-5 b(e)38 b(of)h(or)-5 b(der)39 b FA(1)g Fy(at)f(most.)59 b(Then,)40 b(for)e(al)5 b(l)39 b(initial)f(c)-5 b(onditions)40 b Fz(\030)3390 242 y Fv(\000)3386 306 y Fs(0)3487 280 y Fy(such)118 393 y(that)d Fx(h)p Fz(\030)386 355 y Fv(\000)382 419 y Fs(0)445 393 y Fz(;)p 509 320 55 4 v 15 w(X)567 360 y Fv(\000)627 393 y FA(\()p Fz(y)707 407 y Fs(0)746 393 y Fz(;)15 b(z)828 407 y Fs(0)868 393 y FA(\))903 360 y Fv(\000)p Fs(1)998 393 y Fz(\030)1042 355 y Fv(\000)1038 419 y Fs(0)1101 393 y Fx(i)31 b Fn(6)e Fz(\013)1325 360 y Fs(2)1365 393 y Fz(h)1417 360 y Fs(2)1492 393 y Fy(with)37 b(an)e Fz(\013)c Fx(2)f FA(\(0)p Fz(;)15 b FA(1])p Fy(,)37 b(al)5 b(l)36 b Fz(\015)g Fx(2)29 b FA(\(0)p Fz(;)15 b FA(1)p Fz(=)p FA(2\))p Fy(,)39 b(and)d(al)5 b(l)36 b(su\016ciently)118 506 y(smal)5 b(l)34 b FA(\001)25 b Fz(>)g FA(0)p Fy(,)138 728 y Fr(P)199 691 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)315 661 y Fk(\000)312 712 y Ft(0)368 691 y Fw(;z)421 700 y Ft(0)455 691 y Fw(;y)510 700 y Ft(0)544 691 y Fs(\))575 627 y Fq(n)854 728 y FA(sup)636 805 y Fs(0)p Fo(6)p Fw(s)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)917 816 y Fk(N)979 805 y Fv(^)p Fw(\034)1057 813 y Fl(\021)1096 805 y Fv(^)p Fw(\034)1174 813 y Fl(z)1225 654 y Fq(\012)1268 728 y Fz(\030)1312 691 y Fv(\000)1308 751 y Fw(s)1371 728 y Fz(;)p 1435 655 V 15 w(X)1493 691 y Fv(\000)1553 728 y FA(\()p Fz(y)1633 742 y Fw(s)1669 728 y Fz(;)15 b(z)1751 742 y Fw(s)1789 728 y FA(\))1824 691 y Fv(\000)p Fs(1)1919 728 y Fz(\030)1963 691 y Fv(\000)1959 751 y Fw(s)2022 654 y Fq(\013)2090 728 y Fn(>)25 b Fz(h)2238 691 y Fs(2)2278 627 y Fq(o)628 1002 y Fn(6)724 874 y Fq(\030)830 941 y Fz(t)p 787 982 119 4 v 787 1065 a FA(\001)p Fz(")915 874 y Fq(\031)1248 941 y FA(1)p 978 982 587 4 v 978 1069 a(\(1)c Fx(\000)f FA(2)p Fz(\015)5 b FA(\))1302 1042 y Fs(\()p Fw(n)p Fv(\000)p Fw(q)r Fs(\))p Fw(=)p Fs(2)1589 1002 y FA(exp)1728 874 y Fq(\032)1796 1002 y Fx(\000)p Fz(\015)1930 941 y(h)1982 908 y Fs(2)p 1929 982 95 4 v 1929 1065 a Fz(\033)1984 1039 y Fs(2)2034 902 y Fq(h)2077 1002 y FA(1)20 b Fx(\000)g Fz(\013)2291 965 y Fs(2)2351 1002 y Fx(\000)g(O)2517 929 y Fq(\000)2559 1002 y FA(\001)g(+)g(\(1)h(+)f Fz(\026)p FA(\))p Fz(h)g FA(+)g Fz(h)3243 1016 y Fw(\021)3306 1002 y FA(+)g Fz(h)3449 1016 y Fw(z)3489 929 y Fq(\001)3530 902 y(i)3573 874 y(\033)719 1262 y FA(+)810 1134 y Fq(\030)915 1200 y Fz(t)p 873 1241 119 4 v 873 1324 a FA(\001)p Fz(")1001 1134 y Fq(\031)1069 1262 y FA(e)1109 1224 y Fs(\()p 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Fw(\021)1114 1739 y FA(only)-8 b(.)p 3595 1739 4 62 v 3599 1681 55 4 v 3599 1739 V 3653 1739 4 62 v 259 1927 a(Next,)35 b(w)m(e)e(need)f(to)i(con)m(trol)f(the)g (stopping)e(times)i Fz(\034)2098 1941 y Fw(\021)2172 1927 y FA(and)f Fz(\034)2391 1941 y Fw(z)2431 1927 y FA(.)47 b(Lemma)33 b(3.4)h(holds)d(with)h(minor)118 2040 y(c)m(hanges,)g(incorp)s(orating)c(the)j Fz(z)1242 2054 y Fw(t)1272 2040 y FA(-dep)s(enden)m(t)f(terms.)40 b(W)-8 b(e)32 b(\014nd)d(that)118 2228 y Fm(Lemma)k(5.2.)42 b Fy(L)-5 b(et)33 b Fz(\030)894 2189 y Fv(\000)890 2254 y Fs(0)985 2228 y Fy(satisfy)h Fx(h)p Fz(\030)1348 2189 y Fv(\000)1344 2254 y Fs(0)1407 2228 y Fz(;)p 1471 2155 55 4 v 15 w(X)1529 2195 y Fv(\000)1589 2228 y FA(\()p Fz(y)1669 2242 y Fs(0)1708 2228 y Fz(;)15 b(z)1790 2242 y Fs(0)1831 2228 y FA(\))1866 2195 y Fv(\000)p Fs(1)1960 2228 y Fz(\030)2004 2189 y Fv(\000)2000 2254 y Fs(0)2063 2228 y Fx(i)26 b Fn(6)f Fz(h)2272 2195 y Fs(2)2312 2228 y Fy(.)41 b(Then)201 2485 y Fr(P)262 2448 y Fs(0)p Fw(;)p 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2658 y Fq(\022)2401 2787 y Fz(\037)2458 2749 y Fs(\(1\))2553 2787 y FA(\()p Fz(t)p FA(\))15 b Fz(h)2723 2801 y Fw(\021)2766 2686 y Fq(\020)2820 2787 y FA(1)20 b(+)2988 2725 y Fz(h)3040 2692 y Fs(2)p 2986 2766 95 4 v 2986 2849 a Fz(h)3038 2823 y Fs(2)3038 2871 y Fw(\021)3111 2787 y FA(+)3213 2725 y Fz(h)3265 2692 y Fs(2)3265 2748 y Fw(z)p 3212 2766 V 3212 2849 a Fz(h)3264 2823 y Fs(2)3264 2871 y Fw(\021)3316 2686 y Fq(\021)3370 2658 y(\023\025\033)3553 2787 y Fz(:)3475 2972 y FA(\(5.5\))259 3168 y(The)30 b(next)h(result)e(allo)m(ws)h(to)h (con)m(trol)g(the)f(stopping)f(time)h Fz(\034)2364 3182 y Fw(z)2404 3168 y FA(.)41 b(W)-8 b(e)31 b(de\014ne)1072 3412 y Fz(\037)1129 