Content-Type: multipart/mixed; boundary="-------------9910250341145" This is a multi-part message in MIME format. ---------------9910250341145 Content-Type: text/plain; name="99-406.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-406.keywords" Markov chains, passage times, recurrence criteria ---------------9910250341145 Content-Type: application/postscript; name="tmp.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="tmp.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: wedge.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o tmp.ps wedge %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.10.24:1535 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def 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1585 y Fp(;)8 b(x)791 1592 y Fv(2)810 1585 y Fz(\))k Fq(2)h Ft(Z)917 1566 y Fv(2)946 1585 y Fz(:)f Fp(x)997 1592 y Fv(1)1030 1585 y Fq(\025)h Fz(0)p Fp(;)8 b(x)1148 1592 y Fv(2)1179 1585 y Fq(\025)13 b Fz(0)p Fq(g)f Fz(=)h(\()p Ft(Z)1384 1566 y Fv(+)1410 1585 y Fz(\))1428 1566 y Fv(2)1448 1585 y Fp(;)24 1683 y Fz(that)h(will)j(b)q(e)f(called)h Fu(the)f(sp)n(ac)n(e)e Fz(in)j(the)e(sequel.)21 b(W)l(e)15 b(de\014ne)634 1781 y Fp(@)658 1788 y Fv(1)678 1781 y Ft(X)q Fz(=)e Fq(f)p FA(x)f Fz(=)h(\()p Fp(x)926 1788 y Fv(1)945 1781 y Fp(;)8 b Fz(0\);)g Fp(x)1054 1788 y Fv(1)1084 1781 y Fq(2)13 b Ft(Z)1161 1762 y Fv(+)1187 1781 y Fq(g)g(\022)f Ft(X)24 1879 y Fz(and)627 1977 y Fp(@)651 1984 y Fv(2)671 1977 y Ft(X)r Fz(=)h Fq(f)p FA(x)f Fz(=)h(\(0)p Fp(;)8 b(x)964 1984 y Fv(2)982 1977 y Fz(\);)g Fp(x)1047 1984 y Fv(2)1078 1977 y Fq(2)13 b Ft(Z)1154 1958 y Fv(+)1181 1977 y Fq(g)f(\022)h Ft(X)-8 b Fp(:)24 2090 y Fz(The)14 b(set)h Fp(@)s Ft(X)q Fz(=)e Fp(@)331 2097 y Fv(1)351 2090 y Ft(X)m Fq([)c Fp(@)455 2097 y Fv(2)475 2090 y Ft(X)s Fz(will)17 b(b)q(e)e(called)h Fu(the)g(b)n(oundaries)e Fz(and)1192 2054 y Fy(\016)1185 2094 y Ft(X)1230 2090 y Fz(=)f Ft(X)n Fq(n)8 b Fp(@)s Ft(X)s Fu(the)16 b(interior)f Fz(of)f(the)g(space.)24 2146 y(By)h(abuse)g(of)g(notation,)f(w)o(e)h(use)g(the)g(same)g(sym)o (b)q(ol,)g Ft(X)-9 b Fz(,)12 b(to)i(denote)i(the)f(quadran)o(t)f(\()p Ft(R)1559 2130 y Fv(+)1586 2146 y Fz(\))1604 2130 y Fv(2)1638 2146 y Fz(when)i(w)o(e)e(refer)24 2203 y(to)g(functions)j(de\014ned)g (on)e(the)h(space)f(b)q(ecause)i(w)o(e)e(are)g(in)o(terested)h(ev)o(en) o(tually)g(only)h(on)e(the)g(v)m(alues)i(these)24 2259 y(functions)f(tak)o(e)f(on)h(the)g(p)q(oin)o(ts)h(of)e(the)h(in)o (teger)g(lattice.)23 b(Th)o(us,)16 b Fp(f)j Fq(2)14 b Fp(C)1291 2243 y Fv(2)1310 2259 y Fz(\()p Ft(X)-8 b Fp(;)7 b Ft(R)-5 b Fz(\))o(,)13 b(for)i(instance,)i(will)g(mean)24 2316 y(a)e(t)o(wice)g(con)o(tin)o(uously)h(di\013eren)o(tiable)h(real)e (function)h Fp(f)21 b Fz(on)15 b(\()p Ft(R)1141 2299 y Fv(+)1168 2316 y Fz(\))1186 2299 y Fv(2)1205 2316 y Fz(.)61 2419 y(F)l(or)g(some)f(strictly)i(p)q(ositiv)o(e)g(in)o(teger)g Fp(L)c(>)h Fz(0)i(and)g(ev)o(ery)h FA(x)c Fq(2)h Ft(X)-8 b Fz(,)11 b(denote)16 b(b)o(y)639 2553 y Ft(A)668 2560 y Fo(x)708 2553 y Fz(=)756 2476 y Fn(\()813 2517 y Fq(f)p Fz(0)p Fp(;)8 b Fq(\001)g(\001)g(\001)k Fp(;)c(L)p Fq(g)40 b Fz(if)i FA(x)12 b Fq(2)h Fp(@)s Ft(X)813 2590 y Fq(f)p Fz(0)p Fq(g)180 b Fz(if)42 b FA(x)12 b Fq(2)1221 2554 y Fy(\016)1213 2594 y Ft(X)1246 2590 y Fp(:)24 2690 y Fz(A)k(state)g(of)h(the)g(system)f(will)i(b)q(e)g(parametrised)f(b)o(y) f Fp(u)f Fz(=)h(\()p FA(x)p Fz(;)8 b Fp(\013)p Fz(\))14 b(=)h(\()p Fp(x)1277 2697 y Fv(1)1297 2690 y Fp(;)8 b(x)1344 2697 y Fv(2)1363 2690 y Fz(;)g Fp(\013)p Fz(\))15 b(where)i FA(x)e Fz(=)h(\()p Fp(x)1717 2697 y Fv(1)1736 2690 y Fp(;)8 b(x)1783 2697 y Fv(2)1802 2690 y Fz(\))15 b Fq(2)h Ft(X)24 2746 y Fz(and,)f(for)f(this)i(particular)g FA(x)p Fz(,)e(the)h(co)q(ordinate)h Fp(\013)d Fq(2)g Ft(A)965 2753 y Fo(x)991 2746 y Fz(.)20 b(The)15 b(set)g(of)g(all)h(these)g (states,)636 2844 y Ft(U)t Fz(=)d Fq(f)p Fp(u)f Fz(=)h(\()p FA(x)p Fz(;)8 b Fp(\013)p Fz(\))j(:)i FA(x)f Fq(2)h Ft(X)-8 b Fz(;)7 b Fp(\013)j Fq(2)j Ft(A)1239 2851 y Fo(x)1266 2844 y Fq(g)p Fp(;)p eop %%Page: 3 3 3 2 bop 24 74 a FC(Markov)15 b(chains)g(in)g(a)g(we)n(dge)g(with)f (excitable)h(b)n(oundaries)933 b FB(3)24 169 y Fz(will)17 b(b)q(e)g(called)g Fu(the)g(c)n(on\014gur)n(ation)f(sp)n(ac)n(e)p Fz(;)f(that)g(is)h(to)f(sa)o(y)h Ft(U)7 b Fz(is)16 b(a)g(trivial)h (\014bre)f(bundle)i(with)e(base)g Ft(X)t Fz(and)24 226 y(\014bre)f Ft(A)157 233 y Fo(x)198 226 y Fz(o)o(v)o(er)f(eac)o(h)g FA(x)f Fq(2)f Ft(X)-8 b Fz(.)17 b(A)e(more)f(in)o(tuitiv)o(e)i(w)o(a)o (y)e(of)g(seeing)h(this)h(construction)e(is)i(to)e(sa)o(y)g(that)g(on)g (the)24 295 y(b)q(oundaries)i Fp(@)s Ft(X)t Fz(there)f(is)h(some)f(in)o (ternal)h(degrees)g(of)f(freedom)g(whic)o(h)h(are)f(absen)o(t)h(from)e (the)i(in)o(terior,)1876 258 y Fy(\016)1868 299 y Ft(X)1901 295 y Fz(,)24 351 y(of)e(the)i(space;)f(w)o(e)g(ma)o(y)f(think)i(of)f (these)g(degrees)h(of)f(freedom)g(as)f(colours.)73 432 y Fu(De\014nition)i(1.1)23 b(F)m(or)16 b Fp(u)c Fz(=)h(\()p Fp(x)586 439 y Fv(1)606 432 y Fp(;)8 b(x)653 439 y Fv(2)672 432 y Fz(;)g Fp(\013)p Fz(\))j Fq(2)i Ft(U)-5 b Fu(,)13 b(we)k(c)n(al)r(l)e Fz(canonical)i(pro)s(jections)f FA(X)g Fu(and)g Fp(A)p Fu(,)g(the)h(mappings)699 521 y FA(X)12 b Fz(:)g Ft(U)s Fq(!)5 b Ft(X)21 b Fu(and)33 b Fp(A)12 b Fz(:)h Ft(U)r Fq(!)5 b Ft(A)24 609 y Fu(de\014ne)n(d)15 b(by)462 697 y Fp(u)d Fq(7!)h FA(X)p Fz(\()p Fp(u)p Fz(\))f Fq(\021)h Fz(\()p Fp(x)764 704 y Fv(1)783 697 y Fp(;)8 b(x)830 704 y Fv(2)849 697 y Fz(\))k Fq(2)h Ft(X)22 b Fu(and)32 b Fp(u)13 b Fq(7!)g Fp(A)p Fz(\()p Fp(u)p Fz(\))f Fq(\021)h Fp(\013)g Fq(2)g Ft(A)c Fp(:)24 832 y Fz(W)l(e)16 b(denote)g(as)f(usual)i Fp(X)464 839 y Fv(1)483 832 y Fz(\()p Fp(u)p Fz(\))d(=)g Fp(x)634 839 y Fv(1)669 832 y Fz(and)j Fp(X)797 839 y Fv(2)816 832 y Fz(\()p Fp(u)p Fz(\))c(=)h Fp(x)966 839 y Fv(2)1002 832 y Fz(the)i(horizon)o(tal)g (and)g(v)o(ertical)h(comp)q(onen)o(ts)f(of)f(the)24 889 y(spatial)g(pro)s(jection.)73 1049 y(1.3.)20 b Fu(De\014nition)11 b(and)g(pr)n(op)n(erties)h(of)g(the)f(Markov)h(chain)28 b Fz(W)l(e)11 b(consider)g(a)f(discrete-time,)i(time-homogeneous,)24 1106 y(irreducible,)17 b(ap)q(erio)q(dic,)f Ft(U)-5 b Fz(-v)m(alued)13 b(Mark)o(o)o(v)f(c)o(hain,)j Fp(\030)g Fz(=)e(\()p Fp(\030)1077 1113 y Fs(n)1100 1106 y Fz(\))1118 1113 y Fs(n)p Fy(2)p Fm(N)1189 1106 y Fz(,)h(de\014ned)i(b)o(y)e(the)h (sto)q(c)o(hastic)f(transition)24 1162 y(matrix)794 1250 y FA(P)e Fz(=)h(\()p Fp(P)937 1258 y Fs(u;u)987 1248 y Fr(0)1001 1250 y Fz(\))1019 1258 y Fs(u;u)1069 1248 y Fr(0)1081 1258 y Fy(2)p Fm(U)1130 1250 y Fp(;)24 1338 y Fz(with)318 1427 y Fp(P)347 1434 y Fs(u;u)397 1425 y Fr(0)424 1427 y Fq(\021)g Fp(P)501 1436 y Fv(\()p Fs(x)535 1441 y Fl(1)552 1436 y Fs(;x)582 1441 y Fl(2)599 1436 y Fv(;)p Fs(\013)p Fv(\))p Fs(;)p Fv(\()p Fs(x)690 1425 y Fr(0)690 1447 y Fl(1)706 1436 y Fs(;x)736 1425 y Fr(0)736 1447 y Fl(2)753 1436 y Fv(;)p Fs(\013)786 1427 y Fr(0)796 1436 y Fv(\))825 1427 y Fz(=)g Ft(P)-5 b Fz(\()o Fp(\030)941 1434 y Fs(n)p Fv(+1)1019 1427 y Fz(=)13 b(\()p Fp(x)1111 1408 y Fy(0)1111 1438 y Fv(1)1131 1427 y Fp(;)8 b(x)1178 1408 y Fy(0)1178 1438 y Fv(2)1197 1427 y Fz(;)g Fp(\013)1247 1408 y Fy(0)1258 1427 y Fz(\))p Fq(j)p Fp(\030)1309 1434 y Fs(n)1344 1427 y Fz(=)13 b(\()p Fp(x)1436 1434 y Fv(1)1455 1427 y Fp(;)8 b(x)1502 1434 y Fv(2)1521 1427 y Fz(;)g Fp(\013)p Fz(\)\))p Fp(:)73 1515 y Fu(R)n(emark)17 b(1.1)23 b(It)17 b(is)f(evident)h(that)h(e)n(quivalently)e(to)h(our)h(\014br)n (e)e(bund)r(le)h(description,)h(we)f(c)n(an)f(extend)h(the)24 1571 y(c)n(on\014gur)n(ation)e(sp)n(ac)n(e)h(into)g Ft(X)o Fq(\002)10 b Ft(A)f Fu(,)19 b(wher)n(e)d Ft(A)25 b Fz(=)13 b Fq(f)p Fz(0)p Fp(;)8 b Fq(\001)g(\001)g(\001)k Fp(;)c(L)p Fq(g)p Fu(,)15 b(by)h(r)n(e)n(quiring)456 1674 y Fp(P)485 1684 y Fv(\()p Fs(x)519 1689 y Fl(1)536 1684 y Fs(;x)566 1689 y Fl(2)583 1684 y Fv(;)p Fs(\013)p Fv(\))p Fs(;)p Fv(\()p Fs(x)674 1673 y Fr(0)674 1695 y Fl(1)689 1684 y Fs(;x)719 1673 y Fr(0)719 1695 y Fl(2)736 1684 y Fv(;)p Fs(\013)769 1675 y Fr(0)780 1684 y Fv(\))809 1674 y Fz(=)c(0)33 b Fu(if)f Fz(\()p Fp(x)1016 1656 y Fy(0)1016 1686 y Fv(1)1036 1674 y Fp(;)8 b(x)1083 1656 y Fy(0)1083 1686 y Fv(2)1101 1674 y Fz(\))13 b Fq(2)1182 1638 y Fy(\016)1175 1678 y Ft(X)1240 1674 y Fu(and)33 b Fp(\013)1374 1656 y Fy(0)1398 1674 y Fq(6)p Fz(=)13 b(0)p Fp(:)61 1810 y Fz(The)i(transition)h (probabilities)h(of)e(the)g(Mark)o(o)o(v)f(c)o(hain)i(ha)o(v)o(e)f(v)m (arious)g(prop)q(erties.)73 1891 y Fu(De\014nition)h(1.2)23 b([L)n(ower)16 b(b)n(ounde)n(dness)g(of)h(jumps)g(\(LBJ\)])e(We)i(say)g (the)g(chain)f(has)h Fz(the)f(LBJ)h(prop)q(ert)o(y)24 1948 y Fu(if)77 2090 y(1.)23 b(F)m(or)17 b(jumps)h(starting)f(in)f(the) i(interior)f(of)g(the)h(sp)n(ac)n(e,)f FA(x)d Fq(2)1187 2053 y Fy(\016)1179 2094 y Ft(X)1212 2090 y Fu(,)k(the)f(tr)n(ansition) f(pr)n(ob)n(ability)h(vanishes)137 2146 y(for)h(westb)n(ound)g(or)g (southb)n(ound)g(jumps)g(of)g(length)f(mor)n(e)h(than)g(1)g(and)g(for)g (entering)f(the)h(b)n(oundary)137 2202 y(thr)n(ough)f(c)n(olours)g (di\013er)n(ent)e(fr)n(om)i(0,)f Fz(i.e.)137 2343 y Fp(P)166 2353 y Fv(\()p Fs(x)200 2358 y Fl(1)217 2353 y Fs(;x)247 2358 y Fl(2)264 2353 y Fv(;)p Fs(\013)p Fv(\))p Fs(;)p Fv(\()p Fs(x)355 2341 y Fr(0)355 2363 y Fl(1)371 2353 y Fs(;x)401 2341 y Fr(0)401 2363 y Fl(2)418 2353 y Fv(;)p Fs(\013)451 2343 y Fr(0)462 2353 y Fv(\))490 2343 y Fz(=)538 2250 y Fn(8)538 2291 y(<)538 2373 y(:)599 2287 y Fz(0)42 b Fu(if)f Fz(\()p Fp(x)777 2294 y Fv(1)809 2287 y Fq(\025)13 b Fz(1)p Fp(;)8 b(x)927 2294 y Fv(2)958 2287 y Fq(\025)13 b Fz(1\))32 b Fu(and)h Fp(x)1210 2270 y Fy(0)1210 2299 y Fv(1)1239 2287 y Fq(\000)11 b Fp(x)1311 2294 y Fv(1)1343 2287 y Fp(<)i Fq(\000)p Fz(1)599 2343 y(0)42 b Fu(if)f Fz(\()p Fp(x)777 2350 y Fv(1)809 2343 y Fq(\025)13 b Fz(1)p Fp(;)8 b(x)927 2350 y Fv(2)958 2343 y Fq(\025)13 b Fz(1\))32 b Fu(and)h Fp(x)1210 2327 y Fy(0)1210 2355 y Fv(2)1239 2343 y Fq(\000)11 b Fp(x)1311 2350 y Fv(2)1343 2343 y Fp(<)i Fq(\000)p Fz(1)599 2400 y(0)42 b Fu(if)f Fz(\()p Fp(x)777 2407 y Fv(1)809 2400 y Fz(=)13 b(1)p Fp(;)8 b(x)927 2383 y Fy(0)927 2412 y Fv(1)958 2400 y Fz(=)13 b(0)p Fp(;)8 b(x)1076 2407 y Fv(2)1107 2400 y Fq(6)p Fz(=)13 b(0)p Fp(;)8 b(\013)1228 2383 y Fy(0)1252 2400 y Fq(6)p Fz(=)13 b(0\))32 b Fu(or)h Fz(\()p Fp(x)1492 2407 y Fv(2)1524 2400 y Fz(=)13 b(1)p Fp(;)8 b(x)1642 2383 y Fy(0)1642 2412 y Fv(2)1673 2400 y Fz(=)13 b(0)p Fp(;)8 b(x)1791 2407 y Fv(1)1822 2400 y Fq(6)p Fz(=)13 b(0)p Fp(;)8 b(\013)1943 2383 y Fy(0)1967 2400 y Fq(6)p Fz(=)13 b(0\))p Fp(:)77 2503 y Fu(2.)23 b(F)m(or)g(jumps)g(starting)f (on)h(the)g(b)n(oundaries,)h(westb)n(ound)e(\(horizontal\))g(or)h (southb)n(ound)h(\(vertic)n(al\))137 2559 y(jumps)17 b(ar)n(e)f(uniformly)g(b)n(ounde)n(d,)g Fz(i.e.)21 b Fu(ther)n(e)c(exists)e Fp(K)1098 2566 y Fv(LBJ)1178 2559 y Fp(>)e Fz(0)j Fu(such)g(that)346 2648 y Fp(P)375 2657 y Fv(\()p Fs(x)409 2662 y Fl(1)426 2657 y Fs(;)p Fv(0;)p Fs(\013)p