3374 y Fs(\(1\))1129 3434 y Fw(z)1223 3412 y FA(\()p Fz(t)p FA(\))26 b(=)58 b(sup)1448 3488 y Fs(0)p Fo(6)p Fw(s)p Fo(6)p Fw(t)1666 3288 y Fq(Z)1757 3314 y Fw(s)1716 3494 y Fs(0)1794 3311 y Fq(\020)1889 3412 y FA(sup)1848 3488 y Fw(u)p Fo(6)p Fw(v)r Fo(6)p Fw(s)2068 3412 y Fx(k)p Fz(U)2185 3374 y Fs(0)2225 3412 y FA(\()p Fz(s;)15 b(v)s FA(\))p Fx(k)2470 3311 y Fq(\021)2541 3412 y FA(d)o Fz(u;)807 b FA(\(5.6\))1072 3668 y Fz(\037)1129 3630 y Fs(\(2\))1129 3690 y Fw(z)1223 3668 y FA(\()p Fz(t)p FA(\))26 b(=)58 b(sup)1448 3744 y Fs(0)p Fo(6)p Fw(s)p Fo(6)p Fw(t)1666 3544 y Fq(Z)1757 3570 y Fw(s)1716 3750 y Fs(0)1794 3567 y Fq(\020)1889 3668 y FA(sup)1848 3744 y Fw(u)p Fo(6)p Fw(v)r Fo(6)p Fw(s)2068 3668 y Fx(k)p Fz(U)2185 3630 y Fs(0)2225 3668 y FA(\()p Fz(s;)15 b(v)s FA(\))p Fx(k)2470 3630 y Fs(2)2511 3567 y Fq(\021)2580 3668 y FA(d)p Fz(u;)767 b FA(\(5.7\))118 3949 y(where)30 b Fz(U)453 3916 y Fs(0)523 3949 y FA(is)f(the)i(principal)c(solution)i (of)h Fz(")1658 3925 y FA(_)1639 3949 y Fz(\020)i FA(=)25 b Fz(A)1875 3916 y Fs(0)1915 3949 y FA(\()p Fz(z)1996 3916 y Fs(det)1992 3971 y Fw(t)2099 3949 y Fz(;)15 b(y)2187 3916 y Fs(det)2184 3971 y Fw(t)2289 3949 y Fz(;)g(")p FA(\))p Fz(\020)7 b FA(.)118 4136 y Fm(Lemma)33 b(5.3.)42 b Fy(L)-5 b(et)33 b Fz(\030)894 4098 y Fv(\000)890 4162 y Fs(0)985 4136 y Fy(satisfy)h Fx(h)p Fz(\030)1348 4098 y Fv(\000)1344 4162 y Fs(0)1407 4136 y Fz(;)p 1471 4063 55 4 v 15 w(X)1529 4103 y Fv(\000)1589 4136 y FA(\()p Fz(y)1669 4150 y Fs(0)1708 4136 y Fz(;)15 b(z)1790 4150 y Fs(0)1831 4136 y FA(\))1866 4103 y Fv(\000)p Fs(1)1960 4136 y Fz(\030)2004 4098 y Fv(\000)2000 4162 y Fs(0)2063 4136 y Fx(i)26 b Fn(6)f Fz(h)2272 4103 y Fs(2)2312 4136 y Fy(.)41 b(Then)235 4367 y Fr(P)296 4329 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)412 4299 y Fk(\000)409 4350 y Ft(0)464 4329 y Fw(;z)517 4338 y Ft(0)551 4329 y Fw(;y)606 4338 y Ft(0)640 4329 y Fs(\))672 4266 y Fq(n)954 4367 y FA(sup)732 4443 y Fs(0)p Fo(6)p Fw(s)p Fo(6)p Fw(t)p Fv(^)p Fw(\034)1013 4467 y Fk(B)1055 4453 y(\000)1108 4467 y Ft(\()p Fl(h)p Ft(\))1196 4443 y Fv(^)p Fw(\034)1274 4451 y Fl(\021)1312 4367 y Fx(k)p Fz(z)1399 4381 y Fw(s)1457 4367 y Fx(\000)20 b Fz(z)1594 4329 y Fs(det)1590 4389 y Fw(s)1696 4367 y Fx(k)26 b Fn(>)f Fz(h)1915 4381 y Fw(z)1955 4266 y Fq(o)446 4668 y Fn(6)g FA(2)587 4540 y Fq(\030)693 4607 y Fz(t)p 650 4647 119 4 v 650 4730 a FA(\001)p Fz(")778 4540 y Fq(\031)846 4668 y FA(e)887 4631 y Fw(q)r(=)p Fs(4)1011 4668 y FA(exp)1149 4540 y Fq(\032)1218 4668 y Fx(\000)p Fz(\024)1341 4682 y Fs(0)1390 4607 y Fz("h)1484 4574 y Fs(2)1484 4629 y Fw(z)1525 4607 y FA(\(1)c Fx(\000)f(O)s FA(\(\001)p Fz(")p FA(\)\))p 1390 4647 626 4 v 1528 4756 a Fz(\033)1583 4730 y Fs(2)1623 4756 y Fz(\037)1680 4708 y Fs(\(2\))1680 4768 y Fw(z)1774 4756 y FA(\()p Fz(t)p FA(\))2026 4540 y Fq(\024)2074 4668 y FA(1)g Fx(\000)g(O)2305 4540 y Fq(\022)2372 4668 y Fz(\037)2429 4631 y Fs(\(1\))2429 4691 y Fw(z)2523 4668 y FA(\()p Fz(t)p FA(\))15 b Fz(h)2693 4682 y Fw(z)2734 4567 y Fq(\020)2788 4668 y FA(1)21 b(+)2955 4607 y Fz(h)3007 4574 y Fs(2)p 2955 4647 93 4 v 2955 4730 a Fz(h)3007 4704 y Fs(2)3007 4753 y Fw(z)3077 4668 y FA(+)3178 4597 y Fz(h)3230 4564 y Fs(2)3230 4620 y Fw(\021)p 3178 4647 95 4 v 3179 4730 a Fz(h)3231 4704 y Fs(2)3231 4753 y Fw(z)3282 4567 y Fq(\021)3337 4540 y(\023\025\033)3520 4668 y Fz(:)3475 4866 y FA(\(5.8\))118 5062 y Fg(Pr)n(oof:)45 b FA(The)29 b(pro)s(of)f(is)g(almost)i(iden)m (tical)d(with)h(the)i(pro)s(of)e(of)h(Lemma)h(3.4)g(and)f(Lemma)g(5.2,) i(with)118 5175 y Fz(\033)173 5142 y Fv(0)227 5175 y FA(replaced)f(b)m(y)g Fz(\033)s(=)812 5110 y Fx(p)p 888 5110 43 4 v 65 x Fz(")h FA(and)f Fz(V)50 b FA(replaced)30 b(b)m(y)h Fz(U)1799 5142 y Fs(0)1838 5175 y FA(.)p 3595 5175 4 62 v 3599 5117 55 4 v 3599 5175 V 3653 5175 4 62 v 118 5363 a Fg(Pr)n(oof)j(of)g(Theorem)f(2.7.)46 b FA(W)-8 b(e)27 b(can)g(rep)s(eat)f(the)g(pro)s(of)f(of)h(Corollary)f (3.10)j(in)c(Section)i(3.2,)i(com-)118 5476 y(paring)23 b(the)h(pro)s(cess)f(to)i(di\013eren)m(t)e(deterministic)f(solutions)g (on)h(successiv)m(e)i(time)e(in)m(terv)-5 b(als)23 b(of)h(length)1844 5871 y(37)p eop %%Page: 38 38 38 37 bop 118 280 a Fz(T)13 b FA(.)40 b(The)26 b(only)h(di\013erence)g (lies)f(in)g(new)g(v)-5 b(alues)27 