Fv(\))p Fs(;)p Fv(\()p Fs(x)545 2646 y Fr(0)545 2668 y Fl(1)560 2657 y Fs(;x)590 2646 y Fr(0)590 2668 y Fl(2)607 2657 y Fv(;)p Fs(\013)640 2648 y Fr(0)651 2657 y Fv(\))679 2648 y Fz(=)d(0)32 b Fu(if)h Fp(x)869 2629 y Fy(0)869 2659 y Fv(1)898 2648 y Fq(\000)11 b Fp(x)970 2655 y Fv(1)1002 2648 y Fp(<)i Fq(\000)p Fp(K)1124 2655 y Fv(LBJ)1191 2648 y Fp(;)8 b Fu(for)32 b FA(x)13 b Fq(2)f Fp(@)1407 2655 y Fv(1)1427 2648 y Ft(X)-8 b Fp(;)7 b(x)1509 2655 y Fv(1)1539 2648 y Fp(>)13 b(K)1626 2655 y Fv(LBJ)1692 2648 y Fp(;)137 2736 y Fu(and)346 2824 y Fp(P)375 2834 y Fv(\(0)p Fs(;x)437 2839 y Fl(2)453 2834 y Fv(;)p Fs(\013)p Fv(\))p Fs(;)p Fv(\()p Fs(x)544 2822 y Fr(0)544 2844 y Fl(1)560 2834 y Fs(;x)590 2822 y Fr(0)590 2844 y Fl(2)607 2834 y Fv(;)p Fs(\013)640 2824 y Fr(0)651 2834 y Fv(\))679 2824 y Fz(=)g(0)32 b Fu(if)h Fp(x)869 2805 y Fy(0)869 2835 y Fv(2)898 2824 y Fq(\000)11 b Fp(x)970 2831 y Fv(2)1002 2824 y Fp(<)i Fq(\000)p Fp(K)1124 2831 y Fv(LBJ)1191 2824 y Fp(;)8 b Fu(for)32 b FA(x)13 b Fq(2)f Fp(@)1407 2831 y Fv(2)1427 2824 y Ft(X)-8 b Fp(;)7 b(x)1509 2831 y Fv(2)1539 2824 y Fp(>)13 b(K)1626 2831 y Fv(LBJ)1692 2824 y Fp(:)p eop %%Page: 4 4 4 3 bop 24 74 a FB(4)1327 b(M.)13 b(Menshik)o(o)o(v)h(and)g(D.)f(P)o (etritis)73 169 y Fu(R)n(emark)k(1.2)23 b(Notic)n(e)15 b(that)i(the)f(c)n(ondition)f(of)h(westb)n(ound)g(or)g(southb)n(ound)h (jumps)f(b)n(ounde)n(d)g(by)g(1)g(in)f(the)24 226 y(internal)g(sp)n(ac) n(e)g(is)h(imp)n(ortant)h(and)f(c)n(annot)g(b)n(e)f(r)n(elaxe)n(d)h(in) g(our)h(appr)n(o)n(ach.)73 307 y(De\014nition)f(1.3)23 b([Partial)16 b(sp)n(atial)g(homo)n(geneity)g(\(PSH\)])e(The)i(Markov)g (chain)g(has)g Fz(the)f(PSH)h(prop)q(ert)o(y)24 371 y Fu(if)j(for)h Fp(\013;)8 b(\013)226 355 y Fy(0)256 371 y Fq(2)19 b Ft(A)9 b Fu(,)23 b(ther)n(e)d(exist)f(functions)g Fp(p)822 379 y Fs(\013\013)868 370 y Fr(0)899 371 y Fz(:)f Ft(Z)-5 b Fq(!)m Fz([0)p Fp(;)8 b Fz(1])p Fu(,)19 b Fp(q)1159 379 y Fs(\013\013)1205 370 y Fr(0)1236 371 y Fz(:)f Ft(Z)-5 b Fq(!)n Fz([0)p Fp(;)8 b Fz(1])p Fu(,)18 b Fp(r)1498 347 y Fv(\(1\))1497 386 y Fs(\013\013)1543 377 y Fr(0)1574 371 y Fz(:)g Ft(Z)1638 355 y Fv(+)1677 371 y Fq(\002)13 b Ft(Z)-5 b Fq(!)n Fz([0)p Fp(;)8 b Fz(1])p Fu(,)24 439 y Fp(r)46 415 y Fv(\(2\))45 454 y Fs(\013\013)91 445 y Fr(0)127 439 y Fz(:)23 b Ft(Z)o Fq(\002)15 b Ft(Z)291 423 y Fv(+)325 439 y Fq(!)q Fz([0)p Fp(;)8 b Fz(1])p Fu(,)21 b(and)i Fp(r)615 446 y Fv(0)p Fs(;)p Fv(0)685 439 y Fz(:)h Ft(Z)o Fq(\002)15 b Ft(Z)-6 b Fq(!)n Fz([0)p Fp(;)8 b Fz(1])20 b Fu(such)j(that)g(the)g(sto)n(chastic)e(matrix)j Fp(P)1698 447 y Fs(u;u)1748 438 y Fr(0)1784 439 y Fu(c)n(an)e(b)n(e)24 496 y(de)n(c)n(omp)n(ose)n(d)15 b(into)238 745 y Fp(P)267 755 y Fv(\()p Fs(x)301 760 y Fl(1)318 755 y Fs(;x)348 760 y Fl(2)365 755 y Fv(;)p Fs(\013)p Fv(\))p Fs(;)p Fv(\()p Fs(x)456 743 y Fr(0)456 765 y Fl(1)472 755 y Fs(;x)502 743 y Fr(0)502 765 y Fl(2)519 755 y Fv(;)p Fs(\013)552 745 y Fr(0)563 755 y Fv(\))591 745 y Fz(=)639 543 y Fn(8)639 584 y(>)639 597 y(>)639 611 y(>)639 624 y(>)639 638 y(>)639 652 y(>)639 665 y(>)639 679 y(>)639 693 y(<)639 774 y(>)639 788 y(>)639 802 y(>)639 815 y(>)639 829 y(>)639 843 y(>)639 856 y(>)639 870 y(>)639 884 y(:)700 579 y Fp(p)723 587 y Fs(\013\013)769 577 y Fr(0)782 579 y Fz(\()p Fp(x)826 563 y Fy(0)826 591 y Fv(1)855 579 y Fq(\000)c Fp(x)927 586 y Fv(1)947 579 y Fz(\))195 b Fu(if)33 b FA(x)p Fp(;)8 b FA(x)1298 563 y Fy(0)1320 579 y Fq(2)13 b Fp(@)1387 586 y Fv(1)1407 579 y Ft(X)-8 b Fp(;)7 b(x)1489 586 y Fv(1)1519 579 y Fp(>)13 b(K)1606 586 y Fv(LBJ)700 636 y Fp(q)720 643 y Fs(\013\013)766 634 y Fr(0)779 636 y Fz(\()p Fp(x)823 619 y Fy(0)823 648 y Fv(2)853 636 y Fq(\000)d Fp(x)924 643 y Fv(2)944 636 y Fz(\))198 b Fu(if)33 b FA(x)p Fp(;)8 b FA(x)1298 619 y Fy(0)1320 636 y Fq(2)13 b Fp(@)1387 643 y Fv(2)1407 636 y Ft(X)-8 b Fp(;)7 b(x)1489 643 y Fv(2)1519 636 y Fp(>)13 b(K)1606 643 y Fv(LBJ)700 708 y Fp(r)722 685 y Fv(\(1\))721 721 y Fs(\013)p Fv(0)769 708 y Fz(\()p Fp(x)813 692 y Fy(0)813 721 y Fv(1)843 708 y Fq(\000)d Fp(x)914 715 y Fv(1)934 708 y Fp(;)e(x)981 692 y Fy(0)981 721 y Fv(2)1010 708 y Fq(\000)i Fp(x)1081 715 y Fv(2)1101 708 y Fz(\))41 b Fu(if)33 b FA(x)12 b Fq(2)h Fp(@)1328 715 y Fv(1)1348 708 y Ft(X)-8 b Fp(;)7 b FA(x)1431 692 y Fy(0)1453 708 y Fq(2)1503 672 y Fy(\016)1496 712 y Ft(X)1528 708 y Fp(;)h(\013)1578 692 y Fy(0)1602 708 y Fz(=)13 b(0)700 781 y Fp(r)722 757 y Fv(\(2\))721 794 y Fs(\013)p Fv(0)769 781 y Fz(\()p Fp(x)813 765 y Fy(0)813 793 y Fv(1)843 781 y Fq(\000)d Fp(x)914 788 y Fv(1)934 781 y Fp(;)e(x)981 765 y Fy(0)981 793 y Fv(2)1010 781 y Fq(\000)i Fp(x)1081 788 y Fv(2)1101 781 y Fz(\))41 b Fu(if)33 b FA(x)12 b Fq(2)h Fp(@)1328 788 y Fv(2)1348 781 y Ft(X)-8 b Fp(;)7 b FA(x)1431 765 y Fy(0)1453 781 y Fq(2)1503 745 y Fy(\016)1496 785 y Ft(X)1528 781 y Fp(;)h(\013)1578 765 y Fy(0)1602 781 y Fz(=)13 b(0)700 854 y Fp(r)721 861 y Fv(00)758 854 y Fz(\()p Fp(x)802 837 y Fy(0)802 866 y Fv(1)832 854 y Fq(\000)d Fp(x)903 861 y Fv(1)923 854 y Fp(;)e(x)970 837 y Fy(0)970 866 y Fv(2)999 854 y Fq(\000)i Fp(x)1070 861 y Fv(2)1090 854 y Fz(\))52 b Fu(if)33 b FA(x)12 b Fq(2)1312 818 y Fy(\016)1304 858 y Ft(X)700 910 y Fz(0)437 b Fu(otherwise.)73 997 y(R)n(emark)17 b(1.3)23 b(The)g(existenc)n(e)f(of)i(function)f Fp(r)892 1004 y Fv(00)929 997 y Fz(\()p Fq(\001)p Fp(;)8 b Fq(\001)p Fz(\))21 b Fu(me)n(ans)i(c)n(omplete)g(sp)n(atial)g(homo)n (geneity)g(in)g(the)24 1053 y(interior)17 b(of)h(the)f(quadr)n(ant,)i (the)e(existenc)n(e)f(of)i(functions)f Fp(p)1074 1061 y Fs(\013\013)1120 1051 y Fr(0)1132 1053 y Fz(\()p Fq(\001)p Fz(\))f Fu(and)i Fp(q)1307 1061 y Fs(\013\013)1353 1051 y Fr(0)1366 1053 y Fz(\()p Fq(\001)p Fz(\))e Fu(me)n(ans)h(p)n(artial)g (horizontal)24 1109 y(\(r)n(esp.)22 b(vertic)n(al\))g(homo)n(geneity)h (on)g(the)h(b)n(oundary)g Fp(@)993 1116 y Fv(1)1012 1109 y Ft(X)12 b Fu(\(r)n(esp.)23 b Fp(@)1228 1116 y Fv(2)1248 1109 y Ft(X)-9 b Fu(\))20 b(far)k(fr)n(om)f(the)h(origin)f(\()p Fz(i.e.)41 b Fu(for)24 1166 y Fq(k)p Fp(x)p Fq(k)12 b Fp(>)h(K)195 1173 y Fv(LBJ)261 1166 y Fu(.\))73 1247 y(De\014nition)j(1.4)23 b([Moment)15 b(b)n(ounde)n(dness)f(\()p Fp(\015)s Fu(-MB\)])f(We)i(say)h(the)f(Markov)h(chain)f(has)g Fz(the)f Fp(\015)s Fz(-MB)f(prop-)24 1304 y(ert)o(y)i Fu(if)h(ther)n(e)h(exist)f(a)g Fp(\015)f(>)e Fz(2)j Fu(and)g(a)g(c)n (onstant)g Fp(K)895 1311 y Fv(MB)966 1304 y Fz(=)d Fp(K)1053 1311 y Fv(MB)1112 1304 y Fz(\()p Fp(\015)s Fz(\))e Fp(>)i Fz(0)j Fu(such)h(that,)g(for)f(every)g Fp(u)d Fq(2)g Ft(U)-5 b Fu(,)576 1391 y Ft(E)s Fz(\()p Fq(k)p FA(X)p Fz(\()p Fp(\030)723 1398 y Fs(n)p Fv(+1)793 1391 y Fz(\))10 b Fq(\000)g FA(X)p Fz(\()p Fp(\030)944 1398 y Fs(n)967 1391 y Fz(\))p Fq(k)1008 1372 y Fs(\015)1029 1391 y Fq(j)p Fp(\030)1062 1398 y Fs(n)1098 1391 y Fz(=)j Fp(u)p Fz(\))f Fp(<)h(K)1289 1398 y Fv(MB)1348 1391 y Fp(;)24 1478 y Fu(wher)n(e)j Fq(k)10 b(\001)g(k)16 b Fu(denotes)f(the)i(Euclide)n(an)e (norm)i(in)f Ft(R)907 1462 y Fv(2)924 1478 y Fu(.)73 1560 y(De\014nition)g(1.5)23 b([Zer)n(o)16 b(drift)h(pr)n(op)n(erty)f (\(ZD\)])g(We)g(say)g(that)h(the)g(chain)f(has)g(zer)n(o)g(drift)g(in)g (the)g(interior)24 1631 y(of)g(the)h(sp)n(ac)n(e)e(if,)h(for)h Fp(u)12 b Fz(=)h(\()p Fp(x)532 1638 y Fv(1)552 1631 y Fp(;)8 b(x)599 1638 y Fv(2)618 1631 y Fz(;)g(0\))14 b Fu(and)j Fz(\()p Fp(x)827 1638 y Fv(1)846 1631 y Fp(;)8 b(x)893 1638 y Fv(2)912 1631 y Fz(\))k Fq(2)993 1595 y Fy(\016)985 1635 y Ft(X)1018 1631 y Fu(,)291 1719 y Ft(E)s Fz(\()p Fp(X)375 1726 y Fv(1)397 1719 y Fz(\()p Fp(\030)435 1726 y Fs(n)p Fv(+1)503 1719 y Fz(\))e Fq(\000)g Fp(X)614 1726 y Fv(1)634 1719 y Fz(\()p Fp(\030)672 1726 y Fs(n)695 1719 y Fz(\)\))p Fq(j)p Fp(\030)764 1726 y Fs(n)799 1719 y Fz(=)j Fp(u)p Fz(\))f(=)h Ft(E)r Fz(\()p Fp(X)1034 1726 y Fv(2)1056 1719 y Fz(\()p Fp(\030)1094 1726 y Fs(n)p Fv(+1)1162 1719 y Fz(\))d Fq(\000)h Fp(X)1274 1726 y Fv(2)1293 1719 y Fz(\()p Fp(\030)1331 1726 y Fs(n)1354 1719 y Fz(\)\))p Fq(j)p Fp(\030)1423 1726 y Fs(n)1458 1719 y Fz(=)i Fp(u)p Fz(\))f(=)h(0)p Fp(:)61 1853 y Fz(W)l(e)i (de\014ne)i(\014nally)f(conditional)h(second)f(momen)o(ts)450 1940 y Fp(\025)477 1947 y Fv(1)538 1940 y Fz(=)42 b Ft(E)r Fz(\()p Fp(X)698 1947 y Fv(1)720 1940 y Fz(\()p Fp(\030)758 1947 y Fs(n)p Fv(+1)826 1940 y Fz(\))10 b Fq(\000)h Fp(X)938 1947 y Fv(1)957 1940 y Fz(\()p Fp(\030)995 1947 y Fs(n)1018 1940 y Fz(\)\))1054 1922 y Fv(2)1073 1940 y Fq(j)p Fp(\030)1106 1947 y Fs(n)1142 1940 y Fz(=)i Fp(u)p Fz(\))538 2016 y(=)662 1973 y Fn(X)615 2074 y Fv(\()p Fs(x)649 2063 y Fr(0)649 2085 y Fl(1)666 2074 y Fs(;x)696 2063 y Fr(0)696 2085 y Fl(2)713 2074 y Fv(\))p Fy(2)p Fm(X)781 2016 y Fp(r)802 2023 y Fv(00)839 2016 y Fz(\()p Fp(x)883 1997 y Fy(0)883 2027 y Fv(1)913 2016 y Fq(\000)d Fp(x)984 2023 y Fv(1)1004 2016 y Fp(;)e(x)1051 1997 y Fy(0)1051 2027 y Fv(2)1080 2016 y Fq(\000)i Fp(x)1151 2023 y Fv(2)1171 2016 y Fz(\)\()p Fp(x)1233 1997 y Fy(0)1233 2027 y Fv(1)1262 2016 y Fq(\000)h Fp(x)1334 2023 y Fv(1)1353 2016 y Fz(\))1371 1997 y Fv(2)1403 2016 y Fq(\025)i Fz(0;)450 2195 y Fp(\025)477 2202 y Fv(2)538 2195 y Fz(=)42 b Ft(E)r Fz(\()p Fp(X)698 2202 y Fv(2)720 2195 y Fz(\()p Fp(\030)758 2202 y Fs(n)p Fv(+1)826 2195 y Fz(\))10 b Fq(\000)h Fp(X)938 2202 y Fv(2)957 2195 y Fz(\()p Fp(\030)995 2202 y Fs(n)1018 2195 y Fz(\)\))1054 2177 y Fv(2)1073 2195 y Fq(j)p Fp(\030)1106 2202 y Fs(n)1142 2195 y Fz(=)i Fp(u)p Fz(\))538 2271 y(=)662 2228 y Fn(X)615 2329 y Fv(\()p Fs(x)649 2317 y Fr(0)649 2339 y Fl(1)666 2329 y Fs(;x)696 2317 y Fr(0)696 2339 y Fl(2)713 2329 y Fv(\))p Fy(2)p Fm(X)781 2271 y Fp(r)802 2278 y Fv(00)839 2271 y Fz(\()p Fp(x)883 2252 y Fy(0)883 2282 y Fv(1)913 2271 y Fq(\000)d Fp(x)984 2278 y Fv(1)1004 2271 y Fp(;)e(x)1051 2252 y Fy(0)1051 2282 y Fv(2)1080 2271 y Fq(\000)i Fp(x)1151 2278 y Fv(2)1171 2271 y Fz(\)\()p Fp(x)1233 2252 y Fy(0)1233 2282 y Fv(2)1262 2271 y Fq(\000)h Fp(x)1334 2278 y Fv(2)1353 2271 y Fz(\))1371 2252 y Fv(2)1403 2271 y Fq(\025)i Fz(0;)24 2411 y(and)376 2498 y Fp(\024)42 b Fz(=)g Ft(E)r Fz(\()p Fp(X)604 2505 y Fv(1)626 2498 y Fz(\()p Fp(\030)664 2505 y Fs(n)p Fv(+1)732 2498 y Fz(\))10 b Fq(\000)h Fp(X)844 2505 y Fv(1)863 2498 y Fz(\()p Fp(\030)901 2505 y Fs(n)924 2498 y Fz(\)\)\()p Fp(X)1016 2505 y Fv(2)1035 2498 y Fz(\()p Fp(\030)1073 2505 y Fs(n)p Fv(+1)1141 2498 y Fz(\))e Fq(\000)i Fp(X)1252 2505 y Fv(2)1271 2498 y Fz(\()p Fp(\030)1309 2505 y Fs(n)1332 2498 y Fz(\)\))p Fq(j)p Fp(\030)1401 2505 y Fs(n)1436 2498 y Fz(=)i Fp(u)p Fz(\))444 2574 y(=)568 2531 y Fn(X)521 2632 y Fv(\()p Fs(x)555 2620 y Fr(0)555 2642 y Fl(1)572 2632 y Fs(;x)602 2620 y Fr(0)602 2642 y Fl(2)619 2632 y Fv(\))p Fy(2)p Fm(X)688 2574 y Fp(r)709 2581 y Fv(00)746 2574 y Fz(\()p Fp(x)790 2555 y Fy(0)790 2585 y Fv(1)819 2574 y Fq(\000)d Fp(x)890 2581 y Fv(1)910 2574 y Fp(;)e(x)957 2555 y Fy(0)957 2585 y Fv(2)986 2574 y Fq(\000)j Fp(x)1058 2581 y Fv(2)1077 2574 y Fz(\)\()p Fp(x)1139 2555 y Fy(0)1139 2585 y Fv(1)1168 2574 y Fq(\000)g Fp(x)1240 2581 y Fv(1)1260 2574 y Fz(\)\()p Fp(x)1322 2555 y Fy(0)1322 2585 y Fv(2)1351 2574 y Fq(\000)f Fp(x)1422 2581 y Fv(2)1442 2574 y Fz(\))i Fq(2)h Ft(R)n Fp(:)73 2716 y Fu(De\014nition)j(1.6)23 b([Positive)16 b(de\014niteness)e(of)i(the)h(c)n(ovarianc)n(e)f (\(PDC\)])f(We)i(say)f(the)h(Markov)f(chain)g(has)24 2799 y Fz(the)f(PDC)g(prop)q(ert)o(y)h Fu(if)g(the)g(matrix)668 2735 y Fn(\022)722 2771 y Fp(\025)749 2778 y Fv(1)820 2771 y Fp(\024)732 2827 y(\024)52 b(\025)837 2834 y Fv(2)877 2735 y Fn(\023)927 2799 y Fu(is)16 b(p)n(ositive)f(de\014nite,)h(i.e.)k Fp(\025)1425 2806 y Fv(1)1445 2799 y Fp(\025)1472 2806 y Fv(2)1501 2799 y Fq(\000)10 b Fp(\024)1572 2783 y Fv(2)1605 2799 y Fq(\025)j Fz(0)p Fu(.)p eop %%Page: 5 5 5 4 bop 24 74 a FC(Markov)15 b(chains)g(in)g(a)g(we)n(dge)g(with)f (excitable)h(b)n(oundaries)933 b FB(5)73 169 y Fz(1.4.)20 b Fu(Drifts)g(and)g(line)n(ar)f(tr)n(ansformation)g(of)h(the)g(lattic)n (e)28 b Fz(In)20 b(order)f(to)f(state)g(our)h(main)h(results,)g(t)o(w)o (o)24 226 y(additional)k(notions)f(are)f(needed.)43 b(Consider)24 b Ft(U)920 209 y Fv(\(1\))987 226 y Fz(and)f Ft(U)1119 209 y Fv(\(2)o(\))1186 226 y Fz(the)f(t)o(w)o(o)g(con\014guration)h (spaces)f(whose)24 282 y(spatial)f(comp)q(onen)o(ts)g(are)g(resp)q (ectiv)o(ely)i(the)e(upp)q(er)h(half)f(plane)h Ft(X)1246 266 y Fv(\(1)o(\))1313 282 y Fz(=)g Ft(Z)o Fq(\002)14 b Ft(Z)1497 266 y Fv(+)1545 282 y Fz(and)21 b(the)g(righ)o(t)g(half)24 339 y(plane)c Ft(X)182 322 y Fv(\()o(2\))240 339 y Fz(=)e Ft(Z)323 322 y Fv(+)360 339 y Fq(\002)c Ft(Z)-12 b Fz(.)19 b(The)e(b)q(oundaries)g(of)f(these)g(spaces)h(are)f(resp)q(ectiv)o(ely) h Fp(@)s Ft(X)1498 322 y Fv(\(1)o(\))1556 339 y Fz(=)e Fp(@)1630 346 y Fv(1)1650 339 y Ft(X)1686 322 y Fv(\()o(1\))1744 339 y Fz(=)g Ft(Z)p Fz(and)24 400 y Fp(@)s Ft(X)86 384 y Fv(\(2)o(\))143 400 y Fz(=)e Fp(@)215 407 y Fv(2)235 400 y Ft(X)271 384 y Fv(\()o(2\))327 400 y Fz(=)g Ft(Z)-12 b Fz(.)16 b(Consider)e(no)o(w)f(t)o(w)o(o)f(mo)q(di\014ed)j(Mark)o(o)o (v)c(c)o(hains)j(\()p Fp(\030)1326 377 y Fv(\(1\))1324 406 y Fs(n)1373 400 y Fz(\))1391 407 y Fs(n)p Fy(2)p Fm(N)1475 400 y Fz(and)g(\()p Fp(\030)1602 377 y Fv(\(2\))1600 406 y Fs(n)1649 400 y Fz(\))1667 407 y Fs(n)p Fy(2)p Fm(N)1751 400 y Fz(ev)o(olving)24 457 y(resp)q(ectiv)o(ely)i(in)g Ft(U)360 440 y Fv(\(1)o(\))419 457 y Fz(and)f Ft(U)543 440 y Fv(\(2\))602 457 y Fz(and)h(ha)o(ving)f(complete,)g(rather)g (than)g(just)f(partial,)h(homogeneit)o(y)g(in)h(the)24 519 y(horizon)o(tal,)f(resp)q(ectiv)o(ely)i(the)e(v)o(ertical,)g (directions.)21 b(Therefore,)15 b(the)g(transtion)g(matrix)g(for)g(\()p Fp(\030)1725 495 y Fv(\(1\))1723 525 y Fs(n)1771 519 y Fz(\))1789 526 y Fs(n)p Fy(2)p Fm(N)1875 519 y Fz(is)145 717 y Fp(P)180 693 y Fv(\(1\))174 734 y(\()p Fs(x)208 739 y Fl(1)225 734 y Fs(;x)255 739 y Fl(2)272 734 y Fv(;)p Fs(\013)p Fv(\))p Fs(;)p Fv(\()p Fs(x)363 722 y Fr(0)363 744 y Fl(1)379 734 y Fs(;x)409 722 y Fr(0)409 744 y Fl(2)426 734 y Fv(;)p Fs(\013)459 724 y Fr(0)469 734 y Fv(\))498 717 y Fz(=)546 569 y Fn(8)546 610 y(>)546 624 y(>)546 637 y(>)546 651 y(>)546 664 y(<)546 746 y(>)546 760 y(>)546 774 y(>)546 787 y(>)546 801 y(:)607 605 y Fp(p)630 613 y Fs(\013\013)676 603 y Fr(0)688 605 y Fz(\()p Fp(x)732 588 y Fy(0)732 617 y Fv(1)762 605 y Fq(\000)10 b Fp(x)833 612 y Fv(1)853 605 y Fz(\))196 b(if)31 b FA(x)p Fp(;)8 b FA(x)1201 588 y Fy(0)1223 605 y Fq(2)13 b Fp(@)1290 612 y Fv(1)1310 605 y Ft(X)1346 588 y Fv(\(1)o(\))607 689 y Fp(r)629 665 y Fv(\(1\))628 702 y Fs(\013)p Fv(0)676 689 y Fz(\()p Fp(x)720 673 y Fy(0)720 702 y Fv(1)749 689 y Fq(\000)e Fp(x)821 696 y Fv(1)840 689 y Fp(;)d(x)887 673 y Fy(0)887 702 y Fv(2)916 689 y Fq(\000)j Fp(x)988 696 y Fv(2)1008 689 y Fz(\))41 b(if)31 b FA(x)12 b Fq(2)h Fp(@)s Ft(X)1269 673 y Fv(\(1\))1313 689 y Fp(;)8 b FA(x)1362 673 y Fy(0)1385 689 y Fq(2)1436 653 y Fy(\016)1428 693 y Ft(X)1461 647 y Fv(\(1\))1539 689 y Fz(and)30 b Fp(\013)1671 673 y Fy(0)1696 689 y Fz(=)13 b(0)607 774 y Fp(r)628 781 y Fv(00)665 774 y Fz(\()p Fp(x)709 757 y Fy(0)709 786 y Fv(1)738 774 y Fq(\000)e Fp(x)810 781 y Fv(1)829 774 y Fp(;)d(x)876 757 y Fy(0)876 786 y Fv(2)905 774 y Fq(\000)j Fp(x)977 781 y Fv(2)997 774 y Fz(\))52 b(if)31 b FA(x)12 b Fq(2)1214 738 y Fy(\016)1207 778 y Ft(X)1240 731 y Fv(\(1\))1317 774 y Fz(and)31 b Fp(\013)12 b Fz(=)h Fp(\013)1539 757 y Fy(0)1564 774 y Fz(=)g(0)607 830 y(0)437 b(otherwise,)24 930 y(and)15 b(similarly)i(for)e(the)g(\()p Fp(\030)486 906 y Fv(\(2\))484 935 y Fs(n)533 930 y Fz(\))551 937 y Fs(n)p Fy(2)p Fm(N)636 930 y Fz(c)o(hain)145 1128 y Fp(P)180 1104 y Fv(\(2\))174 1144 y(\()p Fs(x)208 1149 y Fl(1)225 1144 y Fs(;x)255 1149 y Fl(2)272 1144 y Fv(;)p Fs(\013)p Fv(\))p Fs(;)p Fv(\()p Fs(x)363 1133 y Fr(0)363 1155 y Fl(1)379 1144 y Fs(;x)409 1133 y Fr(0)409 1155 y Fl(2)426 1144 y Fv(;)p Fs(\013)459 1135 y Fr(0)469 1144 y Fv(\))498 1128 y Fz(=)546 980 y Fn(8)546 1021 y(>)546 1034 y(>)546 1048 y(>)546 1062 y(>)546 1075 y(<)546 1157 y(>)546 1171 y(>)546 1184 y(>)546 1198 y(>)546 1212 y(:)607 1016 y Fp(q)627 1023 y 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Fq(\000)j Fp(x)977 1192 y Fv(2)997 1185 y Fz(\))52 b(if)31 b FA(x)12 b Fq(2)1214 1149 y Fy(\016)1207 1189 y Ft(X)1240 1142 y Fv(\(2\))1317 1185 y Fz(and)31 b Fp(\013)12 b Fz(=)h Fp(\013)1539 1168 y Fy(0)1564 1185 y Fz(=)g(0)607 1241 y(0)437 b(otherwise.)73 1331 y Fu(De\014nition)16 b(1.7)23 b(Use)d(the)h(symb)n(ol)g Fp(])f Fu(to)h(denote)g(void)h(or)f Fz(\(1\))f Fu(or)h Fz(\(2\))p Fu(.)34 b(F)m(or)21 b(p)n(oints)f FA(x)h Fq(2)g Fp(@)s Ft(X)1751 1314 y Fs(])1784 1331 y Fu(on)g(the)24 1392 y(b)n(oundaries)d(of)g(the)g(sp)n(ac)n(es)f Ft(X)560 1375 y Fs(])573 1392 y Fu(,)h(we)g(de\014ne)f(the)i Fz(e\013ectiv)o(e)e (jump)h(matrices)g FA(Q)1407 1375 y Fs(])1439 1392 y Fz(=)e(\()p Fp(Q)1544 1370 y Fs(])1544 1407 y(\013;\013)1600 1397 y Fr(0)1612 1392 y Fz(\))1630 1399 y Fs(\013;\013)1686 1390 y Fr(0)1696 1399 y Fy(2)p Fm(A)1764 1392 y Fu(b)n(etwe)n(en)24 1448 y(internal)f(states)h(by)723 1538 y Fp(Q)759 1516 y Fs(])759 1553 y(\013;\013)815 1543 y Fr(0)840 1538 y Fz(=)895 1495 y Fn(X)888 1594 y Fo(x)909 1584 y Fr(0)920 1594 y Fy(2)p Fm(X)975 1538 y Fp(P)1010 1516 y Fs(])1004 1555 y Fv(\()p Fo(x)p Fv(;)p Fs(\013)p Fv(\))p Fs(;)p Fv(\()p Fo(x)1131 1545 y Fr(0)1142 1555 y Fv(;)p Fs(\013)1175 1545 y Fr(0)1186 1555 y Fv(\))1202 1538 y Fp(:)61 1713 y Fz(It)g(is)h(eviden)o(t)g(that)f(since)i(the)e FA(P)g Fz(matrix)g(is)h(irreducible,)i(the)d(corresp)q(onding)i FA(Q)e Fz(matrix)g(inherits)i(this)24 1769 y(irreducibili)q(t)o(y)g (prop)q(ert)o(y)l(.)24 b(Ho)o(w)o(ev)o(er,)15 b(irreducibili)q(t)o(y)k (of)c FA(P)i Fz(can)f(only)h(b)q(e)g(arise)g(b)o(y)f(mo)o(v)o(es)f (through)i(0.)23 b(T)l(o)24 1826 y(a)o(v)o(oid)13 b(an)o(y)g(irrelev)m (an)o(t)h(tec)o(hnical)h(complication,)g(w)o(e)e(can)g Fu(imp)n(ose)i(as)f(an)g(additionnal)h(assumption)f(on)h(our)24 1882 y(pr)n(oblem,)h(irr)n(e)n(ducibility)g(of)g(al)r(l)g FA(Q)630 1866 y Fs(])662 1882 y Fu(matric)n(es.)61 1986 y Fz(No)o(w,)i FA(Q)220 1969 y Fs(])255 1986 y Fz(are)g(sto)q(c)o (hastic)g(matrices)h(o)o(v)o(er)e(the)i Fu(\014nite)i Fz(in)o(ternal)e(space)g Ft(A)9 b Fz(,)22 b(irreducible)f(and)e(ap)q (erio)q(dic.)24 2042 y(Hence)d(they)f(admit)g(unique)i(in)o(v)m(arian)o (t)f(probabilities)h Fp(\031)1019 2026 y Fs(])1050 2042 y Fz(o)o(v)o(er)d Ft(A)9 b Fz(,)18 b(all)e(of)f(them)g(b)q(eing)h(ergo) q(dic.)73 2129 y Fu(De\014nition)g(1.8)23 b(F)m(or)14 b(the)g(pr)n(eviously)h(de\014ne)n(d)e(chains)h Fz(\()p Fp(\030)1067 2136 y Fs(n)1090 2129 y Fz(\))1108 2136 y Fs(n)p Fy(2)p Fm(N)1179 2129 y Fu(,)g Fz(\()p Fp(\030)1247 2105 y Fv(\(1\))1245 2135 y Fs(n)1294 2129 y Fz(\))1312 2136 y Fs(n)p Fy(2)p Fm(N)1383 2129 y Fu(,)g(and)h Fz(\()p Fp(\030)1538 2105 y Fv(\(2\))1536 2135 y Fs(n)1585 2129 y Fz(\))1603 2136 y Fs(n)p Fy(2)p Fm(N)1673 2129 y Fu(,)g(c)n(ol)r(le)n (ctively)24 2190 y(denote)n(d)h Fz(\()p Fp(\030)230 2168 y Fs(])228 2196 y(n)251 2190 y Fz(\))269 2197 y Fs(n)p Fy(2)p Fm(N)339 2190 y Fu(,)h(and)f(every)g Fp(u)d Fq(2)g Ft(U)696 2173 y Fs(])708 2190 y Fu(,)k(de\014ne)e(the)h Fz(drift)h Fu(of)f(the)h(chains)f(by)611 2282 y FA(m)655 2263 y Fs(])671 2282 y Fz(\()p Fp(u)p Fz(\))c(=)807 2239 y Fn(X)793 2341 y Fs(u)813 2332 y Fr(0)824 2341 y Fy(2)p Fm(U)875 2332 y Fk(])894 2282 y Fp(P)929 2260 y Fs(])923 2297 y(u;u)973 2288 y Fr(0)995 2245 y Fn(\000)1016 2282 y FA(X)p Fz(\()p Fp(u)1100 2263 y Fy(0)1111 2282 y Fz(\))d Fq(\000)i FA(X)p Fz(\()p Fp(u)p Fz(\))1286 2245 y Fn(\001)1313 2282 y Fp(:)73 2447 y Fu(R)n(emark)17 b(1.4)23 b(It)13 b(is)h(worth)g(noticing)f(that)h(for)g Fp(u)f Fz(=)g(\()p FA(x)p Fp(;)8 b(\013)p Fz(\))p Fu(,)13 b(with)h FA(x)e Fq(2)1289 2411 y Fy(\016)1282 2451 y Ft(X)1315 2404 y Fs(])1331 2447 y Fu(,)i(the)g(drift)g FA(m)1575 2431 y Fs(])1590 2447 y Fz(\()p Fp(u)p Fz(\))e(=)h(0)g Fu(vanishes)24 2504 y(due)18 b(to)g(the)g(zer)n(o)g(drift)g(assumptions)f(valid)h(in)f (the)h(interior)g(of)g(the)g(sp)n(ac)n(e.)25 b(Henc)n(e)17 b(the)h(only)f(inter)n(esting)24 2560 y(values)h(of)g(the)h(drifts)f (ar)n(e)g(on)g(the)h(b)n(oundaries.)26 b(F)m(or)18 b Fp(u)f Fz(=)f(\()p FA(x)p Fp(;)8 b(\013)p Fz(\))p Fu(,)17 b(with)i FA(x)d Fq(2)h Fp(@)s Ft(X)1472 2544 y Fv(\(2)o(\))1516 2560 y Fu(,)i(due)g(to)f(the)h(vertic)n(al)24 2617 y(homo)n(geneity)d (of)g(the)h(chain)f Fp(\030)558 2600 y Fv(\(2\))605 2617 y Fu(,)g(the)h(drift)f(dep)n(ends)g(only)g(on)g Fp(\013)p Fu(,)g(\020.e.)722 2702 y FA(m)766 2683 y Fv(\(2\))813 2702 y Fz(\(0)p Fp(;)8 b(x)901 2709 y Fv(2)919 2702 y Fz(;)g Fp(\013)p Fz(\))k Fq(\021)h FA(m)1091 2683 y Fv(\(2\))1138 2702 y Fz(\()p Fp(\013)p Fz(\))p Fp(:)24 2788 y Fu(Similarly,)18 b FA(m)279 2771 y Fv(\(1\))325 2788 y Fz(\()p Fp(x)369 2795 y Fv(1)389 2788 y Fp(;)8 b Fz(0;)g Fp(\013)p Fz(\))14 b Fq(\021)i FA(m)610 2771 y Fv(\(1\))657 2788 y Fz(\()p Fp(\013)p Fz(\))p Fu(.)26 b(Ther)n(efor)n(e,)18 b(the)g(drifts)h(on)f (e)n(ach)g(b)n(oundary)g(ar)n(e)h(functions)e(only)24 2844 y(of)f(the)h(c)n(olour.)p eop %%Page: 6 6 6 5 bop 24 74 a FB(6)1327 b(M.)13 b(Menshik)o(o)o(v)h(and)g(D.)f(P)o (etritis)73 169 y Fu(De\014nition)j(1.9)23 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1616 y Fp(\030)1411 1597 y Fs([)1409 1627 y(T)1430 1636 y Fk(u)1448 1629 y Fr(00)1485 1616 y Fz(=)i(\()p FA(x)1579 1597 y Fy(0)1590 1616 y Fz(;)8 b(0\))1652 1565 y Fn(\021)353 1748 y Fz(=)42 b FA(m)474 1729 y Fs([)489 1748 y Fz(\()p Fp(\013;)8 b Fq(f)p Fp(A)k Fz(=)g(0)p Fq(g)p Fz(\))430 1828 y(+)551 1785 y Fn(X)473 1889 y Fo(x)494 1879 y Fr(00)514 1889 y Fy(2)p Fs(@)r Fm(X)585 1879 y Fk([)597 1889 y Fs(;\013)630 1879 y Fr(00)650 1889 y Fy(6)p Fv(=0)695 1828 y Fz(\()p FA(x)741 1809 y Fy(0)o(0)772 1828 y Fq(\000)e FA(x)p Fz(\))p Fp(P)898 1809 y Fs([)892 1840 y(u;u)942 1830 y Fr(00)987 1785 y Fn(X)972 1889 y Fo(x)993 1879 y Fr(0)1005 1889 y Fy(2)p Fm(X)1055 1879 y Fk([)1075 1828 y Ft(P)1105 