b(for)g(the)h(exp)s(onen)m(ts)f Fz(\024)2507 247 y Fs(+)2566 280 y FA(\(0\))i(\(resulting)d(from)h (Prop)s(osi-)118 393 y(tion)e(5.1\))i(and)e Fz(\024)701 360 y Fv(0)725 393 y FA(.)39 b(In)24 b(fact,)k(c)m(ho)s(osing)d Fz(h)1514 407 y Fw(\021)1582 393 y FA(prop)s(ortional)e(to)k Fz(h)p FA(,)g Fz(h)2363 407 y Fw(z)2428 393 y FA(prop)s(ortional)d(to)i (\()p Fz(\037)3145 345 y Fs(\(2\))3145 405 y Fw(z)3239 393 y FA(\()p Fz(T)13 b FA(\))p Fz(=")p FA(\))3497 360 y Fs(1)p Fw(=)p Fs(2)3609 393 y Fz(h)118 506 y FA(and,)30 b(\014nally)-8 b(,)29 b Fz(\026)668 473 y Fs(2)738 506 y FA(prop)s(ortional)f(to)j(1)21 b(+)e(\()p Fz(h)1615 520 y Fw(\021)1678 506 y FA(+)h Fz(h)1821 520 y Fw(z)1861 506 y FA(\))p Fz(=h)p FA(,)32 b(sho)m(ws)e(that)218 729 y Fr(P)279 691 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)395 662 y Fk(\000)392 713 y Ft(0)447 691 y Fw(;z)500 700 y Ft(0)534 691 y Fw(;y)589 700 y Ft(0)623 691 y Fs(\))655 628 y Fq(n)832 729 y FA(sup)716 807 y Fs(0)p Fo(6)p Fw(s)p Fo(6)p Fw(T)10 b Fv(^)p Fw(\034)1023 818 y Fk(N)1100 655 y Fq(\012)1142 729 y Fz(\030)1186 691 y Fv(\000)1182 751 y Fw(s)1245 729 y Fz(;)p 1309 656 55 4 v 15 w(X)1367 691 y Fv(\000)1427 729 y FA(\()p Fz(y)1507 743 y Fw(s)1544 729 y Fz(;)15 b(z)1626 743 y Fw(s)1664 729 y FA(\))1699 691 y Fv(\000)p Fs(1)1793 729 y Fz(\030)1837 691 y Fv(\000)1833 751 y Fw(s)1896 655 y Fq(\013)1964 729 y Fn(>)25 b Fz(h)2112 691 y Fs(2)2152 628 y Fq(o)2238 729 y Fn(6)g Fz(C)2399 743 y Fw(n;m;q)r(;\015)2641 729 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))g(e)2911 691 y Fv(\000)p Fw(\024)3007 668 y Ft(+)3057 691 y Fs(\()p Fw(\013)p Fs(\))p Fw(h)3197 668 y Ft(2)3233 691 y Fw(=\033)3310 668 y Ft(2)3350 729 y Fz(;)100 b FA(\(5.9\))118 999 y(v)-5 b(alid)27 b(for)i(all)e Fz(\030)642 961 y Fv(\000)638 1025 y Fs(0)730 999 y FA(satisfying)h Fx(h)p Fz(\030)1213 961 y Fv(\000)1209 1025 y Fs(0)1272 999 y Fz(;)p 1336 926 V 15 w(X)1394 966 y Fv(\000)1454 999 y FA(\()p Fz(y)1534 1013 y Fs(0)1573 999 y Fz(;)15 b(z)1655 1013 y Fs(0)1695 999 y FA(\))1730 966 y Fv(\000)p Fs(1)1825 999 y Fz(\030)1869 961 y Fv(\000)1865 1025 y Fs(0)1928 999 y Fx(i)26 b Fn(6)f Fz(\013)2143 966 y Fs(2)2182 999 y Fz(h)2234 966 y Fs(2)2303 999 y FA(and)j(all)g Fz(T)41 b FA(of)29 b(order)g(1)g(at)g(most.)41 b(Here)263 1249 y Fz(C)328 1263 y Fw(n;m;q)r(;\015)570 1249 y FA(\()p Fz(T)8 b(;)15 b(")p FA(\))27 b(=)906 1120 y Fq(\030)995 1187 y Fz(T)p 968 1228 119 4 v 968 1311 a FA(\001)p Fz(")1097 1120 y Fq(\031\024)1478 1187 y FA(1)p 1208 1228 587 4 v 1208 1315 a(\(1)21 b Fx(\000)f FA(2)p Fz(\015)5 b FA(\))1532 1288 y Fs(\()p Fw(n)p Fv(\000)p Fw(q)r Fs(\))p Fw(=)p Fs(2)1824 1249 y FA(+)20 b(e)1955 1211 y Fs(\()p Fw(n)p Fv(\000)p Fw(q)r Fs(\))p Fw(=)p Fs(4)2231 1249 y FA(+2)15 b(e)2403 1211 y Fw(m=)p Fs(4)2555 1249 y FA(+2)g(e)2727 1211 y Fw(q)r(=)p Fs(4)2836 1120 y Fq(\025)2884 1249 y Fz(;)520 b FA(\(5.10\))544 1458 y Fz(\024)596 1421 y Fs(+)655 1458 y FA(\()p Fz(\013)p FA(\))27 b(=)e Fz(\015)958 1384 y Fq(\002)996 1458 y FA(1)20 b Fx(\000)g Fz(\013)1210 1421 y Fs(2)1270 1458 y Fx(\000)g(O)s FA(\(\001\))h Fx(\000)f(O)s FA(\()p Fz(h)p FA(\))h Fx(\000)f(O)s FA(\()p Fz(h)p FA(\()p Fz(\037)2257 1421 y Fs(\(2\))2352 1458 y FA(\()p Fz(T)13 b FA(\))21 b Fx(_)e Fz(\037)2646 1421 y Fs(\(2\))2646 1481 y Fw(z)2741 1458 y FA(\()p Fz(T)13 b FA(\)\))2912 1421 y Fs(1)p Fw(=)p Fs(2)3022 1458 y Fz(=)3067 1388 y Fx(p)p 3144 1388 43 4 v 3144 1458 a Fz(")p FA(\))3221 1384 y Fq(\003)3259 1458 y Fz(:)145 b FA(\(5.11\))118 1654 y(Similar)27 b(argumen)m(ts)k(as)g(in)e(the)h(pro)s(of)g(of)h (Lemma)f(3.7)i(yield)c(a)j(b)s(ound)d(of)j(the)f(form)704 1868 y Fr(P)765 1831 y Fs(0)p Fw(;)p Fs(\()p Fw(\030)881 1801 y Fk(\000)878 1852 y Ft(0)933 1831 y Fw(;z)986 1840 y Ft(0)1020 1831 y Fw(;y)1075 1840 y Ft(0)1109 1831 y Fs(\))1141 1767 y Fq(n)1202 1794 y(\012)1244 1868 y Fz(\030)1288 1830 y Fv(\000)1284 1897 y Fw(T)1347 1868 y Fz(;)p 1411 1795 55 4 v 15 w(X)1469 1831 y Fv(\000)1529 1868 y FA(\()p Fz(y)1609 1897 y Fw(T)1672 1868 y Fz(;)15 b(z)1754 1897 y Fw(T)1817 1868 y FA(\))1852 1831 y Fv(\000)p Fs(1)1947 1868 y Fz(\030)1991 1830 y Fv(\000)1987 1897 y Fw(T)2050 1794 y Fq(\013)2093 1868 y Fz(;)g(\034)2173 1882 y Fv(N)2270 1868 y Fn(>)25 b Fz(T)2432 1767 y Fq(o)2518 1868 y Fn(6)2632 1845 y Fq(b)2614 