1836 y Fs(u)1125 1826 y Fr(0)n(0)1153 1777 y Fn(\020)1180 1828 y Fp(\030)1202 1809 y Fs([)1200 1839 y(T)1221 1849 y Fk(u)1239 1842 y Fr(00)1276 1828 y Fz(=)j(\()p FA(x)1370 1809 y Fy(0)1381 1828 y Fz(;)8 b(0\))1443 1777 y Fn(\021)430 1969 y Fz(+)551 1925 y Fn(X)473 2029 y Fo(x)494 2020 y Fr(00)514 2029 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2258 y Fn(\020)880 2309 y Fp(\030)902 2290 y Fs([)900 2320 y(T)921 2330 y Fk(u)939 2323 y Fr(00)976 2309 y Fz(=)d(\()p FA(x)1070 2290 y Fy(0)1081 2309 y Fz(;)8 b(0\))1143 2258 y Fn(\021)1181 2309 y Fz(=)13 b(1)p Fp(;)24 2471 y Fz(so)h(that)g(the)g(term)h(I)q(I)g(in)h(the)e (sum)h(ab)q(o)o(v)o(e)f(yields)i FA(m)936 2455 y Fs([)952 2471 y Fz(\()p Fp(\013;)8 b Fq(f)p Fp(A)j Fq(6)p Fz(=)i(0)p Fq(g)p Fz(\).)19 b(T)l(o)14 b(handle)i(the)f(term)f(I)q(I)q(I,)i(sp)q (ecify)g(for)24 2528 y(the)g(momen)o(t)f Fp([)e Fz(=)h(2.)22 b(The)16 b(case)g Fp([)d Fz(=)i(1)g(is)i(treated)e(analogously)l(.)22 b(Denoting)16 b FA(e)1414 2535 y Fv(1)1450 2528 y Fz(and)g FA(e)1563 2535 y Fv(2)1598 2528 y Fz(the)g(unit)h(v)o(ectors)24 2584 y(of)d Ft(Z)109 2568 y Fv(2)125 2584 y Fz(,)h(this)h(term)f(reads) 113 2794 y(I)q(I)q(I)f(=)261 2751 y Fn(X)225 2852 y Fs(x)245 2841 y Fr(00)245 2863 y Fl(2)265 2852 y Fs(;\013)298 2843 y Fr(00)318 2852 y Fy(6)p Fv(=0)380 2751 y Fn(X)370 2852 y Fs(x)390 2841 y Fr(0)390 2863 y 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y Fn(\001)1812 2794 y Fp(:)p eop %%Page: 12 12 12 11 bop 24 74 a FB(12)1306 b(M.)13 b(Menshik)o(o)o(v)h(and)g(D.)f(P)o (etritis)24 169 y Fz(Change)j(the)g(dumm)o(y)g(summation)h(v)m(ariable) g Fp(x)867 153 y Fy(0)867 181 y Fv(2)901 169 y Fq(2)e Ft(Z)q Fz(in)o(to)h Fp(x)1112 153 y Fy(0)1112 181 y Fv(2)1142 169 y Fq(\000)c Fp(x)1215 153 y Fy(0)o(0)1215 181 y Fv(2)1252 169 y Fz(and)17 b(use)f(the)h(v)o(ertical)f(homogeneit)o(y)24 226 y(of)e(the)i Fp(\030)176 209 y Fv(\(2\))238 226 y Fz(c)o(hain)g(to)e(write)i(this)f(term)g(as)314 344 y(I)q(I)q(I)43 b(=)520 301 y Fn(X)484 402 y Fs(x)504 390 y Fr(00)504 412 y Fl(2)524 402 y Fs(;\013)557 392 y Fr(00)577 402 y Fy(6)p Fv(=0)630 344 y Fp(P)659 353 y Fv(\(0)p Fs(;x)721 358 y Fl(2)737 353 y Fv(;)p Fs(\013)p Fv(\))p Fs(;)p Fv(\(0)p Fs(;x)856 342 y Fr(0)n(0)856 364 y Fl(2)874 353 y Fv(;)p Fs(\013)907 344 y Fr(00)927 353 y Fv(\))950 301 y Fn(X)967 400 y Fo(x)988 390 y Fr(0)1024 344 y Ft(P)1054 353 y Fv(\(0)p Fs(;)p Fv(0)o(;)p Fs(\013)1146 344 y 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y(drifts)i(dep)n(ending)f(on)h(the)g(c)n(olour)g(ar)n (e)g(r)n(eplac)n(e)n(d)g(by)g(a)g(weighte)n(d)g(aver)n(age,)g(the)h (weight)f(of)g(e)n(ach)g(c)n(olour)g(b)n(eing)24 2020 y(the)19 b(pr)n(ob)n(ability)g(that)h(the)g(system)e(is)h(on)g(that)h (c)n(olour.)31 b(The)19 b(pr)n(o)n(of)g(of)g(the)h(main)f(the)n(or)n (ems)g(c)n(an)g(also)g(b)n(e)24 2076 y(formulate)n(d)g(in)f(terms)h(of) g(emb)n(e)n(dde)n(d)f(quantities)919 2075 y Fz(~)908 2076 y FA(m)p Fu(;)i(however)f(such)g(pr)n(o)n(ofs)g(should)g(b)n(e)f (less)f(formal)i(and)24 2132 y(ther)n(efor)n(e)d(mor)n(e)g(di\016cult)h (to)f(che)n(ck)g(sinc)n(e)f(they)i(lie)e(on)h(intuitive)g(ide)n(as.)24 2307 y FA(3.)23 b(Harmonic)18 b(functions)g(in)g(the)g(w)o(edge)73 2453 y Fz(3.1.)i Fu(Contr)n(ol)r(ling)f(jumps)i(of)g(smo)n(oth)g (functions)28 b Fz(Let)20 b(\()p Fp(\030)1097 2460 y Fs(n)1120 2453 y Fz(\))1138 2460 y Fs(n)p Fy(2)p Fm(N)1229 2453 y Fz(b)q(e)h(the)f(previously)i(de\014ned)f(Mark)o(o)o(v)24 2509 y(c)o(hain)f(in)g Ft(U)11 b 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(incremen)o(t)24 2735 y(for)e(some)h(smo)q(oth)g(function)h Fp(f)5 b Fz(,)15 b(namely)220 2844 y Ft(E)r Fz(\()p Fp(f)5 b Fz(\()p FA(Y)q Fz(\()p Fp(\021)392 2851 y Fs(n)p Fv(+1)462 2844 y Fz(\)\))k Fq(\000)i Fp(f)5 b Fz(\()p FA(Y)q Fz(\()p Fp(\021)680 2851 y Fs(n)702 2844 y Fz(\)\))p Fq(j)p FA(Y)q Fz(\()p Fp(\021)833 2851 y Fs(n)855 2844 y Fz(\))12 b(=)h FA(y)q Fz(\))f(=)h Ft(E)r Fz(\()p Fp(f)5 b Fz(\()p FA(y)13 b Fz(+)d Fc(\022)1243 2851 y Fs(n)p Fv(+1)1312 2844 y Fz(\))g Fq(\000)g Fp(f)5 b Fz(\()p FA(y)q Fz(\))p Fq(j)p FA(Y)q Fz(\()p Fp(\021)1572 2851 y Fs(n)1593 2844 y Fz(\))12 b(=)h FA(y)q Fz(\))p eop %%Page: 13 13 13 12 bop 24 74 a FC(Markov)15 b(chains)g(in)g(a)g(we)n(dge)g(with)f (excitable)h(b)n(oundaries)912 b FB(13)24 169 y Fz(b)o(y)15 b(using)h(T)l(a)o(ylor)e(expansion)338 259 y Fp(f)5 b Fz(\()p FA(y)10 b Fz(+)g Fc(\022)q Fz(\))41 b(=)h Fp(f)5 b Fz(\()p FA(y)q Fz(\))k(+)i(\()p Fq(r)p Fp(;)d Fc(\022)o Fz(\))p Fp(f)d Fz(\()p FA(y)q Fz(\))k(+)i Fp(R)1078 266 y Fv(1)1097 259 y 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Fz(\(\012)p Fp(;)8 b Fq(F)d Fp(;)j Fz(\()o Fq(F)1449 535 y Fs(n)1469 528 y Fz(\))p Fp(;)g Ft(P)-6 b Fz(\))13 b Fu(b)n(e)j(a)h(\014ltr)n(e)n(d)e (pr)n(ob)n(a-)24 585 y(bility)e(sp)n(ac)n(e,)h Fz(\()p Fp(X)322 592 y Fs(n)345 585 y Fz(\))363 592 y Fs(n)401 585 y Fu(a)g(r)n(e)n(al-value)n(d)g(adapte)n(d)h(pr)n(o)n(c)n(ess)e (such)h(that)h Fp(X)1207 592 y Fv(0)1239 585 y Fz(=)e Fp(x)p Fu(,)i(with)f Fp(x)f(>)g(K)k Fu(for)e(some)f(p)n(ositive)24 641 y Fp(K)s Fu(.)20 b(L)n(et)15 b Fp(\034)198 648 y Fs(K)244 641 y Fz(=)e(inf)5 b Fq(f)p Fp(n)12 b Fq(\025)h Fz(0)g(:)f Fp(X)557 648 y Fs(n)593 641 y Fq(\024)h Fp(K)s Fq(g)p Fu(,)i(Assume)f(that)j(ther)n(e)e(exist)g(p)n(ositive)g(c)n (onstants)f Fp(\025)e(>)h Fz(0)i Fu(and)g Fp(p)1810 648 y Fv(0)1843 641 y Fp(>)e Fz(0)24 698 y Fu(such)j(that,)h(for)f(al)r(l)h Fp(n)p Fu(,)f(we)g(have)h Ft(E)r Fq(j)p Fp(X)679 705 y Fs(n)705 698 y Fq(j)718 681 y Fv(2)p Fs(p)754 686 y Fl(0)785 698 y Fp(<)c Fq(1)k Fu(and)475 807 y Ft(E)r Fz(\()p Fp(X)562 785 y Fv(2)p 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Fu(Su\016cient)d(c)n(ondition)g(for)g(non-existenc)n (e)f(of)h(moments)g(of)h(the)f(p)n(assage)g(time)73 1331 y(The)n(or)n(em)g(4.3)h(\(Asp)n(andiiar)n(ov,)e(Iasno)n(gor)n(o)n (dski,)g(Menshikov\))21 b(Supp)n(ose)11 b Fz(\()p Fp(X)1413 1338 y Fs(n)1436 1331 y Fz(\))1454 1338 y Fs(n)p Fy(2)p Fm(N)1536 1331 y Fu(and)h Fz(\()p Fp(Y)1664 1338 y Fs(n)1688 1331 y Fz(\))1706 1338 y Fs(n)p Fy(2)p Fm(N)1788 1331 y Fu(ar)n(e)g(adapte)n(d)24 1388 y(pr)n(o)n(c)n(esses)k(to)i(some)h (\014ltr)n(ation)e Fz(\()p Fq(F)633 1395 y Fs(n)656 1388 y Fz(\))674 1395 y Fs(n)p Fy(2)p Fm(N)763 1388 y Fu(and)h(take)g (values)g(in)g(an)g(unb)n(ounde)n(d)g(subset)f(of)i Ft(R)1666 1371 y Fv(+)1693 1388 y Fu(.)26 b(L)n(et)18 b Fp(K)j Fu(b)n(e)24 1444 y(some)d(lar)n(ge)h(c)n(onstant)f(and)g(de\014ne)g Fp(\033)690 1451 y Fs(K)742 1444 y Fz(=)g(inf)t Fq(f)p Fp(n)f Fq(\025)h Fz(1)f(:)g Fp(X)1078 1451 y Fs(n)1118 1444 y Fq(\024)h Fp(K)s Fq(g)h Fu(and)g Fp(\034)1366 1451 y Fs(K)1417 1444 y Fz(=)f(inf)t Fq(f)p Fp(n)f Fq(\025)h Fz(1)f(:)g Fp(Y)1741 1451 y Fs(n)1782 1444 y Fq(\024)h Fp(K)s Fq(g)p Fu(.)24 1500 y(Supp)n(ose)e(that)h Fp(Y)316 1507 y Fv(0)348 1500 y Fz(=)c Fp(y)i(>)e(K)19 b Fu(and)d(that)h(ther)n (e)f(exist)g(p)n(ositive)g(c)n(onstants)f Fp(c)1323 1507 y Fv(1)1342 1500 y Fu(,)i Fp(c)1393 1507 y Fv(2)1412 1500 y Fu(,)f(and)g Fp(B)r Fu(,)h(such)g(that)77 1648 y(1.)23 b Fp(K)s(B)15 b(<)e(X)314 1655 y Fv(0)346 1648 y Fz(=)g Fp(x)g Fq(\024)g Fp(B)r(y)r Fu(,)77 1751 y(2.)23 b(for)17 b(al)r(l)f Fp(n)p Fu(,)g Fp(X)370 1758 y Fs(n)406 1751 y Fq(\024)d Fp(B)r(Y)516 1758 y Fs(n)541 1751 y Fu(,)77 1854 y(3.)23 b Ft(E)s Fz(\()p Fp(Y)222 1837 y Fv(2)209 1866 y Fs(n)p Fv(+1)290 1854 y Fq(\000)11 b Fp(Y)372 1837 y Fv(2)362 1865 y Fs(n)392 1854 y Fq(jF)438 1861 y Fs(n)461 1854 y Fz(\))h Fq(\025)h(\000)p Fp(c)594 1861 y Fv(1)630 1854 y Fu(on)j Fq(f)p Fp(\034)738 1861 y Fs(K)784 1854 y Fp(>)d(n)p Fq(g)p Fu(,)k(and)77 1956 y(4.)23 b(for)17 b(some)f Fp(r)d(>)g Fz(1)p Fu(,)j Ft(E)s Fz(\()p Fp(Y)546 1940 y Fv(2)p Fs(r)533 1968 y(n)p Fv(+)q(1)614 1956 y Fq(\000)11 b Fp(Y)696 1940 y Fv(2)p Fs(r)686 1968 y(n)733 1956 y Fq(jF)779 1963 y Fs(n)802 1956 y Fz(\))h Fq(\024)h Fp(c)900 1963 y Fv(2)919 1956 y Fp(Y)956 1940 y Fv(2)p Fs(r)q Fy(\000)p Fv(2)945 1968 y Fs(n)1054 1956 y Fu(on)j Fq(f)p Fp(\034)1162 1963 y Fs(K)1208 1956 y Fp(>)d(n)p Fq(g)p Fu(.)24 2111 y(If)j(for)h(some)g(p)n(ositive)f Fp(p)449 2118 y Fv(0)469 2111 y Fu(,)h(the)g(pr)n(o)n(c)n(ess)f Fz(\()p Fp(X)791 2089 y Fv(2)p Fs(p)827 2094 y Fl(0)787 2121 y Fs(n)p Fy(^)p Fs(\033)852 2127 y Fk(K)r(B)907 2111 y Fz(\))925 2118 y Fs(n)965 2111 y Fu(is)g(a)h(submartingale,)g (then)g(for)g(al)r(l)g Fp(p)c(>)h(p)1709 2118 y Fv(0)1729 2111 y Fu(,)j(the)g Fp(p)p Fu(-th)24 2167 y(moment)f(do)n(es)g(not)g (exist,)g(i.e.)g Ft(E)r Fp(\034)630 2145 y Fs(p)625 2182 y(K)675 2167 y Fz(=)d Fq(1)p Fu(.)73 2327 y Fz(4.2.)20 b Fu(Construction)15 b(of)h(Lyapunov)g(function)f(for)h(the)g(we)n(dge) f(pr)n(oblem)28 b Fz(W)l(e)15 b(are)f(seeking)h(for)f(a)h(function)24 2384 y Fp(f)25 b Fz(:)c Ft(V)q Fq(!)5 b Ft(R)234 2367 y Fv(+)281 2384 y Fz(suc)o(h)20 b(that)f Fp(f)5 b Fz(\()p Fp(\021)559 2391 y Fs(n)582 2384 y Fz(\))20 b(is)g(a)g(sup)q (ermartingale.)36 b(Let)20 b Fp(h)1193 2391 y Fs(\014)1237 2384 y Fz(:)h Ft(Y)n Fq(!)5 b Ft(R)1400 2367 y Fv(+)1446 2384 y Fz(b)q(e)21 b(a)f(p)q(ositiv)o(e)h(harmonic)24 2440 y(function)16 b(o)o(v)o(er)e Ft(Y)s Fz(of)h(order)g Fp(\014)r Fz(.)61 2544 y(Since)i(ev)o(ery)e(harmonic)g(function)h(of)f (order)g Fp(\014)i Fz(can)f(b)q(e)g(written)f(as)506 2660 y Fp(h)532 2667 y Fs(\014)555 2660 y Fz(\()p Fp(y)595 2667 y Fv(1)615 2660 y Fp(;)8 b(y)658 2667 y Fv(2)677 2660 y Fz(\))k(=)h(\()p Fp(y)797 2641 y Fv(2)795 2671 y(1)827 2660 y Fz(+)d Fp(y)896 2641 y Fv(2)894 2671 y(2)916 2660 y Fz(\))934 2641 y Fs(\014)r(=)p Fv(2)1000 2660 y Fz(cos\()p Fp(\014)f Fz(arctan)1253 2629 y Fp(y)1275 2636 y Fv(2)p 1253 2649 43 2 v 1253 2691 a Fp(y)1275 2698 y Fv(1)1310 2660 y Fq(\000)i Fp(\014)1382 2667 y Fv(1)1401 2660 y Fz(\))p Fp(;)24 2788 y Fz(the)k(p)q(ositivit)o(y)h (condition)g(implies)i(certain)d(limits)i(on)e(the)g(range)g(of)f (parametres)h Fp(\014)i Fz(and)e Fp(\014)1645 2795 y Fv(1)1665 2788 y Fz(.)k(In)d(order)f(to)24 2844 y(sp)q(ecify)h(b)q(oth) f(these)h(parametres)e(w)o(e)h(note)g Fp(h)827 2851 y Fs(\014)r(;\014)879 2856 y Fl(1)898 2844 y Fz(\()p Fp(y)938 2851 y Fv(1)957 2844 y Fp(;)8 b(y)1000 2851 y Fv(2)1020 2844 y Fz(\))15 b(for)f(the)h(previous)h(function)g(in)g(the)g(sequel.) p eop %%Page: 16 16 16 15 bop 24 74 a FB(16)1306 b(M.)13 b(Menshik)o(o)o(v)h(and)g(D.)f(P)o (etritis)73 169 y Fu(The)n(or)n(em)j(4.4)23 b(L)n(et)16 b Fp(v)g Fz(=)e(\()p FA(y)q Fp(;)8 b(\013)p Fz(\))k Fq(2)i Ft(V)-6 b Fu(,)14 b(wher)n(e)j FA(y)d Fq(2)h Fp(@)986 179 y Fs([)1001 169 y Ft(Y)5 b Fu(is)17 b(a)g(b)n(oundary)g(p)n(oint.) 23 b(Supp)n(ose)17 b(that)g(for)g(some)24 226 y Fp(\017)c(>)g Fz(0)629 316 y(\()p Fq(r)p Fp(h)711 323 y Fs(\014)r(;\014)763 328 y Fl(1)781 316 y Fz(\()p FA(y)q Fz(\))p Fp(;)p 867 290 30 2 v 8 w FA(n)894 297 y Fs([)910 316 y Fz(\))f Fp(<)h Fq(\000)p Fp(\017;)8 b Fu(for)33 b Fp([)13 b Fz(=)g(1)p Fp(;)8 b Fz(2)p Fp(:)24 406 y Fu(Then,)14 b(ther)n(e)i(exists)e(a)i (function)f Fp(f)i Fz(:)c Ft(V)q Fq(!)5 b Ft(R)10 b Fu(with)16 b Fp(f)i Fq(2)13 b Fp(C)1031 389 y Fv(3)1050 406 y Fz(\()p Ft(V)-6 b Fp(;)7 b Ft(R)1157 389 y Fv(+)1183 406 y Fz(\))15 b Fu(and)g(a)h(c)n(onstant)e Fp(K)i(>)d Fz(0)i Fu(such)g(that)h(for)24 462 y Fq(k)p FA(y)q Fq(k)f Fu(su\016ciently)g(lar)n(ge,)77 593 y(1.)23 b(the)17 b(se)n(quenc)n(e)d Fz(\()p Fp(f)5 b Fz(\()p Fp(\021)483 600 y Fs(n)506 593 y Fz(\)\))542 600 y Fs(n)581 593 y Fu(is)16 b(a)g(str)n(ong)g(sup)n(ermartingale)g (ne)n(ar)g(the)g(b)n(oundary)h(i.e.)401 683 y Fp(D)q Fz(\()p Fp(v)r Fz(\))12 b Fq(\021)h Ft(E)r Fz(\()p Fp(f)5 b Fz(\()p Fp(\021)673 690 y Fs(n)p Fv(+1)743 683 y Fz(\))10 b Fq(\000)h Fp(f)5 b Fz(\()p Fp(\021)885 690 y Fs(n)908 683 y Fz(\))p Fq(j)p Fp(\021)962 690 y Fs(n)997 683 y Fz(=)13 b Fp(v)r Fz(\))f Fp(<)g Fq(\000)p Fp(K)s(\017)p Fq(k)p FA(y)q Fq(k)1316 664 y Fs(\014)r Fy(\000)p Fv(1)1385 683 y Fp(;)c Fu(for)32 b FA(y)13 b Fq(2)g Fp(@)s Ft(Y)-10 b Fp(;)77 789 y Fu(2.)23 b(the)15 b(se)n(quenc)n(e)f Fz(\()p Fp(f)5 b Fz(\()p Fp(\021)481 796 y Fs(n)503 789 y Fz(\)\))539 796 y Fs(n)577 789 y Fu(is)14 b(almost)h(a)g(martingale,) g(in)f(the)h(interior)g(sp)n(ac)n(e)f(i.e.)20 b(for)15 b Fp(v)f Fz(=)f(\()p FA(y)q Fp(;)8 b Fz(0\))p Fu(,)13 b(with)137 861 y FA(y)g Fq(2)229 825 y Fy(\016)221 865 y Ft(Y)254 861 y Fu(,)618 951 y Ft(E)r Fz(\()p Fp(f)5 b Fz(\()p Fp(\021)731 958 y Fs(n)p Fv(+1)802 951 y Fz(\))10 b Fq(\000)g Fp(f)5 b Fz(\()p Fp(\021)943 958 y Fs(n)966 951 y Fz(\))p Fq(j)p Fp(\021)1020 958 y Fs(n)1055 951 y Fz(=)13 b Fp(v)r Fz(\))f(=)h Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)1335 932 y Fs(\014)r Fy(\000)p Fv(2)1402 951 y Fz(\))p Fp(:)24 1082 y Fz(T)l(o)i(pro)o(v)o(e)f(this)i(theorem,)e (w)o(e)h(need)h(a)f(lemma)h(from)e(elemen)o(tary)i(linear)g(algebra.)73 1163 y Fu(L)n(emma)g(4.1)23 b(L)n(et)13 b Fz(\()p Fp(Q)447 1171 y Fs(\013;\013)503 1161 y Fr(0)515 1163 y Fz(\))g Fu(b)n(e)g(the)g(sto)n(chastic)g(matrix)h(of)f(an)g(er)n(go)n(dic)g (Markov)g(chain)g(on)g(the)h(\014nite)e(sp)n(ac)n(e)24 1219 y Ft(A)28 b Fu(and)17 b Fz(\()p Fp(\031)206 1226 y Fs(\013)231 1219 y Fz(\))249 1226 y Fs(\013)290 1219 y Fu(its)f(invariant)h(pr)n(ob)n(ability.)22 b(L)n(et)17 b Fz(\()p Fp(\027)911 1226 y Fs(\013)935 1219 y Fz(\))953 1226 y Fs(\013)995 1219 y Fu(b)n(e)f(a)h(given)f(ve)n(ctor)h(on)g Ft(A)9 b Fu(.)25 b(Then)17 b(the)g(set)f(of)h(line)n(ar)24 1276 y(ine)n(qualities)e(for)h(the)h(variables)e Fz(\()p Fp(c)635 1283 y Fs(\013)660 1276 y Fz(\))678 1283 y Fs(\013)594 1372 y Fp(\027)616 1379 y Fs(\013)651 1372 y Fq(\000)c Fp(c)717 1379 y Fs(\013)751 1372 y Fz(+)805 1329 y Fn(X)797 1428 y Fs(\013)820 1419 y Fr(0)831 1428 y Fy(2)p Fm(A)886 1372 y Fp(Q)922 1380 y Fs(\013;\013)978 1371 y Fr(0)990 1372 y Fp(c)1010 1380 y Fs(\013)1033 1371 y Fr(0)1058 1372 y Fp(<)i Fq(\000)p Fp(\017;)41 b(\013)13 b Fq(2)g Ft(A)8 b Fp(;)24 1512 y Fu(de\014nes)15 b(a)h(non-void)g(subset)g(of)g Ft(R)626 1495 y Fm(A)666 1512 y Fu(if,)g(and)g(only)g(if,)969 1478 y Fn(P)1017 1525 y Fs(\013)p Fy(2)p Fm(A)1097 1512 y Fp(\031)1123 1519 y Fs(\013)1147 1512 y Fp(\027)1169 1519 y Fs(\013)1207 1512 y Fp(<)d Fq(\000)p Fp(\017)p Fu(.)39 1641 y(Pr)n(o)n(of)j(of)g(the)n(or)n(em)h(4.4.)36 b Fz(De\014ne)16 b(a)f(function)h Fp(f)i Fz(:)12 b Ft(V)q Fq(!)6 b Ft(R)k Fz(b)o(y)374 1831 y Fp(f)5 b Fz(\()p Fp(v)r Fz(\))12 b(=)521 1724 y Fn(8)521 1765 y(>)521 1779 y(<)521 1861 y(>)521 1874 y(:)582 1783 y Fp(h)608 1790 y Fs(\014)r(;\014)660 1795 y Fl(1)679 1783 y Fz(\()p FA(y)q Fz(\))d(+)h Fp(c)818 1790 y Fv(0)837 1783 y Fp(h)863 1790 y Fs(\014)r Fy(\000)p Fv(1)p Fs(;\016)956 1795 y Fl(1)975 1783 y Fz(\()p FA(y)q Fz(\))50 b(if)30 b Fp(v)15 b Fz(=)e(\()p FA(y)q Fp(;)8 b Fz(0\))p Fp(;)f FA(y)j Fq(2)1449 1747 y Fy(\016)1441 1787 y Ft(Y)582 1839 y Fp(h)608 1846 y Fs(\014)r(;\014)660 1851 y Fl(1)679 1839 y Fz(\()p FA(y)q Fz(\))f(+)h Fp(a)822 1846 y Fs(\013)847 1839 y Fp(h)873 1846 y Fs(\014)r Fy(\000)p Fv(1)p Fs(;\016)966 1851 y Fl(1)984 1839 y Fz(\()p FA(y)q Fz(\))41 b(if)30 b Fp(v)15 b Fz(=)e(\()p FA(y)q Fp(;)8 b(\013)p Fz(\))p Fp(;)g FA(y)i Fq(2)j Fp(@)1472 1846 y Fv(1)1492 1839 y Ft(Y)582 1896 y Fp(h)608 1903 y Fs(\014)r(;\014)660 1908 y Fl(1)679 1896 y Fz(\()p FA(y)q Fz(\))c(+)h Fp(c)818 1903 y Fs(\013)842 1896 y Fp(h)868 1903 y Fs(\014)r Fy(\000)p Fv(1)p Fs(;\016)961 1908 y Fl(1)980 1896 y Fz(\()p FA(y)q Fz(\))45 b(if)30 b Fp(v)15 b Fz(=)e(\()p FA(y)q Fp(;)8 b(\013)p Fz(\))p Fp(;)g FA(y)i Fq(2)j Fp(@)1472 1903 y Fv(2)1492 1896 y Ft(Y)-10 b Fp(:)24 1987 y Fz(Then)16 b(for)g Fp(v)g Fz(=)e(\()p FA(y)q Fp(;)8 b(\013)p Fz(\))15 b(with)h Fp(y)g Fq(2)f Fp(@)642 1994 y Fv(2)662 1987 y Ft(Y)-9 b Fz(,)13 b(w)o(e)i(compute)i(the)f(conditional)i(incremen)o (t)f Fp(D)q Fz(\()p Fp(v)r Fz(\))c(=)i Ft(E)r Fz(\()p Fp(f)5 b Fz(\()p Fp(\021)1779 1994 y Fs(n)p Fv(+1)1850 1987 y Fz(\))10 b Fq(\000)24 2044 y Fp(f)5 b Fz(\()p Fp(\021)92 2051 y Fs(n)114 2044 y Fz(\))p Fq(j)p Fp(\021)168 2051 y Fs(n)207 2044 y Fz(=)17 b Fp(v)r Fz(\).)26 b(F)l(or)17 b(de\014niteness,)i(w)o(e)e(consider)h(the)g(case)f FA(y)g Fq(2)g Fp(@)1225 2051 y Fv(2)1245 2044 y Ft(Y)-10 b Fz(.)24 b(The)18 b(computation)f(for)g(the)g(case)24 2100 y FA(y)c Fq(2)f Fp(@)131 2107 y Fv(1)151 2100 y Ft(Y)s Fz(is)k(carried)g(out)f (along)g(the)g(same)g(lines.)264 2190 y Fp(D)q Fz(\()p Fp(v)r Fz(\))41 b(=)h Fp(P)510 2199 y Fs(v)q(;)p Fv(\()p Fo(y)574 2190 y Fr(0)585 2199 y Fv(;0\))628 2190 y Fz([)p Fp(h)667 2197 y Fs(\014)691 2190 y Fz(\()p FA(y)738 2171 y Fy(0)748 2190 y Fz(\))10 b(+)g Fp(c)841 2197 y Fv(0)861 2190 y Fp(h)887 2197 y Fs(\014)r Fy(\000)p Fv(1)956 2190 y Fz(\()p FA(y)1003 2171 y Fy(0)1013 2190 y Fz(\))g Fq(\000)g Fp(h)1112 2197 y Fs(\014)1136 2190 y Fz(\()p FA(y)q Fz(\))f Fq(\000)h Fp(c)1275 2197 y Fs(\013)1300 2190 y Fp(h)1326 2197 y Fs(\014)r Fy(\000)p Fv(1)1395 2190 y Fz(\()p FA(y)q Fz(\)])404 2272 y(+)528 2229 y 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16 bop 24 74 a FC(Markov)15 b(chains)g(in)g(a)g(we)n(dge)g(with)f (excitable)h(b)n(oundaries)912 b FB(17)24 169 y Fz(But)14 b(no)o(w,)g(for)f(v)o(ery)i(large)f(\014xed)h Fq(k)p FA(y)q Fq(k)p Fz(,)e(w)o(e)h(are)g(in)h(the)g(situation)f(of)g(the)h (previous)g(lemma;)f(w)o(e)h(can)f(alw)o(a)o(ys)24 232 y(c)o(ho)q(ose)h(constan)o(ts)f(\()p Fp(c)407 239 y Fs(\013)431 232 y Fz(\))449 239 y Fs(\013)489 232 y Fz(so)h(that)669 210 y Fs(D)q Fv(\()p Fs(v)q Fv(\))p 648 222 118 2 v 648 250 a Fy(k)p Fo(y)q Fy(k)706 241 y Fk(\014)q Fr(\000)p Fl(1)783 232 y Fp(<)e Fq(\000)p Fp(\017)j Fz(for)e(ev)o(ery)h Fp(\013)h Fz(pro)o(vided)g(that)719 286 y Fn(\022)758 319 y Fq(r)p Fp(h)822 326 y Fs(\014)r(;\014)874 331 y Fl(1)892 319 y Fz(\()p FA(y)q Fz(\))p 758 340 199 2 v 786 381 a Fq(k)p FA(y)q Fq(k)861 368 y Fs(\014)r Fy(\000)p Fv(1)961 350 y Fp(;)p 982 324 30 2 v 8 w FA(n)1010 331 y Fv(\(2\))1057 286 y Fn(\023)1103 350 y Fp(<)d Fq(\000)p Fp(\017:)24 466 y Fz(In)k(a)f(completely)h(analogous)g(w)o(a)o(y)l(,)e 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b(that)f(it)h(is)g(enough)h(to)e(see)h (that)f FA(n)p Fz(\()p FA(y)q Fp(;)8 b Fz(0\))15 b(=)i(0)h(for)f FA(y)h Fq(2)1876 789 y Fy(\016)1868 829 y Ft(Y)1901 825 y Fz(,)24 881 y(thanks)e(to)h(the)g(zero)f(drift)h(prop)q(ert)o(y)g(in) h(the)f(in)o(terior)g(of)f(the)h(space.)26 b(Hence)18 b(the)e(dominan)o(t)h(part)g(of)f(the)24 938 y(conditional)g(incremen)o (t)g(is)g(just)f Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)735 921 y Fs(\014)r Fy(\000)p Fv(2)803 938 y Fz(\).)73 1053 y Fu(R)n(emark)i(4.1)23 b(Notic)n(e)13 b(that)g(the)g(p)n(art)h(due)f (to)g(the)h(c)n(ontribution)e(of)h(the)h(function)e Fp(h)1497 1060 y Fs(\014)r Fy(\000)p Fv(1)p Fs(;\016)1590 1065 y Fl(1)1622 1053 y Fu(is)g(sub)n(dominent)24 1109 y(and)i(ne)n(e)n(d)g (not)h(b)n(e)f(p)n(ositive.)20 b(It)14 b(is)g(only)h(the)g(function)f Fp(f)5 b Fz(\()p Fp(v)r Fz(\))14 b Fu(that)h(is)g(r)n(e)n(quir)n(e)n(d) f(to)h(b)n(e)f(p)n(ositive)g(for)h(lar)n(ge)g Fq(k)p FA(y)q Fq(k)p Fu(.)24 1166 y(This)g(al)r(lows)h(to)h(cho)n(ose)f(the)g 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1548 y Fz(4.3.)k Fu(R)n(e)n(curr)n(enc)n(e)12 b(and)h(er)n(go)n(dic)g(pr)n(op) n(erties)g(of)h(the)f(chain)28 b Fz(W)l(e)12 b(are)g(no)o(w)f(able)i (to)e(pro)o(v)o(e)h(the)g(main)g(results)24 1604 y(of)i(this)i(pap)q (er.)61 1746 y Fu(Pr)n(o)n(of)j(of)g(the)h(the)n(or)n(em)f(1.1.)49 b Fz(First)19 b(examine)g(the)f(case)h(where)f Fp(\037)h Fz(=)1334 1725 y Fs( )1358 1714 y Fl(1)1376 1725 y Fv(+)p Fs( )1427 1714 y Fl(2)p 1334 1736 111 2 v 1377 1762 a Fs( )1468 1746 y Fp(>)f Fz(0.)29 b(Cho)q(ose)18 b Fp(\014)1773 1753 y Fv(1)1811 1746 y Fp(<)g( )1894 1753 y Fv(1)24 1820 y Fz(and)e Fp(\014)139 1827 y Fv(2)173 1820 y Fp(<)f( )253 1827 y Fv(2)288 1820 y Fz(so)h(that)f(0)f Fp(<)h(\014)i Fz(=)628 1799 y Fs(\014)648 1804 y Fl(1)665 1799 y Fv(+)p Fs(\014)712 1804 y Fl(2)p 628 1809 102 2 v 667 1835 a Fs( )749 1820 y Fp(<)e(\037)g Fq(\021)896 1799 y Fs( )919 1804 y Fl(1)937 1799 y Fv(+)p Fs( )987 1804 y Fl(2)p 896 1809 109 2 v 938 1835 a Fs( )1010 1820 y Fz(.)23 b(Cho)q(ose)16 b(some)g Fp(p)1343 1827 y Fv(0)1378 1820 y Fz(with)h(0)d Fp(<)h(p)1593 1827 y Fv(0)1627 1820 y Fq(\024)1681 1797 y Fv(min)o(\()p Fs(\014)r(;\015)r Fv(\))p 1681 1809 138 2 v 1741 1835 a(2)1840 1820 y Fz(and)24 1876 y(consider)h(the)f(functions)475 2009 y Fp(f)5 b Fz(\()p Fp(v)r Fz(\))12 b(=)622 1902 y Fn(8)622 1943 y(>)622 1956 y(<)622 2038 y(>)622 2052 y(:)683 1960 y Fp(h)709 1967 y Fs(\014)r(;\014)761 1972 y Fl(1)780 1960 y Fz(\()p FA(y)q Fz(\))d(+)h Fp(c)919 1967 y Fv(0)938 1960 y Fp(h)964 1967 y Fs(\014)r Fy(\000)p Fv(1)p Fs(;\014)1061 1972 y Fl(1)1080 1960 y Fz(\()p FA(y)q Fz(\))50 b(if)42 b FA(y)13 b Fq(2)1354 1924 y Fy(\016)1347 1964 y Ft(Y)683 2017 y Fp(h)709 2024 y Fs(\014)r(;\014)761 2029 y Fl(1)780 2017 y Fz(\()p FA(y)q Fz(\))c(+)h Fp(c)919 2024 y Fs(\013)944 2017 y Fp(h)970 2024 y Fs(\014)r Fy(\000)p Fv(1)p Fs(;\014)1067 2029 y Fl(1)1085 2017 y Fz(\()p FA(y)q Fz(\))45 b(if)d FA(y)13 b Fq(2)g Fp(@)1371 2024 y Fv(2)1391 2017 y Ft(Y)683 2073 y Fp(h)709 2080 y Fs(\014)r(;\014)761 2085 y Fl(1)780 2073 y Fz(\()p FA(y)q Fz(\))c(+)h Fp(a)923 2080 y Fs(\013)948 2073 y Fp(h)974 2080 y Fs(\014)r Fy(\000)p Fv(1)p Fs(;\014)1071 2085 y Fl(1)1090 2073 y Fz(\()p FA(y)q Fz(\))40 b(if)i FA(y)13 b Fq(2)g Fp(@)1371 2080 y Fv(1)1391 2073 y Ft(Y)-10 b Fp(;)24 2160 y Fz(and)15 b Fp(g)r Fz(\()p Fp(v)r Fz(\))c(=)i Fp(f)282 2143 y Fv(2)p Fs(p)318 2148 y Fl(0)335 2143 y Fs(=\014)376 2160 y Fz(\()p Fp(v)r Fz(\).)61 2264 y(Near)g(the)h(b)q (oundary)l(,)g(it)g(is)g(enough)g(to)f(consider)h(the)g(\014rst)f (order)g(T)l(a)o(ylor)g(expansion)i(of)e(the)h(conditional)24 2320 y(incremen)o(t.)29 b(In)19 b(the)f(in)o(terior)h(of)f(the)g (space,)h(w)o(e)f(need)h(to)f(con)o(tin)o(ue)g(up)h(to)f(the)g(second)h (order.)28 b(In)19 b(fact,)24 2376 y(near)c(the)g(b)q(oundary)l(,)268 2471 y Fp(D)q Fz(\()p Fp(v)r Fz(\))d(=)432 2440 y(2)p Fp(p)478 2447 y Fv(0)p 432 2461 66 2 v 450 2502 a Fp(\014)502 