1868 y Fz(C)c FA(e)2741 1831 y Fv(\000)p Fw(\024)2837 1807 y Fk(0)2859 1831 y Fw(h)2900 1807 y Ft(2)2934 1831 y Fw(=\033)3011 1807 y Ft(2)3051 1868 y Fz(;)353 b FA(\(5.12\))118 2082 y(where)164 2278 y Fz(\024)216 2241 y Fv(0)264 2278 y FA(=)25 b Fz(\015)412 2204 y Fq(\002)450 2278 y FA(1)9 b Fx(\000)g(O)s FA(\(\001\))g Fx(\000)g(O)s FA(\()p Fz(h)p FA(\))g Fx(\000)g(O)s FA(\()p Fz(h)p FA(\()p Fz(\037)1434 2241 y Fs(\(2\))1531 2278 y FA(\()p Fz(T)k FA(\))c Fx(_)g Fz(\037)1803 2241 y Fs(\(2\))1803 2301 y Fw(z)1897 2278 y FA(\()p Fz(T)k FA(\)\))2068 2241 y Fs(1)p Fw(=)p Fs(2)2179 2278 y Fz(=)2224 2208 y Fx(p)p 2300 2208 43 4 v 70 x Fz(")p FA(\))c Fx(\000)g(O)s FA(\(e)2617 2241 y Fv(\000)p Fs(2)p Fw(K)2767 2250 y Ft(0)2801 2241 y Fw(T)f(=")2937 2278 y Fz(=)p FA(\(1)h Fx(\000)g FA(2)p Fz(\015)c FA(\)\))3318 2204 y Fq(\003)3358 2278 y Fz(:)46 b FA(\(5.13\))118 2474 y(In)38 b(order)h(for)g(the)g(estimates)h (\(5.9\))h(and)e(\(5.12\))i(to)f(b)s(e)e(useful,)i(w)m(e)g(need)e(to)i (tak)m(e)h Fz(T)52 b FA(of)39 b(order)g Fz(")p FA(.)118 2587 y(Ho)m(w)m(ev)m(er,)30 b(this)25 b(leads)h(to)h(an)f(error)g(term) h(of)f(order)g(1)h(in)e(the)i(exp)s(onen)m(t)f Fz(\024)2665 2554 y Fv(0)2689 2587 y FA(,)h(whic)m(h)e(is)h(due)g(to)h(the)f(fact) 118 2700 y(that)36 b Fz(\030)364 2662 y Fv(\000)360 2724 y Fw(t)458 2700 y FA(has)f(to)s(o)h(little)e(time)h(to)h(relax)f(to)h (the)f(adiabatic)g(manifold.)53 b(In)35 b(order)g(to)h(\014nd)d(the)j (b)s(est)118 2813 y(compromise,)29 b(w)m(e)g(tak)m(e)h Fz(T)38 b FA(=)25 b Fz(\022)s(")j FA(and)g(optimize)g(o)m(v)m(er)i Fz(\022)s FA(.)39 b(Assume)28 b(w)m(e)h(are)g(in)e(the)i(w)m(orst)g (case,)h(when)118 2937 y Fx(k)p Fz(U)235 2904 y Fs(0)275 2937 y Fx(k)h FA(gro)m(ws)g(exp)s(onen)m(tially)d(lik)m(e)i(e)1374 2904 y Fw(K)1434 2913 y Ft(+)1484 2904 y Fw(t=")1582 2937 y FA(.)41 b(Then)29 b Fz(\037)1942 2889 y Fs(\(2\))1942 2949 y Fw(z)2036 2937 y FA(\()p Fz(T)13 b FA(\))31 b(is)e(of)i(the)f (order)g Fz("\022)18 b FA(e)2936 2904 y Fs(2)p Fw(K)3031 2913 y Ft(+)3081 2904 y Fw(\022)3120 2937 y FA(.)41 b(The)30 b(c)m(hoice)1323 3152 y(e)1364 3114 y Fv(\000)p Fw(\022)1483 3152 y FA(=)1579 3078 y Fq(\002)1617 3152 y Fz(h)p FA(\(1)21 b Fx(\000)f FA(2)p Fz(\015)5 b FA(\))1993 3078 y Fq(\003)2032 3101 y Fs(1)p Fw(=)p Fs(\(2)p Fw(K)2224 3110 y Ft(0)2259 3101 y Fs(+)p Fw(K)2374 3110 y Ft(+)2424 3101 y Fs(\))3429 3152 y FA(\(5.14\))118 3358 y(yields)33 b(an)i(almost)g(optimal)e (error)i(term)f(of)h(order)g Fz(h)1996 3325 y Fw(\027)2039 3358 y FA(\(1)24 b Fx(\000)f FA(2)p Fz(\015)5 b FA(\))2369 3325 y Fs(1)p Fv(\000)p Fw(\027)2503 3358 y Fx(j)p FA(log)r(\()p Fz(h)p FA(\(1)25 b Fx(\000)d FA(2)p Fz(\015)5 b FA(\)\))p Fx(j)3123 3325 y Fs(1)p Fw(=)p Fs(2)3235 3358 y FA(,)36 b(with)d Fz(\027)39 b FA(=)118 3487 y(2)p Fz(K)240 3501 y Fs(0)280 3487 y Fz(=)p FA(\(2)p Fz(K)482 3501 y Fs(0)531 3487 y FA(+)8 b Fz(K)687 3501 y Fs(+)746 3487 y FA(\).)39 b(The)24 b(smaller)f Fz(K)1411 3501 y Fs(+)1470 3487 y FA(,)i(i.)15 b(e.,)26 b(the)f(slo)m(w)m(er)f Fz(\037)2177 3439 y Fs(\(2\))2177 3499 y Fw(z)2271 3487 y FA(\()p Fz(t)p FA(\))h(gro)m(ws,)h(the)f(closer)f Fz(\027)30 b FA(is)23 b(to)i(one.)p 3595 3487 4 62 v 3599 3429 55 4 v 3599 3487 V 3653 3487 4 62 v 118 3729 a Fp(5.2)112 b(The)38 b(reduced)g(system)118 3901 y FA(Giv)m(en)30 b(the)h(SDE)f(\(5.1\))r(,)g(w)m(e)h(call)897 4135 y(d)p Fz(z)994 4097 y Fs(0)990 4157 y Fw(t)1059 4135 y FA(=)1165 4073 y(1)p 1165 4114 46 4 v 1167 4197 a Fz(")1240 4111 y FA(^)1220 4135 y Fz(f)1275 4097 y Fs(0)1314 4135 y FA(\(0)p Fz(;)15 b(z)1480 4097 y Fs(0)1476 4157 y Fw(t)1521 4135 y Fz(;)g(y)1609 4097 y Fs(0)1606 4157 y Fw(t)1648 4135 y Fz(;)g(")p FA(\))g(d)q Fz(t)20 b FA(+)2018 4073 y Fz(\033)p 1986 4114 119 4 v 1986 4132 a Fx(p)p 2062 4132 43 4 v 66 x Fz(")2132 4112 y Fq(b)2114 4135 y Fz(F)2185 4097 y Fs(0)2225 4135 y FA(\(0)p Fz(;)15 b(z)2391 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Fv(\000)3101 5509 y Fx(k)21 b FA(+)g Fx(k)p Fz(\020)7 b Fx(k)p FA(,)33 b(while)118 5622 y Fx(k)181 5599 y Fq(e)163 5622 y Fz(F)14 b Fx(k)27 b FA(is)e(b)s(ounded.)37 b(The)27 b(matrices)f Fz(A)1411 5589 y Fv(\000)1470 5622 y FA(,)i Fz(B)j FA(and)26 b Fz(C)33 b FA(are)27 b(those)g(de\014ned)e(in)g(\(2.66\))s(,)i(\(2.76\)) i(and)d(\(2.77\))r(.)1844 5871 y(38)p eop %%Page: 39 39 39 38 bop 259 280 a FA(F)-8 b(or)29 b(a)f(giv)m(en)g(con)m(tin)m(uous)g (sample)f(path)h Fx(f)p Fz(\020)1784 247 y Fs(0)1777 303 y Fw(t)1823 280 y FA(\()p Fz(!)s FA(\))p Fx(g)1998 294 y Fw(t)p Fo(>)p Fs(0)2147 280 y FA(of)35 b(\(5.16\))r(,)29 b(w)m(e)f(denote)h(b)m(y)e Fz(U)3148 294 y Fw(!)3227 280 y FA(and)g Fx(V)3457 294 y Fw(!)3535 280 y FA(the)118 393 y(principal)g(solutions)i(of)h Fz(")1042 369 y FA(_)1022 393 y Fz(\030)1066 360 y Fv(\000)1151 393 y FA(=)25 b Fz(A)1315 360 y Fv(\000)1374 393 y FA(\()p Fz(\020)1456 360 y Fs(0)1449 416 y Fw(t)1496 393 y FA(\()p Fz(!)s FA(\))p Fz(;)15 b(")p FA(\))p Fz(\030)1787 360 y Fv(\000)1878 393 y FA(and)29 b Fz(")2115 369 y FA(_)2096 393 y Fz(\020)j FA(=)25 b Fz(B)5 b FA(\()p Fz(\020)2420 360 y Fs(0)2413 416 y Fw(t)2459 393 y FA(\()p Fz(!)s FA(\))p Fz(;)15 b(")p FA(\))p Fz(\020)7 b FA(.)42 b(If)30 b(w)m(e)h(further)e(de\014ne) 973 648 y Fx(S)1028 662 y Fw(!)1078 648 y FA(\()p Fz(t;)15 b(s)p FA(\))26 b(=)1396 587 y(1)p 1396 627 46 4 v 1398 711 a Fz(")1466 525 y Fq(Z)1557 551 y Fw(t)1517 731 y(s)1602 648 y Fx(V)1658 662 y Fw(!)1708 648 y FA(\()p Fz(t;)15 b(u)p FA(\))p Fz(C)7 b FA(\()p Fz(\020)2057 611 y Fs(0)2050 671 y Fw(u)2097 648 y FA(\()p Fz(!)s FA(\))p Fz(;)15 b(")p FA(\))p Fz(U)2406 662 y Fw(!)2458 648 y FA(\()p Fz(u;)g(s)p FA(\))g(d)q Fz(u;)622 b FA(\(5.17\))118 893 y(w)m(e)31 b(can)g(write)e(the)i(solution)e(of)37 b(\(5.16\))c(as)742 1148 y Fz(\020)782 1162 y Fw(t)812 1148 y FA(\()p Fz(!)s FA(\))26 b(=)1105 1087 y Fz(\033)p 1074 1127 119 4 v 1074 1145 a Fx(p)p 1149 1145 43 4 v 1149 1211 a Fz(")1217 1024 y Fq(Z)1308 1051 y Fw(t)1267 1231 y 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Fz(!)s FA(\))1084 1660 y(+)1185 1599 y(1)p 1185 1639 46 4 v 1187 1723 a Fz(")1255 1536 y Fq(Z)1346 1563 y Fw(t)1306 1743 y Fs(0)1391 1660 y Fx(V)1447 1674 y Fw(!)1497 1660 y FA(\()p Fz(t;)g(s)p FA(\))p Fz(c)p FA(\()p Fz(\030)1801 1623 y Fv(\000)1797 1683 y Fw(s)1861 1660 y FA(\()p Fz(!)s FA(\))p Fz(;)g(\020)2071 1674 y Fw(s)2109 1660 y FA(\()p Fz(!)s FA(\))p Fz(;)g(\020)2326 1623 y Fs(0)2319 1683 y Fw(s)2366 1660 y FA(\()p Fz(!)s FA(\))p Fz(;)g(")p FA(\))g(d)q Fz(s)1084 1914 y FA(+)1185 1853 y(1)p 1185 1893 V 1187 1977 a Fz(")1255 1790 y Fq(Z)1346 1817 y Fw(t)1306 1997 y Fs(0)1391 1914 y Fx(S)1446 1928 y Fw(!)1496 1914 y FA(\()p Fz(t;)g(s)p FA(\))p Fz(b)p FA(\()p Fz(\030)1800 1877 y Fv(\000)1796 1937 y Fw(s)1860 1914 y FA(\()p Fz(!)s FA(\))p Fz(;)g(\020)2070 1928 y Fw(s)2108 1914 y FA(\()p Fz(!)s FA(\))p Fz(;)g(\020)2325 1877 y Fs(0)2318 1937 y Fw(s)2365 1914 y FA(\()p Fz(!)s FA(\))p Fz(;)g(")p FA(\))g(d)q Fz(s:)682 b FA(\(5.18\))118 2159 y(Concerning)29 b(the)i(\014rst)e(t)m(w)m(o)j(summands)d(in)g (\(5.18\))r(,)i(note)g(that)g(the)f(iden)m(tities)930 2356 y Fx(V)986 2370 y Fw(!)1036 2356 y FA(\()p Fz(t;)15 b(s)p FA(\))26 b(=)e Fx(V)1399 2370 y Fw(!)1450 2356 y FA(\()p Fz(t;)15 b FA(0\))p Fx(V)1694 2370 y Fw(!)1745 2356 y FA(\()p Fz(s;)g FA(0\))1943 2318 y Fv(\000)p Fs(1)2039 2356 y Fz(;)930 2503 y Fx(S)985 2517 y Fw(!)1036 2503 y FA(\()p Fz(t;)g(s)p FA(\))26 b(=)e Fx(S)1398 2517 y Fw(!)1449 2503 y FA(\()p Fz(t;)15 b FA(0\))p Fz(U)1699 2517 y Fw(!)1751 2503 y FA(\()p Fz(s;)g FA(0\))1949 2466 y Fv(\000)p Fs(1)2065 2503 y FA(+)k Fx(V)2211 2517 y Fw(!)2262 2503 y FA(\()p Fz(t;)c FA(0\))p Fx(S)2505 2517 y Fw(!)2556 2503 y FA(\()p Fz(s;)g FA(0\))2754 2466 y Fv(\000)p Fs(1)3429 2430 y FA(\(5.19\))118 2694 y(allo)m(w)38 b(to)g(rewrite)g(the)g(sto)s(c)m(hastic)h(in)m(tegrals)f(in)e(suc)m(h)i (a)g(w)m(a)m(y)h(that)g(the)f(in)m(tegrands)g(are)g(adapted)118 2807 y(with)29 b(resp)s(ect)i(to)g(the)f(\014ltration)f(generated)j(b)m (y)e Fx(f)p 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Fz(t;)15 b(s)p FA(\))p Fz(;)3429 3303 y FA(\(5.20\))118 3572 y(uniformly)28 b(in)h Fz(")p FA(,)i(whenev)m(er)f Fz(s)25 b Fn(6)g Fz(t)g Fn(6)g Fz(\034)10 b FA(\()p Fz(!)s FA(\),)31 b(and)f(de\014ne)656 3815 y Fz(\037)713 3778 y Fs(\()p Fw(i)p Fs(\))796 3815 y FA(\()p Fz(t)p FA(\))c(=)58 b(sup)1021 3892 y Fs(0)p Fo(6)p Fw(s)p Fo(6)p Fw(t)1249 3754 y FA(1)p 1249 3794 V 1251 3878 a Fz(")1320 3692 y Fq(Z)1411 3718 y Fw(s)1370 3898 y Fs(0)1463 3815 y Fz(#)p FA(\()p Fz(s;)15 b(u)p FA(\))1722 3778 y Fw(i)1765 3815 y FA(d)p Fz(u;)921 b(i)26 b FA(=)f(1)p