2471 y(f)534 2430 y Fl(2)p Fk(p)565 2437 y Fl(0)p 534 2441 49 2 v 549 2461 a Fk(\014)588 2448 y Fy(\000)p Fv(1)635 2471 y Fz(\()p Fp(v)r Fz(\)\()p Fq(r)p Fp(f)5 b Fz(\()p Fp(v)r Fz(\))p Fp(;)j Ft(E)p Fz(\()p Fc(\022)931 2478 y Fs(n)p Fv(+1)999 2471 y Fq(j)p Fp(\021)1035 2478 y Fs(n)1071 2471 y Fz(=)13 b Fp(v)r Fz(\))c(+)i Ft(E)r Fz(\()p Fp(R)1296 2478 y Fv(1)1318 2471 y Fz(\()p Fp(f)1363 2452 y Fv(2)p Fs(p)1399 2457 y Fl(0)1415 2452 y Fs(=\014)1456 2471 y Fz(;)d FA(y)q Fp(;)g Fc(\022)1552 2478 y Fs(n)p Fv(+1)1621 2471 y Fz(\)\))p Fp(:)24 2584 y Fz(No)o(w,)15 b(the)i(c)o(hoice)g Fp(\014)383 2591 y Fv(1)417 2584 y Fp(<)e( )497 2591 y Fv(1)533 2584 y Fz(and)i Fp(\014)649 2591 y Fv(2)683 2584 y Fp(<)e( )763 2591 y Fv(2)798 2584 y Fz(guaran)o(tees)h(that)g(\()1166 2562 y Fy(r)p Fs(f)t Fv(\()p Fs(v)q Fv(\))p 1149 2574 132 2 v 1149 2600 a Fy(kr)p Fs(f)t Fv(\()p Fs(v)q Fv(\))p Fy(k)1285 2584 y Fp(;)8 b Ft(E)q Fz(\()p Fc(\022)1380 2591 y Fs(n)p Fv(+1)1448 2584 y Fq(j)p Fp(\021)1484 2591 y Fs(n)1522 2584 y Fz(=)15 b Fp(v)r Fz(\)\))e Fq(\024)i(\000)p Fp(C)s(\017)p Fz(,)i(for)f FA(y)24 2647 y Fz(large)f(near)g 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y Fs(n)p Fv(+1)1749 2744 y Fz(\))1767 2725 y Fv(2)1786 2744 y Fq(j)p Fp(\021)1822 2751 y Fs(n)1857 2744 y Fz(=)13 b Fp(v)r Fz(\)\))240 2844 y(+)p Ft(E)s Fz(\()p Fp(R)356 2851 y Fv(2)377 2844 y Fz(\()p Fp(f)422 2826 y Fv(2)p Fs(p)458 2831 y Fl(0)475 2826 y Fs(=\014)516 2844 y Fz(;)8 b FA(y)q Fp(;)g Fc(\022)612 2851 y Fs(n)p Fv(+1)680 2844 y Fz(\))p Fp(:)p eop %%Page: 18 18 18 17 bop 24 74 a FB(18)1306 b(M.)13 b(Menshik)o(o)o(v)h(and)g(D.)f(P)o (etritis)24 169 y Fz(No)o(w)18 b(the)i(\014rst)f(order)g(term)g(in)h (the)g(T)l(a)o(ylor)f(expansion)h(v)m(anishes)h(due)f(to)e(the)i(zero)f (drift)h(condition)h(in)24 226 y(the)14 b(in)o(terior)i(of)e(the)h (space)g(while)h(the)f(exp)q(ectation)g(of)f(the)h(square)g(of)f(the)h (scalar)f(pro)q(duct)h(in)h(the)f(second)24 282 y(order)h(term)g(is)i (p)q(ositiv)o(e)f(and)g(of)g(the)g(order)f Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)945 266 y Fv(2)p Fs(p)981 271 y Fl(0)997 266 y Fy(\000)p Fv(2)1044 282 y Fz(\).)24 b(Moreo)o(v)o(er,)15 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y(Therefore)g(the)h(c)o(hoice)g Fp( )497 787 y Fv(1)541 803 y Fp(<)i(\014)627 810 y Fv(1)669 803 y Fz(and)d Fp( )796 787 y Fv(2)840 803 y Fp(<)j(\014)926 810 y Fv(2)968 803 y Fz(guaran)o(tees)d(that)g(near)g(the)h(b)q (oundary)g(the)f(scalar)24 859 y(pro)q(duct)c(\()238 837 y Fy(r)p Fs(f)t Fv(\()p Fs(v)q Fv(\))p 220 849 132 2 v 220 876 a Fy(kr)p Fs(f)t Fv(\()p Fs(v)q Fv(\))p Fy(k)356 859 y Fp(;)8 b Ft(E)r Fz(\()p Fc(\022)451 866 y Fs(n)p Fv(+1)520 859 y Fq(j)p Fp(\021)556 866 y Fs(n)597 859 y Fz(=)18 b Fp(v)r Fz(\)\))f Fq(\024)i(\000)p Fp(C)s(\017)g Fz(and)f(since)i(2)p Fp(p)1141 866 y Fv(0)1160 859 y Fp(=\014)h Fz(is)e(alw)o(a)o(ys)e(p)q(ositiv)o(e,)j(the)f(conditional) 24 922 y(incremen)o(t)e Fp(D)q Fz(\()p Fp(v)r Fz(\))e Fp(<)g Fz(0.)25 b(In)17 b(the)g(in)o(terior)g(of)g(the)g(space,)g (again)g(the)g(\014rst)f(term)h(of)f(the)h(T)l(a)o(ylor)f(expansion)24 979 y(v)m(anishes)i(due)g(to)e(the)h(zero)g(drift)g(condition)h(and)f (the)g(second)h(term)e(is)i(negativ)o(e)f(b)q(ecause)h(0)d 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Fp(f)300 2002 y Fs(s)318 2019 y Fz(,)g(with)g Fp(s)e Fz(=)g(1)p Fp(=\014)r Fz(,)i(satis\014es)g(the)g(conditions)h(of)f (prop)q(osition)g(3.1.)22 b(F)l(ollo)o(wing)16 b(the)g(lines)i(of)24 2075 y(the)d(pro)q(of)g(of)f(theorem)h(6)g(of)g([2)o(])g(it)h(is)f (then)h(p)q(ossible)h(to)d(sho)o(w)h(that)f(the)i(pro)q(cess)f Fp(X)1509 2082 y Fs(n)1545 2075 y Fz(=)e Fp(f)1620 2059 y Fv(1)p Fs(=\014)1679 2075 y Fz(\()p Fp(\021)1720 2082 y Fs(n)1743 2075 y Fz(\))i(v)o(eri\014es)491 2165 y Ft(E)s Fz([)p Fp(X)574 2143 y Fv(2)p Fs(p)610 2148 y Fl(0)570 2178 y Fs(n)p Fv(+1)651 2165 y Fq(\000)10 b Fp(X)738 2146 y Fv(2)p Fs(p)774 2151 y Fl(0)734 2176 y Fs(n)792 2165 y Fq(jF)838 2172 y Fs(n)861 2165 y Fz(])i Fq(\024)h(\000)p Fp(\025X)1038 2146 y Fv(2)p Fs(p)1074 2151 y Fl(0)1090 2146 y Fy(\000)p Fv(2)1034 2176 y Fs(n)1167 2165 y Fz(on)31 b Fq(f)p Fp(\034)1289 2172 y Fs(K)1335 2165 y Fp(>)13 b(n)p Fq(g)p Fp(:)24 2255 y Fz(This)i(remark)g(allo)o(ws)g(to)g (conclude.)73 2374 y Fu(Cor)n(ol)r(lary)i(4.1)23 b(If)15 b Fp( )436 2381 y Fv(1)472 2374 y Fu(and)h Fp( )590 2381 y Fv(2)626 2374 y Fu(b)n(elong)f(to)i Fz(])9 b Fq(\000)i Fp(\031)r(=)p Fz(2)p Fp(;)d(\031)r(=)p Fz(2[)13 b Fu(and)1174 2353 y Fs( )1197 2358 y Fl(1)1214 2353 y Fv(+)p Fs( )1264 2358 y Fl(2)p 1174 2363 109 2 v 1216 2389 a Fs( )1300 2374 y Fp(>)g Fz(2)p Fu(,)j(the)g(chain)g(is)g(er)n(go)n(dic.)61 2553 y(Pr)n(o)n(of)e(of)g(the)n(or)n(em)g(1.3.)33 b Fz(W)l(e)13 b(use)g(the)f(criterion)i(of)e(non-existence)i(4.3.)k(Let)13 b Fp(\037)g Fz(=)1506 2532 y Fs( )1529 2537 y Fl(1)1546 2532 y Fv(+)p Fs( )1596 2537 y Fl(2)p 1506 2543 V 1548 2569 a Fs( )1632 2553 y Fz(b)q(e)g(determined)24 2623 y(b)o(y)j(the)g(geometry)f(and)h(c)o(ho)q(ose)g(parametres)g Fp(\014)863 2630 y Fv(1)896 2623 y Fp(>)f( )976 2630 y Fv(1)995 2623 y Fz(,)h Fp(\014)1050 2630 y Fv(2)1083 2623 y Fp(>)f( )1163 2630 y Fv(2)1182 2623 y Fz(,)h Fp(\014)g Fz(=)1308 2602 y Fs(\014)1328 2607 y Fl(1)1345 2602 y Fv(+)p Fs(\014)1392 2607 y Fl(2)p 1308 2612 102 2 v 1346 2639 a Fs( )1429 2623 y Fp(>)e(\037)p Fz(.)23 b(Consider)17 b(functions)24 2698 y Fp(g)46 2705 y Fs(\016)o(;\016)87 2710 y Fl(1)123 2698 y Fz(=)i Fp(h)203 2705 y Fs(\016)o(;\016)244 2710 y Fl(1)275 2698 y Fz(+)13 b Fp(ch)369 2705 y Fs(\016)q Fy(\000)p Fv(1)p Fs(;\016)457 2710 y Fl(1)494 2698 y Fz(and)19 b(pro)q(cesses)g Fp(X)824 2705 y Fs(n)865 2698 y Fz(=)g Fp(g)943 2674 y Fv(1)p Fs(=\014)941 2713 y(\014)r(;\014)993 2718 y Fl(1)1011 2698 y Fz(\()p Fp(\021)1052 2705 y Fs(n)1075 2698 y Fz(\))f(and)h Fp(y)1225 2705 y Fs(n)1267 2698 y Fz(=)g Fp(g)1345 2674 y Fv(1)p Fs(=\014)1343 2713 y(\037; )1398 2718 y Fl(1)1417 2698 y Fz(\()p Fp(\021)1458 2705 y Fs(n)1480 2698 y Fz(\).)30 b(Observ)o(e)19 b(that)f(b)q(oth)24 2754 y(pro)q(cesses)d Fp(X)258 2761 y Fs(n)297 2754 y Fz(and)g Fp(Y)411 2761 y Fs(n)450 2754 y Fz(satisfy)g(conditions)h(of)f (applicabili)q(t)o(y)i(of)e(the)g(criterion)h(4.3,)e(namely)i(that)451 2844 y Ft(E)r Fz(\()p Fp(Y)535 2826 y Fv(2)522 2856 y Fs(n)q Fv(+1)604 2844 y Fq(\000)10 b Fp(Y)686 2826 y Fv(2)675 2856 y Fs(n)705 2844 y Fq(jF)751 2851 y Fs(n)774 2844 y Fz(\))j(=)g(\()p Fq(r)p Fp(f)5 b Fz(\()p FA(y)q Fz(\))p Fp(;)j Ft(E)p Fz(\()p Fc(\022)1094 2851 y Fs(n)p Fv(+1)1163 2844 y Fq(jF)1209 2851 y Fs(n)1231 2844 y Fz(\)\))i(+)g Fp(r)1343 2851 y Fv(1)1363 2844 y Fz(\()p Fp(f)5 b Fz(;)j FA(y)q Fz(\))p Fp(;)p eop %%Page: 19 19 19 18 bop 24 74 a FC(Markov)15 b(chains)g(in)g(a)g(we)n(dge)g(with)f (excitable)h(b)n(oundaries)912 b FB(19)24 177 y Fz(with)21 b Fp(f)28 b Fz(=)23 b Fp(g)265 153 y Fv(2)p Fs(=\037)263 191 y(\037; )318 196 y Fl(1)337 177 y Fz(.)39 b(But)21 b Fp(f)26 b Fz(satis\014es)c(prop)q(erties)g(of)f(prop)q(osition)h(3.1) e(with)i Fp(\032)g Fz(=)i(2)p Fp(\037=\037)f Fz(=)g(2.)38 b(Hence,)24 240 y Fp(r)45 247 y Fv(1)64 240 y Fz(\()p Fp(f)5 b Fz(;)j FA(y)q Fz(\))18 b(=)j Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)381 223 y Fs(\032)p Fy(\000)p Fv(2)445 240 y Fz(\))f(and)g(although)g Fq(kr)p Fp(f)5 b Fz(\()p FA(y)q Fz(\))p Fq(k)18 b Fz(=)j Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)1150 223 y Fs(\032)p Fy(\000)p Fv(1)1214 240 y Fz(\),)g(the)f(dominan)o(t)g(part)f(of)g(the)h(scalar)24 296 y(pro)q(duct)e(is)h(in)g(fact)e(strictly)i(zero)f(since)h(w)o(e)f (ha)o(v)o(e)g(c)o(hosen)g(the)h(parametres)e(of)h(the)g(harmonic)h (function)24 353 y(in)d(suc)o(h)f(a)g(w)o(a)o(y)f(as)h(to)g(giv)o(e)g (a)g(gradien)o(t)g(at)g(the)g(b)q(oundaries)i(exactly)e(p)q(erp)q (endicul)q(ar)i(to)e(the)g(mean)h(drifts.)24 409 y(Hence,)22 b Ft(E)r Fz(\()p Fp(Y)262 393 y Fv(2)249 421 y Fs(n)q Fv(+1)334 409 y Fq(\000)14 b Fp(Y)420 393 y Fv(2)409 420 y Fs(n)439 409 y Fq(jF)485 416 y Fs(n)508 409 y Fz(\))21 b(=)h Fq(O)q Fz(\(1\))e(whic)o(h)h(means)f(that)g(it)g(can)h(b)q(e)g(b) q(ounded)h(from)d(b)q(elo)o(w)i(b)o(y)g(some)24 466 y(constan)o(t.)61 569 y(Similarly)l(,)c(for)e Fp(r)e(>)g Fz(1)i(and)g Fp(f)j Fz(=)13 b Fp(g)653 545 y Fv(2)p Fs(r)q(=\037)651 584 y(\037; )706 589 y Fl(1)442 675 y Ft(E)s Fz(\()p Fp(Y)527 656 y Fv(2)p Fs(r)514 686 y(n)p Fv(+)q(1)595 675 y Fq(\000)e Fp(Y)677 656 y Fv(2)p Fs(r)667 686 y(n)714 675 y Fq(jF)760 682 y Fs(n)783 675 y Fz(\))h(=)h(\()p Fq(r)p Fp(f)5 b Fz(\()p FA(y)q Fz(\))p Fp(;)j Ft(E)p Fz(\()p Fc(\022)1103 682 y Fs(n)p Fv(+1)1171 675 y Fq(jF)1217 682 y Fs(n)1240 675 y Fz(\)\))h(+)i Fp(r)1352 682 y Fv(1)1371 675 y Fz(\()p Fp(f)5 b Fz(;)j FA(y)q Fz(\))p Fp(:)24 771 y Fz(Again)18 b(the)f(function)i Fp(f)k Fz(satis\014es)17 b(the)h(conditions)h(of)e (prop)q(osition)i(3.1)d(with)i Fp(\032)e Fz(=)i(2)p Fp(r)q(\037=\037)e Fz(=)h(2)p Fp(r)h Fz(and)g(the)24 827 y(dominan)o(t)f(part)f(of)g(the)h (scalar)g(pro)q(duct)g(v)m(anishes.)26 b(T)l(o)17 b(conclude,)h(it)f (is)g(enough)h(to)e(pro)o(v)o(e)g(that)g Fp(X)1812 805 y Fv(2)p Fs(p)1848 810 y Fl(0)1808 833 y Fs(n)1883 827 y Fz(is)24 889 y(a)h(submartingale)i(for)e Fp(p)458 896 y Fv(0)495 889 y Fz(=)g Fp(\037=)p Fz(2.)28 b(But)18 b Fp(X)797 867 y Fv(2)p Fs(p)833 872 y Fl(0)793 895 y Fs(n)868 889 y Fz(=)g Fp(f)5 b Fz(\()p Fp(\021)989 896 y Fs(n)1012 889 y Fz(\))17 b(with)h Fp(f)23 b Fz(=)17 b Fp(g)1274 865 y Fv(2)p Fs(p)1310 870 y Fl(0)1326 865 y Fs(=\014)1272 904 y(\014)r(;\014)1324 909 y Fl(1)1385 889 y Fz(that)g(satis\014es)i(conditions)g(of)24 946 y(prop)q(osition)d(3.1)e(with)h Fp(\032)e Fz(=)g(2)p Fp(p)570 953 y Fv(0)589 946 y Fp(\014)r(=\014)h Fz(=)f(2)p Fp(p)774 953 y Fv(0)794 946 y Fz(.)61 1049 y(On)j(the)f(b)q(oundaries,) h(it)g(is)f(easily)i(established)f(that)f Fp(X)1049 1027 y Fv(2)p Fs(p)1085 1032 y Fl(0)1045 1055 y Fs(n)1118 1049 y Fz(is)h(a)f(submartingale.)20 b(In)c(fact,)425 1146 y Ft(E)s Fz(\()p Fp(X)513 1124 y Fv(2)p Fs(p)549 1129 y Fl(0)509 1159 y Fs(n)p Fv(+1)590 1146 y Fq(\000)10 b Fp(X)677 1127 y Fv(2)p Fs(p)713 1132 y Fl(0)673 1157 y Fs(n)731 1146 y Fq(jF)777 1153 y Fs(n)800 1146 y Fz(\))i(=)h(\()p Fq(r)p Fp(f)5 b Fz(\()p FA(y)q Fz(\))p Fp(;)j Ft(E)p Fz(\()p Fc(\022)1120 1153 y Fs(n)p Fv(+1)1188 1146 y Fq(jF)1234 1153 y Fs(n)1257 1146 y Fz(\)\))h(+)i Fp(r)1369 1153 y Fv(1)1388 1146 y Fz(\()p Fp(f)5 b Fz(;)j FA(y)q Fz(\))p Fp(;)24 1274 y Fz(and)16 b Fq(r)p Fp(f)21 b Fz(=)249 1253 y Fv(2)p Fs(p)285 1258 y Fl(0)p 249 1263 53 2 v 264 1289 a Fs(\014)307 1274 y Fp(g)336 1222 y Fl(2)p Fk(p)367 1229 y Fl(0)p 335 1233 49 2 v 350 1253 a Fk(\014)388 1240 y Fy(\000)p Fv(1)329 1288 y Fs(\014)r(;\014)381 1293 y Fl(1)436 1274 y Fz(.)j(But)16 b(\()p Fq(r)p Fp(g)642 1281 y Fs(\014)r(;\014)694 1286 y Fl(1)712 1274 y Fp(;)p 733 1248 30 2 v 8 w FA(n)761 1257 y Fs([)777 1274 y Fz(\))f Fp(>)g(\017)i Fz(and)g Fq(kr)p Fp(f)5 b Fz(\()p FA(y)q Fz(\))p Fq(k)13 b