Fz(;)15 b FA(2)p Fz(;)307 b FA(\(5.21\))656 4071 y Fz(\037)713 4023 y Fs(\()p Fw(i)p Fs(\))713 4100 y Fw(C)796 4071 y FA(\()p Fz(t)p FA(\))26 b(=)58 b(sup)1021 4148 y Fs(0)p Fo(6)p Fw(s)p Fo(6)p Fw(t)1249 4010 y FA(1)p 1249 4050 V 1251 4134 a Fz(")1320 3948 y Fq(Z)1411 3974 y Fw(s)1370 4154 y Fs(0)1447 3970 y Fq(\020)1543 4071 y FA(sup)1502 4148 y Fw(u)p Fo(6)p Fw(v)r Fo(6)p Fw(s)1737 4071 y Fz(#)1791 4085 y Fw(C)1850 4071 y FA(\()p Fz(s;)15 b(v)s FA(\))2050 4034 y Fw(i)2079 3970 y Fq(\021)2148 4071 y FA(d)p Fz(u;)538 b(i)26 b FA(=)f(1)p Fz(;)15 b FA(2)p Fz(:)307 b FA(\(5.22\))259 4325 y(The)30 b(follo)m(wing)f(prop)s (osition)f(establishes)h(a)i(lo)s(cal)e(v)m(ersion)h(of)h(Theorem)f (2.8.)118 4508 y Fm(Prop)s(osition)39 b(5.4.)k Fy(L)-5 b(et)35 b FA(\001)g Fy(b)-5 b(e)35 b(su\016ciently)f(smal)5 b(l,)37 b(\014x)e(times)g Fz(s)28 b(<)h(t)35 b Fy(such)g(that)h Fz(t)21 b Fx(\000)h Fz(s)28 b FA(=)h(\001)p Fz(")p Fy(,)35 b(and)118 4621 y(assume)42 b(that)h(ther)-5 b(e)42 b(exists)g(a)g(c)-5 b(onstant)43 b Fz(#)1646 4635 y Fs(0)1727 4621 y Fz(>)e FA(0)g Fy(such)h(that)h Fz(#)p FA(\()p Fz(u;)15 b(s)p FA(\))41 b Fn(6)g Fz(#)2802 4635 y Fs(0)2883 4621 y Fy(and)h Fz(#)3122 4635 y Fw(C)3181 4621 y FA(\()p Fz(u;)15 b(s)p FA(\))42 b Fn(6)f Fz(#)3594 4635 y Fs(0)3634 4621 y Fy(,)118 4734 y(whenever)30 b Fz(u)25 b Fx(2)g FA([)p Fz(s;)15 b(t)p FA(])p Fy(.)41 b(Then)29 b(ther)-5 b(e)30 b(exist)f(c)-5 b(onstants)31 b Fz(\024)2021 4748 y Fs(0)2061 4734 y Fz(;)15 b(h)2153 4748 y Fs(0)2219 4734 y Fz(>)25 b FA(0)k Fy(such)g(that)i(for)e(al)5 b(l)30 b Fz(h)c Fn(6)e Fz(h)3268 4748 y Fs(0)3308 4734 y FA([)p Fz(\037)3390 4701 y Fs(\(1\))3485 4734 y FA(\()p Fz(t)p FA(\))13 b Fx(_)118 4863 y Fz(\037)175 4815 y Fs(\(1\))175 4891 y Fw(C)269 4863 y FA(\()p Fz(t)p FA(\)])397 4830 y Fv(\000)p Fs(1)492 4863 y Fy(,)364 5123 y Fr(P)425 5085 y Fs(0)p Fw(;)p Fs(0)519 5022 y Fq(n)702 5123 y FA(sup)580 5200 y Fw(s)p Fv(^)p Fw(\034)8 b Fo(6)p Fw(u)f Fz(h)1310 5022 y Fq(o)1396 5123 y Fn(6)g FA(2)15 b(e)1593 5085 y Fs(\()p Fw(m)p Fs(+)p Fw(q)r Fs(\))p Fw(=)p Fs(4)1889 5123 y FA(exp)2028 5022 y Fq(n)2088 5123 y Fx(\000)p Fz(\024)2211 5137 y Fs(0)2262 5061 y Fz(h)2314 5028 y Fs(2)p 2261 5102 95 4 v 2261 5185 a Fz(\033)2316 5159 y Fs(2)2709 5061 y FA(1)p 2375 5102 713 4 v 2375 5211 a Fz(\037)2432 5163 y Fs(\(2\))2432 5239 y Fw(C)2527 5211 y FA(\()p Fz(t)p FA(\))20 b(+)g Fz(h)2793 5184 y Fs(2)2833 5211 y Fz(\037)2890 5184 y Fs(\(2\))2984 5211 y FA(\()p Fz(t)p FA(\))3098 5022 y Fq(o)3158 5123 y Fz(:)246 b FA(\(5.23\))118 5396 y Fg(Pr)n(oof:)45 b FA(The)30 b(pro)s(of)f(follo)m(ws)g(along)i (the)f(lines)e(of)j(the)f(pro)s(of)f(of)i(Lemma)f(3.4,)h(the)g(main)e (di\013erence)118 5509 y(lying)20 b(in)h(the)h(fact)g(that)h(the)e(sto) s(c)m(hastic)i(in)m(tegrals)e(in)g(\(5.18\))j(in)m(v)m(olv)m(e)e(the)g (principal)c(solutions)i Fz(U)3456 5523 y Fw(!)3507 5509 y FA(,)j Fx(V)3611 5523 y Fw(!)118 5622 y FA(dep)s(ending)32 b(on)i(the)g(realization)g(of)g(the)h(pro)s(cess.)52 b(Ho)m(w)m(ev)m(er,)37 b(the)e(existence)g(of)f(the)h(deterministic) 1844 5871 y(39)p eop %%Page: 40 40 40 39 bop 118 280 a FA(b)s(ound)40 b(\(5.20\))45 b(allo)m(ws)c(for)h(a) h(similar)c(conclusion.)75 b(In)42 b(particular,)i(the)e(\014rst)f(and) h(second)g(term)118 393 y(in)29 b(\(5.18\))k(create)f(resp)s(ectiv)m(e) e(con)m(tributions)f(of)i(the)f(form)1219 651 y(e)1259 613 y Fs(\()p Fw(m)p Fs(+)p Fw(q)r Fs(\))p Fw(=)p Fs(4)1555 651 y FA(exp)1694 550 y Fq(n)1755 651 y Fx(\000)2109 589 y Fz(H)2192 556 y Fs(2)2185 614 y(0)p 1836 630 670 4 v 1836 717 a FA(16)p Fz(\033)1981 691 y Fs(2)2021 717 y Fz(h)2073 691 y Fs(2)2113 717 y Fz(M)2211 686 y Fs(2)2201 743 y(1)2250 717 y Fz(\037)2307 691 y Fs(\(2\))2402 717 y FA(\()p Fz(t)p FA(\))2515 550 y Fq(o)3429 651 y FA(\(5.24\))1219 914 y(e)1259 876 y Fs(\()p Fw(m)p Fs(+)p Fw(q)r Fs(\))p Fw(=)p Fs(4)1555 914 y FA(exp)1694 813 y Fq(n)1755 914 y Fx(\000)2063 852 y Fz(H)2146 819 y Fs(2)2139 877 y(1)p 1836 893 578 4 v 1836 1001 a FA(16)p Fz(\033)1981 975 y Fs(2)2021 1001 y Fz(M)2119 970 y Fs(2)2109 1027 y(1)2159 1001 y Fz(\037)2216 953 y Fs(\(2\))2216 1030 y Fw(C)2310 1001 y FA(\()p Fz(t)p FA(\))2423 813 y Fq(o)3429 914 y FA(\(5.25\))118 1210 y(to)22 b(the)f(probabilit)m(y)f(\(5.23\))r(.)38 b(The)20 b(third)g(and)g(fourth)g(term)h(only)g(cause)g(corrections)h (of)f(order)f Fz(h\037)3463 1177 y Fs(\(1\))3558 1210 y FA(\()p Fz(t)p FA(\))118 1339 y(and)30 b Fz(h\037)404 1291 y Fs(\(1\))404 1368 y Fw(C)498 1339 y FA(\()p Fz(t)p FA(\))h(in)e(the)i(exp)s(onen)m(t.)p 3595 1339 4 62 v 3599 1281 55 4 v 3599 1339 V 3653 1339 4 62 v 259 1526 a(No)m(w)c(Theorem)f(2.8)h(follo)m(ws)e(from)g(Prop)s(osition)f(5.4)j (b)m(y)f(using)f(a)h(partition)f(of)h(the)g(in)m(terv)-5 b(al)25 b([0)p Fz(;)15 b(t)p FA(])118 1639 y(in)m(to)30 b(smaller)f(in)m(terv)-5 b(als)30 b(of)g(length)g(\001)p Fz(")p FA(.)118 1914 y FB(References)160 2095 y FD([1])45 b(L.)27 b(Arnold.)37 b FC(R)l(andom)30 b(Dynamic)l(al)g(Systems)p FD(.)37 b(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1998.)160 2244 y([2])45 b(R.)d(Azencott.)79 b(P)n(etites)41 b(p)r(erturbations)g (al)n(\023)-39 b(eatoires)39 b(des)j(syst)n(\022)-39 b(emes)40 b(dynamiques:)65 b(d)n(\023)-39 b(ev)n(elopp)r(emen)n(ts)293 2344 y(asymptotiques.)36 b FC(Bul)t(l.)31 b(Sci.)f(Math.)h(\(2\))p FD(,)d(109:253{308,)23 b(1985.)160 2493 y([3])45 b(R.)28 b(Bellman.)36 b FC(Intr)l(o)l(duction)29 b(to)h(Matrix)g(A)n(nalysis)p FD(.)38 b(McGra)n(w{Hill,)26 b(New)i(Y)-7 b(ork,)27 b(1960.)160 2641 y([4])45 b(E.)27 b(Beno)-9 b(^)-32 b(\020t,)26 b(editor.)37 b FC(Dynamic)30 b(Bifur)l(c)l(ations)p FD(,)f(Berlin,)e(1991.)f (Springer-V)-7 b(erlag.)160 2790 y([5])45 b(N.)f(Berglund)e(and)i(B.)f (Gen)n(tz.)84 b(A)44 b(sample-paths)f(approac)n(h)e(to)i(noise-induced) g(sync)n(hronization:)293 2890 y(Sto)r(c)n(hastic)27 b(resonance)f(in)i(a)f(double-w)n(ell)g(p)r(oten)n(tial.)36 b(T)-7 b(o)28 b(app)r(ear)e(in)i(Ann.)g(Appl.)h(Probab.)293 2989 y(Av)-5 b(ailable)27 b(at)h Fb(http://arXiv.or)o(g/a)o(bs)o(/m)o (ath)o(.P)o(R/0)o(01)o(22)o(67)p FD(,)21 b(2000.)160 3138 y([6])45 b(N.)24 b(Berglund)f(and)h(B.)g(Gen)n(tz.)30 b(The)24 b(e\013ect)h(of)f(additiv)n(e)f(noise)g(on)h(dynamical)f(h)n (ysteresis.)29 b FC(Nonline)l(arity)p FD(,)293 3238 y(15\(3\):605{632,) 23 b(2002.)160 3387 y([7])45 b(N.)25 b(Berglund)f(and)g(B.)h(Gen)n(tz.) 32 b(P)n(ath)n(wise)23 b(description)h(of)h(dynamic)f(pitc)n(hfork)h (bifurcations)f(with)h(addi-)293 3486 y(tiv)n(e)d(noise.)28 b FC(Pr)l(ob)l(ab.)f(The)l(ory)g(R)l(elate)l(d)e(Fields)p FD(,)g(122\(3\):341{388,)19 b(2002.)27 b(DOI)c(10.1007/s004400)o(10)o (01)o(74)o(.)160 3635 y([8])45 b(H.)24 b(Crauel)e(and)h(F.)h(Flandoli.) 30 b(A)n(ttractors)22 b(for)h(random)f(dynamical)h(systems.)29 b FC(Pr)l(ob)l(ab.)f(The)l(ory)f(R)l(elate)l(d)293 3735 y(Fields)p FD(,)i(100\(3\):365{393,)23 b(1994.)160 3883 y([9])45 b(M.)33 b(V.)h(Da)n(y)-7 b(.)52 b(On)33 b(the)h(exp)r(onen)n (tial)e(exit)h(la)n(w)g(in)g(the)h(small)e(parameter)g(exit)h(problem.) 53 b FC(Sto)l(chastics)p FD(,)293 3983 y(8:297{323,)24 b(1983.)118 4132 y([10])45 b(M.)28 b(V.)g(Da)n(y)-7 b(.)36 b(On)27 b(the)h(exit)g(la)n(w)f(from)g(saddle)g(p)r(oin)n(ts.)37 b FC(Sto)l(chastic)30 b(Pr)l(o)l(c)l(ess.)h(Appl.)p FD(,)e(60:287{311,) 23 b(1995.)118 4281 y([11])45 b(N.)34 b(F)-7 b(enic)n(hel.)55 b(Geometric)34 b(singular)e(p)r(erturbation)i(theory)f(for)g(ordinary)f (di\013eren)n(tial)i(equations.)54 b FC(J.)293 4380 y(Di\013er)l (ential)30 b(Equations)p FD(,)e(31\(1\):53{98,)d(1979.)118 4529 y([12])45 b(W.)23 b(H.)f(Fleming)h(and)f(M.)h(R.)g(James.)k (Asymptotic)c(series)e(and)h(exit)h(time)g(probabilities.)k FC(A)n(nn.)e(Pr)l(ob)l(ab.)p FD(,)293 4629 y(20\(3\):1369{1384,)d (1992.)118 4777 y([13])45 b(M.)26 b(I.)h(F)-7 b(reidlin.)35 b(On)26 b(stable)h(oscillations)e(and)h(equilibriums)g(induced)h(b)n(y) f(small)g(noise.)35 b FC(J.)28 b(Stat.)h(Phys.)p FD(,)293 4877 y(103:283{300,)23 b(2001.)118 5026 y([14])45 b(M.)31 b(I.)h(F)-7 b(reidlin)31 b(and)g(A.)h(D.)g(W)-7 b(en)n(tzell.)48 b FC(R)l(andom)33 b(Perturb)l(ations)g(of)h(Dynamic)l(al)g(Systems)p FD(.)48 b(Springer-)293 5125 y(V)-7 b(erlag,)26 b(New)i(Y)-7 b(ork,)27 b(second)g(edition,)h(1998.)118 5274 y([15])45 b(I.)20 b(S.)h(Grad)l(\024)-37 b(ste)-9 b(\025)-32 b(\020n.)22 b(Application)e(of)g(A.)h(M.)f(Ly)n(apuno)n(v's)f(theory)g(of)h (stabilit)n(y)g(to)h(the)f(theory)g(of)g(di\013eren)n(tial)293 5374 y(equations)j(with)i(small)g(co)r(e\016cien)n(ts)f(in)h(the)f (deriv)-5 b(ativ)n(es.)31 b FC(Mat.)d(Sb)l(ornik)f(N.)g(S.)p FD(,)f(32\(74\):263{286,)20 b(1953.)118 5523 y([16])45 b(R.)c(Hab)r(erman.)76 b(Slo)n(wly)40 b(v)-5 b(arying)40 b(jump)i(and)f(transition)f(phenomena)g(asso)r(ciated)g(with)h (algebraic)293 5622 y(bifurcation)27 b(problems.)36 b FC(SIAM)30 b(J.)f(Appl.)i(Math.)p FD(,)e(37\(1\):69{106,)24 b(1979.)1844 5871 y FA(40)p eop %%Page: 41 41 41 40 bop 118 280 a FD([17])45 b(C.)19 b(K.)g(R.)g(T.)g(Jones.)j (Geometric)d(singular)e(p)r(erturbation)i(theory)-7 b(.)22 b(In)d FC(Dynamic)l(al)k(systems)f(\(Monte)l(c)l(atini)293 380 y(T)-6 b(erme,)30 b(1994\))p FD(,)g(pages)c(44{118.)f(Springer,)h (Berlin,)i(1995.)118 529 y([18])45 b(Y.)38 b(Kifer.)67 b(The)38 b(exit)g(problem)g(for)f(small)h(random)f(p)r(erturbations)g (of)h(dynamical)f(systems)g(with)i(a)293 629 y(h)n(yp)r(erb)r(olic)27 b(\014xed)g(p)r(oin)n(t.)37 b FC(Isr)l(ael)31 b(J.)e(Math.)p FD(,)h(40\(1\):74{96,)24 b(1981.)118 779 y([19])45 b(N.)29 b(R.)h(Leb)r(o)n(vitz)e(and)h(R.)g(J.)g(Sc)n(haar.)40 b(Exc)n(hange)27 b(of)i(stabilities)g(in)h(autonomous)e(systems.)40 b FC(Studies)31 b(in)293 878 y(Appl.)g(Math.)p FD(,)e(54\(3\):229{260,) 23 b(1975.)118 1028 y([20])45 b(N.)28 b(R.)f(Leb)r(o)n(vitz)g(and)g(R.) h(J.)f(Sc)n(haar.)35 b(Exc)n(hange)26 b(of)h(stabilities)h(in)f (autonomous)f(systems.)h(II.)h(Vertical)293 1127 y(bifurcation.)36 b FC(Studies)30 b(in)g(Appl.)h(Math.)p FD(,)e(56\(1\):1{50,)c(1976/77.) 118 1277 y([21])45 b(E.)18 b(F.)h(Mishc)n(henk)n(o)f(and)h(N.)g(K.)g (Rozo)n(v.)i FC(Di\013er)l(ential)h(e)l(quations)g(with)h(smal)t(l)f(p) l(ar)l(ameters)g(and)h(r)l(elaxation)293 1376 y(oscil)t(lations)p FD(.)78 b(Plen)n(um)40 b(Press,)i(New)f(Y)-7 b(ork,)43 b(1980.)74 b(T)-7 b(ranslated)40 b(from)g(the)h(Russian)f(b)n(y)h(F.)g (M.)f(C.)293 1476 y(Go)r(o)r(dsp)r(eed.)118 1625 y([22])45 b(A.)30 b(I.)g(Ne)-9 b(\025)-32 b(\020sh)n(tadt.)43 b(P)n(ersistence)29 b(of)h(stabilit)n(y)f(loss)g(for)h(dynamical)f(bifurcations)h(I.)44 b FC(Di\013er)l(ential)32 b(Equa-)293 1725 y(tions)p FD(,)c(23:1385{1391,)22 b(1987.)118 1874 y([23])45 b(A.)28 b(I.)g(Ne)-9 b(\025)-32 b(\020sh)n(tadt.)37 b(P)n(ersistence)26 b(of)i(stabilit)n(y)g(loss)f(for)h(dynamical)f(bifurcations)g(I)r(I.)39 b FC(Di\013er)l(ential)30 b(Equa-)293 1974 y(tions)p FD(,)e(24:171{176,)23 b(1988.)118 2123 y([24])45 b(L.)23 b(S.)g(P)n(on)n(try)n(agin)d(and)i(L.)h(V.)g(Ro)r(dygin.)29 b(Appro)n(ximate)22 b(solution)g(of)h(a)f(system)h(of)g(ordinary)e (di\013eren)n(tial)293 2223 y(equations)h(in)n(v)n(olving)g(a)h(small)f (parameter)g(in)i(the)f(deriv)-5 b(ativ)n(es.)29 b FC(Soviet)d(Math.)h (Dokl.)p FD(,)e(1:237{240,)20 b(1960.)118 2373 y([25])45 b(B.)40 b(Sc)n(hmalfu\031.)76 b(In)n(v)-5 b(arian)n(t)39 b(attracting)h(sets)g(of)h(nonlinear)e(sto)r(c)n(hastic)h(di\013eren)n (tial)h(equations.)74 b(In)293 2472 y(H.)31 b(Langer)f(and)h(V.)h (Nollau,)g(editors,)f FC(Markov)k(pr)l(o)l(c)l(esses)f(and)g(c)l(ontr)l (ol)f(the)l(ory)p FD(,)g(v)n(olume)d(54)h(of)g FC(Math.)293 2572 y(R)l(es.)p FD(,)c(pages)g(217{228,)d(Berlin,)j(1989.)f(Ak)-5 b(ademie-V)e(erlag.)36 b(Gau\031ig,)26 b(1988.)118 2721 y([26])45 b(A.)31 b(N.)g(Tihono)n(v.)44 b(Systems)31 b(of)g(di\013eren)n(tial)f(equations)g(con)n(taining)f(small)i (parameters)e(in)h(the)h(deriv)-5 b(a-)293 2821 y(tiv)n(es.)36 b FC(Mat.)31 b(Sb)l(ornik)f(N.)g(S.)p FD(,)e(31:575{586,)23 b(1952.)118 3194 y(Nils)28 b(Berglund)118 3294 y Fa(Dep)-6 b(ar)g(tment)33 b(of)f(Ma)-6 b(thema)g(tics,)32 b(ETH)f(Z)1672 3287 y(\177)1669 3294 y(urich)118 3394 y FD(ETH)d(Zen)n(trum,)f(8092)f (Z)r(\177)-44 b(uric)n(h,)26 b(Switzerland)118 3493 y FC(and)118 3593 y Fa(FR)n(UMAM,)k(CPT{CNRS)i(Luminy)118 3693 y FD(Case)27 b(907,)f(13288)f(Marseille)i(Cedex)g(9,)h(F)-7 b(rance)118 3792 y FC(and)118 3892 y Fa(PHYMA)g(T,)29 b(Universit)983 3885 y(\023)983 3892 y(e)j(de)g(Toulon)118 3991 y FD(B.)14 b(P)-7 b(.)27 b(132,)f(83957)g(La)h(Garde)g(Cedex,)g(F) -7 b(rance)118 4091 y FC(E-mail)31 b(addr)l(ess:)40 b Fb(berglund@cpt.un)o(iv-)o(mr)o(s.f)o(r)118 4290 y FD(Barbara)25 b(Gen)n(tz)118 4390 y Fa(Weierstra\031)34 b(Institute)d(f)n(or)h (Applied)g(Anal)-6 b(ysis)31 b(and)g(Stochastics)118 4490 y FD(Mohrenstra\031e)26 b(39,)g(10117)g(Berlin,)h(German)n(y)118 4589 y FC(E-mail)k(addr)l(ess:)40 b Fb(gentz@wias-berl)o(in.)o(de)1844 5871 y FA(41)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0203301608631--