Fz(=)j Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)1355 1257 y Fv(2)p Fs(p)1391 1262 y Fl(0)1406 1257 y Fy(\000)p Fv(1)1453 1274 y Fz(\).)24 b(Hence)18 b(the)f(\014rst)f (term)24 1330 y(dominates)d(the)h(subleading)h(remainder)f Fp(r)773 1337 y Fv(1)792 1330 y Fz(.)19 b(F)l(or)13 b FA(y)h Fz(b)q(eing)h(in)f(the)f(in)o(terior)h(space,)g(it)f(is)h (enough)g(to)f(pursue)24 1387 y(the)i(T)l(a)o(ylor)g(expansion)h(up)f (to)g(second)h(order)f(b)q(ecause)h(the)f(drift)h(v)m(anishes)g(in)g (the)f(in)o(terior.)21 b(This)15 b(yields)116 1507 y Ft(E)r Fz(\()p Fp(f)5 b Fz(\()p Fp(\021)229 1514 y Fs(n)p Fv(+1)300 1507 y Fz(\))10 b Fq(\000)g Fp(f)5 b Fz(\()p Fp(\021)441 1514 y Fs(n)464 1507 y Fz(\))p Fq(j)p Fp(\021)518 1514 y Fs(n)553 1507 y Fz(=)13 b(\()p FA(y)q Fp(;)8 b Fz(0\)\))39 b(=)849 1476 y(2)p Fp(p)895 1483 y Fv(0)p 849 1497 66 2 v 867 1538 a Fp(\014)919 1507 y Fz(\()942 1476 y(2)p Fp(p)988 1483 y Fv(0)p 942 1497 V 960 1538 a Fp(\014)1022 1507 y Fq(\000)10 b Fz(1\))p Fp(g)1137 1456 y Fl(2)p Fk(p)1168 1463 y Fl(0)p 1136 1466 49 2 v 1151 1487 a Fk(\014)1190 1474 y Fy(\000)p Fv(2)1130 1522 y Fs(\014)r(;\014)1182 1527 y Fl(1)1237 1507 y Fz([\()1273 1476 y Fp(@)s(g)1322 1483 y Fs(\014)r(;\014)1374 1488 y Fl(1)p 1272 1497 119 2 v 1297 1538 a Fp(@)s(y)1346 1545 y Fv(1)1396 1507 y Fz(\))1414 1488 y Fv(2)1443 1507 y Fz(+)1494 1476 y Fp(@)s(g)1543 1483 y Fs(\014)r(;\014)1595 1488 y Fl(1)p 1494 1497 V 1519 1538 a Fp(@)s(y)1568 1545 y Fv(2)1618 1507 y Fz(\))1636 1488 y Fv(2)1655 1507 y Fz(])844 1633 y(+2)907 1602 y(2)p Fp(p)953 1609 y Fv(0)p 907 1623 66 2 v 925 1664 a Fp(\014)977 1633 y(g)1006 1582 y Fl(2)p Fk(p)1037 1589 y Fl(0)p 1005 1592 49 2 v 1020 1613 a Fk(\014)1059 1600 y Fy(\000)p Fv(1)999 1648 y Fs(\014)r(;\014)1051 1653 y Fl(1)1106 1633 y Fz([\()1142 1602 y Fp(@)1169 1585 y Fv(2)1188 1602 y Fp(g)1210 1609 y Fs(\014)r(;\014)1262 1614 y Fl(1)p 1141 1623 139 2 v 1175 1665 a Fp(@)s(y)1226 1650 y Fv(2)1224 1678 y(1)1285 1633 y Fz(\))f(+)1363 1602 y Fp(@)1390 1585 y Fv(2)1409 1602 y Fp(g)1431 1609 y Fs(\014)r(;\014)1483 1614 y Fl(1)p 1363 1623 V 1397 1665 a Fp(@)s(y)1448 1650 y Fv(2)1446 1678 y(2)1506 1633 y Fz(\)])h(+)g Fp(r)1613 1640 y Fv(2)1632 1633 y Fz(\()p Fp(g)1679 1582 y Fl(2)p Fk(p)1710 1589 y Fl(0)p 1678 1592 49 2 v 1693 1613 a Fk(\014)1672 1648 y Fs(\014)r(;\014)1724 1653 y Fl(1)1742 1633 y Fz(;)e FA(y)q Fz(\))p Fp(:)24 1790 y Fz(No)o(w,)18 b Fp(r)165 1797 y Fv(2)184 1790 y Fz(\()p Fp(g)231 1738 y Fl(2)p Fk(p)262 1745 y Fl(0)p 230 1749 V 245 1769 a Fk(\014)224 1805 y Fs(\014)r(;\014)276 1810 y Fl(1)294 1790 y Fz(;)8 b FA(y)q Fz(\))p Fp(:)16 b Fz(=)i Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)574 1773 y Fv(2)p Fs(p)610 1778 y Fl(0)626 1773 y Fy(\000)p Fs(\015)675 1790 y Fz(\))g(while)i(the)f(other)f (terms)g(are)g(of)g(order)g Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)1553 1773 y Fv(2)p Fs(p)1589 1778 y Fl(0)1605 1773 y Fy(\000)p Fv(2)1652 1790 y Fz(\).)29 b(Hence)19 b(the)24 1857 y(pro)q(cess)14 b(is)h(a)e(submartingale)i(pro)o(vided)g(that)e (the)h(n)o(umerical)i(co)q(e\016cien)o(t)1341 1837 y Fv(2)p Fs(p)1377 1842 y Fl(0)p 1341 1847 53 2 v 1356 1873 a Fs(\014)1406 1857 y Fq(\000)8 b Fz(1)13 b Fp(>)g Fz(0)g(what)h(happ)q(ens)h(for)24 1914 y(2)p Fp(p)70 1921 y Fv(0)102 1914 y Fp(>)e(\014)r Fz(.)19 b(This)d(remark)f(ends)h (the)f(pro)q(of)g(of)f(the)i(theorem.)24 2127 y FA(5.)23 b(The)c(mo)q(del)g(of)g(a)g(Mark)o(o)o(v)e(c)o(hain)j(in)f(t)o(w)o(o)f (adjacen)o(t)i(w)o(edges)d(with)j(p)q(ermeable)f(in)o(terface)88 2184 y(and)f(excitable)h(b)q(oundaries)73 2330 y Fz(5.1.)h Fu(Intr)n(o)n(duction)15 b(of)h(the)f(mo)n(del)28 b Fz(In)15 b(this)g(section,)g(the)g(space)f Ft(X)s Fz(will)i(b)q(e)g(comp)q(osed) e(b)o(y)h(t)o(w)o(o)e(adjacen)o(t)24 2386 y(quadran)o(ts,)h Fu(i.e.)22 b Ft(X)r Fz(=)14 b Ft(Z)-5 b Fq(\002)10 b Ft(Z)549 2370 y Fv(+)589 2386 y Fq(\021)k Ft(X)673 2370 y Fy(\000)710 2386 y Fq([)d Ft(X)787 2370 y Fv(+)829 2386 y Fz(with)16 b Ft(X)969 2370 y Fy(\000)1009 2386 y Fz(=)e Ft(Z)1091 2370 y Fy(\000)1128 2386 y Fq(\002)d Ft(Z)1207 2370 y Fv(+)1234 2386 y Fz(,)k Ft(X)1298 2370 y Fv(+)1338 2386 y Fz(=)f Ft(Z)1420 2370 y Fv(+)1457 2386 y Fq(\002)d Ft(Z)1536 2370 y Fv(+)1563 2386 y Fz(,)k Fp(@)1615 2393 y Fv(1)1635 2386 y Ft(X)r Fz(=)f Ft(Z)1763 2370 y Fy(\000)1800 2386 y Fq([)d Ft(Z)1874 2370 y Fv(+)1901 2386 y Fz(,)24 2443 y(and)k Fp(@)136 2450 y Fv(2)156 2443 y Ft(X)192 2426 y Fy(\000)231 2443 y Fz(=)e Fp(@)303 2450 y Fv(2)323 2443 y Ft(X)358 2426 y Fv(+)398 2443 y Fz(=)g Ft(Z)479 2426 y Fv(+)505 2443 y Fz(.)20 b(The)c(in)o(ternal)g 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b(that)24 2225 y(this)e(k)o(eeps)h(their)f (direction)i(unc)o(hanged.)61 2329 y(W)l(e)d(in)o(tro)q(duce,)g(as)f(w) o(e)g(did)i(in)f(Section)h(1.4,)d(transformed)h(c)o(hains)h(that)f(are) g(small)h(p)q(erturbations)g(of)f(the)24 2391 y(c)o(hain)g Fp(\030)h Fz(when)f(observ)o(ed)g(far)e(from)h(the)h(origin.)19 b(These)13 b(c)o(hains)g(are)f(\()p Fp(\030)1245 2374 y Fv(\(2\))1292 2391 y Fz(\))1310 2398 y Fs(n)1333 2391 y Fz(,)h(\()p Fp(\030)1399 2367 y Fv(\(1)p Fs(;)p Fy(\000)p Fv(\))1397 2397 y Fs(n)1483 2391 y Fz(\))1501 2398 y Fs(n)1524 2391 y Fz(,)g(and)f(\()p Fp(\030)1675 2367 y Fv(\(1)p Fs(;)p Fv(+\))1673 2397 y Fs(n)1759 2391 y Fz(\))1777 2398 y Fs(n)1800 2391 y Fz(;)h(they)24 2447 y(corresp)q(ond)k(to)f(mo)q(dels)i(with)f(complete)g(v)o(ertical)h (homogeneit)o(y)e(or)h(complete)g(horizon)o(tal)g(homogeneit)o(y)24 2504 y(in)g(the)h(left)f(or)f(righ)o(t)h(quadran)o(ts.)25 b(Therefore,)17 b(the)g(sym)o(b)q(ol)g Fp(])g Fz(tak)o(es)f(no)o(w)h 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Fz(and)h(since)g(it)p eop %%Page: 22 22 22 21 bop 24 74 a FB(22)1306 b(M.)13 b(Menshik)o(o)o(v)h(and)g(D.)f(P)o (etritis)24 169 y Fz(is)g(a)g(sto)q(c)o(hastic)h(matrix)e(on)i(a)f (\014nite)h(space,)f(it)h(admits)f(an)g(ergo)q(dic)i(in)o(v)m(arian)o (t)e(probabilit)o(y)i Fp(\031)1657 153 y Fs(])1672 169 y Fz(.)k(W)l(e)14 b(extend)24 226 y(also)h(the)g(notion)g(of)g(the)h (restricted)f(drift)g(in)o(tro)q(duced)i(in)f(de\014nition)h(2.3)d(to) 604 328 y FA(m)648 309 y Fs(])664 328 y Fz(\()p Fp(u;)8 b(S)s Fz(\))i(=)843 284 y Fn(X)836 383 y Fs(u)856 374 y Fr(0)868 383 y Fy(2)p Fs(S)922 328 y Fp(P)957 305 y Fs(])951 342 y(u;u)1001 333 y Fr(0)1016 328 y Fz(\()p FA(X)p Fz(\()p Fp(u)1118 309 y Fy(0)1128 328 y Fz(\))g Fq(\000)g FA(X)p Fz(\()p Fp(u)p Fz(\)\))p Fp(;)24 479 y Fz(v)m(alid)16 b(for)f(ev)o(ery)g(subset)h Fp(S)f Fq(\022)e Ft(U)589 463 y Fs(])602 479 y Fz(.)20 b(W)l(e)15 b(shall)i(b)q(e)f(in)o (terested)f(in)o(to)g(t)o(w)o(o)f(particular)i(restricted)g(drifts)487 586 y FA(m)531 568 y Fv(\(2\))578 586 y Fz(\()p Fp(\013;)8 b Fq(\025)p Fz(\))k(=)812 543 y Fn(X)759 644 y Fs(x)779 633 y Fr(0)779 655 y Fl(1)796 644 y Fy(\025)p Fv(0)p Fs(;x)871 633 y Fr(0)871 655 y Fl(2)888 644 y Fs(;\013)921 635 y Fr(0)939 586 y Fp(P)974 562 y Fv(\(2\))968 603 y(\()p Fs(x)1002 608 y Fl(1)1019 603 y Fs(;x)1049 608 y Fl(2)1066 603 y Fs(;\013)p Fv(\);\()p Fs(x)1157 592 y Fr(0)1157 614 y Fl(1)1173 603 y Fs(;x)1203 592 y Fr(0)1203 614 y Fl(2)1220 603 y Fs(;\013)1253 594 y Fr(0)1264 603 y Fv(\))1279 586 y Fz(\()p FA(x)1325 568 y Fy(0)1346 586 y Fq(\000)f FA(x)p Fz(\))p Fp(;)24 741 y Fz(and)462 842 y FA(m)506 824 y Fv(\(2\))552 842 y Fz(\()p Fp(\013;)d(<)p Fz(\))k(=)812 799 y Fn(X)733 900 y Fs(x)753 889 y Fr(0)753 911 y Fl(1)770 900 y Fs(<)p Fv(0)p Fs(;x)845 889 y Fr(0)845 911 y Fl(2)862 900 y Fs(;\013)895 891 y Fr(0)906 900 y Fv(=)p Fs(L)965 842 y Fp(P)1000 818 y Fv(\(2\))994 859 y(\()p Fs(x)1028 864 y Fl(1)1045 859 y Fs(;x)1075 864 y Fl(2)1092 859 y Fs(;\013)p Fv(\);\()p Fs(x)1183 848 y Fr(0)1183 870 y Fl(1)1199 859 y Fs(;x)1229 848 y Fr(0)1229 870 y Fl(2)1246 859 y Fs(;\013)1279 850 y Fr(0)1289 859 y Fv(\))1305 842 y Fz(\()p FA(x)1351 824 y Fy(0)1372 842 y Fq(\000)f FA(x)p Fz(\))p Fp(;)24 1006 y Fz(together)j(with)i(the)f(standard)g(drifts)g FA(m)739 990 y Fv(\(1)p Fs(;)p Fy(\000)p Fv(\))823 1006 y Fz(\()p Fp(\013)p Fz(\))g(and)g FA(m)1035 990 y Fv(\(1)p Fs(;)p Fv(+\))1119 1006 y Fz(\()p Fp(\013)p Fz(\).)73 1088 y Fu(De\014nition)h(5.1)23 b(L)n(et)11 b Fp(\031)473 1071 y Fs([)500 1088 y Fu(b)n(e)g(the)h(stationary)g(pr)n(ob)n(ability)f (for)h(the)g(matrix)g FA(Q)1366 1071 y Fs([)1394 1088 y Fu(for)g Fp([)g Fq(2)h(f)p Fz(\(2\))p Fp(;)8 b Fz(\(1)p Fp(;)f Fq(\000)p Fz(\))p Fp(;)h Fz(\()o(1)p Fp(;)f Fz(+\))p Fq(g)p Fu(.)24 1144 y(De\014ne)15 b(the)i(aver)n(age)f(drifts)g(on)g (the)h(b)n(oundaries)f(by)p 776 1225 44 2 v 776 1250 a FA(m)820 1231 y Fs([)848 1250 y Fz(=)898 1207 y Fn(X)896 1306 y Fs(\013)p Fy(2)p Fm(A)974 1250 y Fp(\031)1002 1231 y Fs([)1000 1261 y(\013)1024 1250 y FA(m)1068 1231 y Fs([)1084 1250 y Fz(\()p Fp(\013)p Fz(\))p Fp(;)24 1393 y Fu(and)g(the)g(aver)n(age)h(r)n(estricte)n(d)e(drifts)p 682 1474 V 682 1500 a FA(m)725 1481 y Fv(\(2\))772 1500 y Fz(\()p Fp(<)p Fz(\))e(=)906 1456 y Fn(X)904 1555 y Fs(\013)p Fy(2)p Fm(A)981 1500 y Fp(\031)1009 1481 y Fs([)1007 1511 y(\013)1032 1500 y FA(m)1076 1481 y Fv(\(2\))1123 1500 y Fz(\()p Fp(\013;)8 b(<)p Fz(\))p Fp(;)24 1643 y Fu(and)p 682 1719 V 682 1745 a FA(m)725 1726 y Fv(\(2\))772 1745 y Fz(\()p Fq(\025)p Fz(\))13 b(=)906 1701 y Fn(X)904 1800 y Fs(\013)p Fy(2)p Fm(A)981 1745 y Fp(\031)1009 1726 y Fs([)1007 1756 y(\013)1032 1745 y FA(m)1076 1726 y Fv(\(2\))1123 1745 y Fz(\()p Fp(\013;)8 b Fq(\025)p Fz(\))p Fp(:)24 1935 y Fz(W)l(e)15 b(are)g(no)o(w)f(able)i(to)f (describ)q(e)i(the)e(geometric)g(settings)h(of)e(the)i(problem.)61 2039 y(De\014ne,)f(for)g(ev)o(ery)g Fp(\014)g Fq(2)e Ft(R)n Fz(,)e(the)16 b(v)o(ectors)533 2165 y FA(g)559 2172 y Fy(\000)589 2165 y Fz(\()p Fp(\014)r 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2540 219 2 v 328 2567 a(cos\()p Fs(\014)r( )435 2571 y Fr(\000)460 2567 y Fy(\000)p Fs( )510 2573 y Fk(L)534 2567 y Fv(\))552 2550 y Fz(.)73 2632 y Fu(De\014nition)16 b(5.2)23 b(F)m(or)16 b(every)g Fp(\014)f Fq(2)e Ft(R)m Fu(,)h(de\014ne)p 477 2697 39 2 v 477 2734 a Fp(D)p Fz(\()p Fp(\014)r Fz(\))f(=)f Fp(\025c)p Fz(\()p Fp(\014)r Fz(\)\()p FA(g)794 2741 y Fy(\000)822 2734 y Fz(\()p Fp(\014)r Fz(\))p Fp(;)p 907 2708 30 2 v 8 w FA(n)935 2715 y Fv(\(2\))982 2734 y Fz(\()p Fp(<)p Fz(\)\))e(+)g(\()p FA(g)1170 2741 y Fv(+)1200 2734 y Fz(\()p Fp(\014)r Fz(\))p Fp(;)p 1285 2708 V 8 w FA(n)1312 2715 y Fv(\(2\))1359 2734 y Fz(\()p Fq(\025)p Fz(\)\))p Fp(:)24 2835 y Fu(This)15 b(quantity)i(is)f(c)n(al) r(le)n(d)f(the)h(me)n(an)g(gain)g(on)g(the)h(interfac)n(e.)p eop %%Page: 23 23 23 22 bop 24 74 a FC(Markov)15 b(chains)g(in)g(a)g(we)n(dge)g(with)f (excitable)h(b)n(oundaries)912 b FB(23)73 169 y Fu(R)n(emark)17 b(5.1)23 b(Notic)n(e)d(that)p 569 133 39 2 v 22 w Fp(D)q Fz(\()p Fp(\014)r Fz(\))g Fu(is)g(a)i(pur)n(ely)e(ge)n(ometric)n(al)h (quantity)g(that)g(c)n(an)g(b)n(e)f(c)n(ompute)n(d)h(in)g(an)24 226 y(elementary)16 b(way)g(out)h(of)g(the)f(tr)n(ansition)g(pr)n(ob)n (abilities)f(of)h(the)h(Markov)f(chain.)61 354 y Fz(Denote)h Fp(\014)245 361 y Fv(min)323 354 y Fz(=)g(max)o(\()482 332 y Fy(\000)p Fs(\031)q(=)p Fv(2+)p Fs( )616 337 y Fl(0)p 482 344 152 2 v 534 370 a Fs( )557 374 y Fl(+)639 354 y Fp(;)664 332 y Fy(\000)p Fs(\031)q(=)p Fv(2+)p Fs( )798 338 y Fk(L)p 664 344 158 2 v 719 370 a Fs( )742 374 y Fr(\000)827 354 y Fz(\))g(and)h Fp(\014)979 361 y Fv(max)1063 354 y Fz(=)g(min\()1214 332 y Fs(\031)q(=)p Fv(2+)p Fs( )1321 337 y Fl(0)p 1214 344 125 2 v 1252 370 a Fs( )1275 374 y Fl(+)1343 354 y Fp(;)1369 332 y Fs(\031)q(=)p Fv(2+)p Fs( )1476 338 y Fk(L)p 1369 344 131 2 v 1409 370 a Fs( )1432 374 y Fr(\000)1504 354 y Fz(\).)27 b(Then)18 b Fp(\014)1709 361 y Fv(min)1786 354 y Fq(\024)g Fz(0)e Fq(\024)24 411 y Fp(\014)50 418 y Fv(max)132 411 y Fz(and)g(for)e(all)i(ph)o(ysical)h(c)o(hoices)f(of)f (the)g(angles)h Fp( )978 418 y Fv(0)997 411 y Fz(,)f Fp( )1055 418 y Fs(L)1080 411 y Fz(,)g Fp( )1138 418 y Fv(+)1167 411 y Fz(,)g(and)g Fp( )1313 418 y Fy(\000)1342 411 y Fz(,)g(strict)g(inequalities)j(hold.)73 492 y Fu(The)n(or)n(em)e (5.1)23 b(Cho)n(ose)15 b(a)h Fp(\014)565 499 y Fv(0)597 492 y Fq(2)d Fz(\()p Fp(\014)684 499 y Fv(min)744 492 y Fp(;)8 b(\014)791 499 y Fv(max)858 492 y Fz(\))15 b Fu(such)g(that)p 1085 456 39 2 v 16 w Fp(D)q Fz(\()p Fp(\014)1168 499 y Fv(0)1187 492 y Fz(\))e(=)f(0)p Fu(,)k(with)p 1416 456 V 16 w Fp(D)p Fz(\()p Fp(\014)1500 473 y Fy(\000)1498 505 y Fv(0)1530 492 y Fz(\))c Fp(<)h Fz(0)i Fu(and)p 1734 456 V 16 w Fp(D)p Fz(\()p Fp(\014)1818 473 y Fv(+)1816 505 y(0)1848 492 y Fz(\))d Fp(>)24 549 y Fz(0)p Fu(.)77 679 y(1.)23 b(Such)17 b(a)f Fp(\014)313 686 y Fv(0)349 679 y Fu(always)g(exists.)77 769 y(2.)23 b(If)16 b Fp(\014)211 776 y Fv(0)243 769 y Fp(<)d Fz(0)j Fu(then)g(the)h(Markov)f(chain)g(is) g(tr)n(ansient.)77 859 y(3.)23 b(If)16 b Fp(\014)211 866 y Fv(0)243 859 y Fp(>)d Fz(0)j Fu(then)g(the)h(Markov)f(chain)g(is) g(r)n(e)n(curr)n(ent.)77 948 y(4.)23 b(If)16 b Fp(\014)211 955 y Fv(0)243 948 y Fp(>)d Fz(0)p Fu(,)j(let)g Fp(p)431 955 y Fv(0)463 948 y Fz(=)516 926 y Fv(min)o(\()p Fs(\014)609 931 y Fl(0)626 926 y Fs(;\015)r Fv(\))p 516 938 154 2 v 584 964 a(2)674 948 y Fu(.)21 b(Then)16 b(for)g(every)g Fp(p)d(<)g(p)1127 955 y Fv(0)1163 948 y Fu(we)j(have)865 1038 y Ft(E)r Fp(\034)917 1016 y Fs(p)912 1053 y(K)962 1038 y Fq(\024)d Fp(C)s Fq(k)p FA(x)p Fq(k)1120 1020 y Fv(2)p Fs(p)1156 1025 y Fl(0)1173 1038 y Fp(:)77 1154 y Fu(5.)23 b(L)n(et)16 b Fp(p)239 1161 y Fv(0)271 1154 y Fz(=)324 1132 y Fv(min)o(\()p Fs(\014)417 1137 y Fl(0)434 1132 y Fs(;\015)r Fv(\))p 324 1144 V 392 1170 a(2)482 1154 y Fu(.)21 b(If)16 b Fp(p)588 1161 y Fv(0)620 1154 y Fp(<)d(\015)s(=)p Fz(2)i Fu(then)h(for)g(every)h Fp(p)12 b(>)h(p)1156 1161 y Fv(0)1176 1154 y Fu(,)924 1244 y Ft(E)r Fp(\034)976 1222 y Fs(p)971 1258 y(K)1021 1244 y Fz(=)g Fq(1)p Fp(:)73 1327 y Fu(Cor)n(ol)r(lary)k(5.1)23 b(If)15 b Fz(2)p Fp( )459 1334 y Fv(+)498 1327 y Fq(\000)c Fp( )574 1334 y Fv(0)606 1327 y Fp(>)i(\031)r(=)p Fz(2)i Fu(or)i Fz(2)p Fp( )855 1334 y Fy(\000)894 1327 y Fq(\000)10 b Fp( )969 1334 y Fs(L)1007 1327 y Fp(>)j(\031)r(=)p Fz(2)p Fu(,)j(then)g(the)g(chain)g(c)n(annot)g(b)n(e)g(er)n(go)n(dic.) 61 1493 y(Pr)n(o)n(of.)f Fz(If)g(one)g(of)g(these)g(conditions)g(is)h (v)o(eri\014ed,)f(then)g Fp(\014)1054 1500 y Fv(max)1134 1493 y Fp(<)e Fz(2.)20 b(Hence)15 b Fp(p)1395 1500 y Fv(0)1427 1493 y Fq(\024)1480 1473 y Fs(\014)1500 1478 y Fl(0)p 1480 1483 38 2 v 1490 1509 a Fv(2)1535 1493 y Fp(<)e Fz(1.)19 b(Assertion)c(5)g(of)24 1550 y(the)g(previous)g (theorem)g(establishes)h(that)e Ft(E)s Fp(\034)830 1557 y Fs(K)879 1550 y Fz(=)f Fq(1)j Fz(prev)o(en)o(ting)f(th)o(us)g(the)g (c)o(hain)g(from)g(b)q(eing)h(ergo)q(dic.)61 1691 y(The)f(pro)q(of)g (of)g(the)g(theorem)g(is)h(based)f(on)h(the)f(follo)o(wing)h(prop)q (osition.)73 1772 y Fu(Pr)n(op)n(osition)g(5.1)23 b(L)n(et)15 b Fp(h)502 1779 y Fs(\032;\016)549 1772 y Fz(\()p FA(y)q Fz(\))c(=)i(\()p Fp(y)715 1756 y Fv(2)713 1784 y(1)744 1772 y Fz(+)c Fp(y)812 1756 y Fv(2)810 1784 y(2)832 1772 y Fz(\))850 1756 y Fs(\032=)p Fv(2)913 1772 y Fz(cos)o(\()p Fp(\032)f Fz(arctan)1161 1752 y Fs(y)1178 1757 y Fl(2)p 1161 1762 35 2 v 1161 1788 a Fs(y)1178 1793 y Fl(1)1210 1772 y Fq(\000)h Fp(\016)r Fz(\))p Fu(.)20 b(F)m(or)c Fp(v)e Fz(=)f(\()p FA(y)q Fz(;)8 b Fp(\013)p Fz(\))p Fu(,)14 b(let)h Fp(f)1725 1779 y Fs(\014)1762 1772 y Fz(:)d Ft(V)r Fq(!)5 b Ft(R)24 1829 y Fu(b)n(e)15 b(de\014ne)n(d)h(by) 369 2051 y Fp(f)5 b Fz(\()p Fp(v)r Fz(\))12 b(=)516 1876 y Fn(8)516 1917 y(>)516 1930 y(>)516 1944 y(>)516 1958 y(>)516 1971 y(>)516 1985 y(>)516 1999 y(<)516 2080 y(>)516 2094 y(>)516 2108 y(>)516 2121 y(>)516 2135 y(>)516 2149 y(>)516 2162 y(:)577 1915 y Fp(h)603 1922 y Fs(\014)r(; )658 1927 y Fl(0)677 1915 y Fz(\()p FA(y)q Fz(\))d(+)h Fp(a)820 1922 y Fs(\013)845 1915 y Fp(h)871 1922 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )971 1927 y Fl(0)990 1915 y Fz(\()p FA(y)q Fz(\))215 b Fu(if)41 b FA(y)13 b Fq(2)g Fp(@)1447 1922 y Fv(1)1467 1915 y Ft(Y)1503 1898 y Fv(+)577 1994 y Fp(h)603 2001 y Fs(\014)r(; )658 2006 y Fl(0)677 1994 y Fz(\()p FA(y)q Fz(\))c(+)h Fp(a)820 2001 y Fv(0)840 1994 y Fp(h)866 2001 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )966 2006 y Fl(0)985 1994 y Fz(\()p FA(y)q Fz(\))220 b Fu(if)41 b FA(y)13 b Fq(2)1431 1958 y Fy(\016)1423 1998 y Ft(Y)1456 1952 y Fv(+)577 2051 y Fp(h)603 2058 y Fs(\014)r(; )658 2063 y Fl(0)677 2051 y Fz(\()p FA(y)q Fz(\))c(+)h Fp(b)816 2058 y Fs(\013)840 2051 y Fp(h)866 2058 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )966 2063 y Fl(0)985 2051 y Fz(\()p FA(y)q Fz(\))220 b Fu(if)41 b FA(y)13 b Fq(2)g Fp(@)1447 2058 y Fv(2)1467 2051 y Ft(V)577 2130 y Fp(c)p Fz(\()p Fp(\014)r Fz(\))p Fp(h)687 2137 y Fs(\014)r(; )742 2143 y Fk(L)766 2130 y Fz(\()p FA(y)q Fz(\))c(+)h Fp(c)905 2137 y Fs(L)932 2130 y Fz(~)-24 b Fp(c)p Fz(\()p Fp(\014)r Fz(\))p Fp(h)1041 2137 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )1141 2143 y Fk(L)1165 2130 y Fz(\()p FA(y)q Fz(\))40 b Fu(if)h FA(y)13 b Fq(2)1431 2094 y Fy(\016)1423 2134 y Ft(Y)1456 2088 y Fy(\000)577 2187 y Fp(c)p Fz(\()p Fp(\014)r Fz(\))p Fp(h)687 2194 y Fs(\014)r(; )742 2200 y Fk(L)766 2187 y Fz(\()p FA(y)q Fz(\))c(+)h Fp(c)905 2194 y Fs(\013)931 2187 y Fz(~)-24 b Fp(c)o Fz(\()p Fp(\014)r Fz(\))p Fp(h)1039 2194 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )1139 2200 y Fk(L)1164 2187 y Fz(\()p FA(y)q Fz(\))41 b Fu(if)g FA(y)13 b Fq(2)g Fp(@)1447 2194 y Fv(1)1467 2187 y Ft(Y)1503 2170 y Fy(\000)1530 2187 y Fp(:)24 2281 y Fu(If)p 69 2245 39 2 v 13 w Fp(D)q Fz(\()p Fp(\014)r Fz(\))f Fp(<)h Fq(\000)p Fp(\017)p Fu(,)i(then)f(it)g(is)g(always)g(p)n(ossible)f(to)h(cho)n(ose)g(the)g (p)n(ar)n(ametr)n(es)g Fz(\()p Fp(a)1346 2288 y Fs(\013)1370 2281 y Fz(\))p Fu(,)g Fz(\()p Fp(b)1454 2288 y Fs(\013)1478 2281 y Fz(\))p Fu(,)g Fz(\()p Fp(c)1562 2288 y Fs(\013)1587 2281 y Fz(\))p Fu(,)g Fp(c)p Fz(\()p Fp(\014)r Fz(\))p Fu(,)f(and)i Fz(~)-24 b Fp(c)p Fz(\()p Fp(\014)r Fz(\))24 2338 y Fu(in)15 b(such)i(a)f(manner)g(that)h Fp(D)q Fz(\()p Fp(v)r Fz(\))12 b(=)h Ft(E)r Fz(\()p Fp(f)5 b Fz(\()p Fp(\021)755 2345 y Fs(n)p Fv(+1)825 2338 y Fz(\))10 b Fq(\000)h Fp(f)5 b Fz(\()p Fp(\021)967 2345 y Fs(n)990 2338 y Fz(\))p Fq(j)p Fp(\021)1044 2345 y Fs(n)1079 2338 y Fz(=)13 b Fp(v)r Fz(\))p Fu(,)i(for)i(lar)n(ge)e Fq(k)p FA(y)q Fq(k)p Fu(,)92 2469 y Fq(\017)22 b Fu(is)16 b(a)h(str)n(ong)e (sup)n(ermartingale)h(ne)n(ar)g(the)g(b)n(oundaries,)g(i.e.)821 2558 y Fp(D)q Fz(\()p Fp(v)r Fz(\))11 b Fq(\024)i(\000)p Fp(K)s(\017)p Fq(k)p FA(y)q Fq(k)1149 2540 y Fs(\014)r Fy(\000)p Fv(1)1217 2558 y Fp(;)92 2665 y Fq(\017)22 b Fu(is)f(an)g(\\almost)g(martingale")h(\(with)f(r)n(esp)n(e)n(ct)f(to) h(its)g(values)g(ne)n(ar)f(the)i(b)n(oundaries\))e(i.e.)h(for)g FA(y)h Fq(2)145 2707 y Fy(\016)137 2747 y Ft(Y)170 2700 y Fy(\000)210 2743 y Fq([)258 2707 y Fy(\016)250 2747 y Ft(Y)283 2700 y Fv(+)312 2743 y Fu(,)820 2833 y Fq(j)p Fp(D)q Fz(\()p Fp(v)r Fz(\))p Fq(j)10 b Fz(=)j Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)1133 2814 y Fs(\014)r Fy(\000)p Fv(2)1201 2833 y Fz(\))p Fp(:)p eop %%Page: 24 24 24 23 bop 24 74 a FB(24)1306 b(M.)13 b(Menshik)o(o)o(v)h(and)g(D.)f(P)o (etritis)61 169 y Fu(Pr)n(o)n(of.)j Fz(The)g(pro)q(of)e(follo)o(ws)i (the)f(same)g(lines)i(as)e(the)g(pro)q(of)g(of)g(theorem)g(4.4)f(in)i (the)g(\014rst)f(mo)q(del.)21 b(Let)15 b(us)24 226 y(c)o(hec)o(k)h(the) h(sup)q(ermartingale)g(condition)h(only)f(on)f Fp(@)953 233 y Fv(2)973 226 y Ft(Y)-9 b Fz(.)20 b(The)d(other)f(cases)g(are)g (treated)g(in)h(a)f(completely)24 282 y(iden)o(tical)22 b(manner.)34 b(W)l(e)20 b(remark)g(\014rst)g(that)f(w)o(e)h(can)g(alw)o (a)o(ys)f(c)o(ho)q(ose)h(the)h(set)e(of)h(parametres)f(so)h(that)24 339 y Fp(a)48 346 y Fv(0)86 339 y Fz(=)f Fp(b)160 346 y Fv(0)197 339 y Fz(and)g Fp(b)309 346 y Fs(L)353 339 y Fz(=)g Fp(c)427 346 y Fs(L)472 339 y Fz(since)g(the)g(v)o(ectors)f (\()p Fp(a)868 346 y Fs(\013)892 339 y Fz(\),)h(\()p Fp(b)980 346 y Fs(\013)1004 339 y Fz(\),)g(and)g(\()p Fp(c)1184 346 y Fs(\013)1208 339 y Fz(\))f(are)h(de\014ned)h(only)f(mo) q(dulo)g(a)g(global)24 395 y(m)o(ultiplicativ)o(e)e(factor.)61 499 y(F)l(or)e Fp(v)f Fq(2)f Fp(@)246 506 y Fv(2)266 499 y Ft(V)301 482 y Fv(\(2\))361 499 y Fz(and)i Fq(k)p FA(y)q Fq(k)f Fz(large)h(enough,)g(compute)243 602 y Fp(D)q Fz(\()p Fp(v)r Fz(\))40 b(=)486 559 y Fn(X)459 663 y Fs(v)477 654 y Fr(0)489 663 y Fy(2)p Fm(V)539 653 y Fl(\(2\))585 602 y Fp(P)620 578 y Fv(\(2\))614 617 y Fs(v)q(v)650 607 y Fr(0)668 602 y Fz(\()p Fp(f)5 b Fz(\()p Fp(v)755 583 y Fy(0)766 602 y Fz(\))k Fq(\000)i Fp(f)5 b Fz(\()p Fp(v)r Fz(\)\))382 735 y(=)526 692 y Fn(X)459 796 y Fs(\013)482 787 y Fr(0)493 796 y Fs(;)p Fo(y)525 787 y Fr(0)536 796 y Fy(2)p Fm(Y)587 786 y Fl(\(2)p Fk(;)p Fl(+\))665 735 y Fp(P)700 711 y Fv(\(2\))694 750 y Fs(v)q(v)730 741 y Fr(0)748 735 y Fz(\()p Fp(f)g Fz(\()p Fp(v)835 716 y Fy(0)846 735 y Fz(\))10 b Fq(\000)g Fp(f)5 b Fz(\()p Fp(v)r Fz(\)\))k(+)1171 692 y Fn(X)1078 796 y Fs(\013)1101 787 y Fr(0)1112 796 y Fv(=)p Fs(L;)p Fo(y)1195 787 y Fr(0)1207 796 y Fy(2)p Fm(Y)1257 786 y Fl(\(2)p Fk(;)p Fr(\000)p Fl(\))1336 735 y Fp(P)1371 711 y Fv(\(2\))1365 750 y Fs(v)q(v)1401 741 y Fr(0)1419 735 y Fz(\()p Fp(f)c Fz(\()p Fp(v)1506 716 y Fy(0)1517 735 y Fz(\))10 b Fq(\000)g Fp(f)5 b Fz(\()p Fp(v)r Fz(\)\))382 855 y(=)42 b Fp(D)497 862 y Fv(1)527 855 y Fz(+)10 b Fp(D)610 862 y Fv(2)630 855 y Fp(:)24 950 y Fz(No)o(w,)321 1051 y Fp(D)359 1058 y Fv(1)420 1051 y Fz(=)563 1008 y Fn(X)496 1112 y Fs(\013)519 1102 y Fr(0)530 1112 y Fs(;)p Fo(y)562 1102 y Fr(0)573 1112 y Fy(2)p Fm(Y)624 1102 y Fl(\(2)p Fk(;)p Fl(+\))702 1051 y Fp(P)737 1027 y Fv(\(2\))731 1066 y Fs(v)q(v)767 1056 y Fr(0)785 1051 y Fz(\()p Fp(h)829 1058 y Fs(\014)r(; )884 1063 y Fl(0)903 1051 y Fz(\()p FA(y)950 1032 y Fy(0)960 1051 y Fz(\))g(+)h Fp(b)1054 1058 y Fs(\013)1077 1049 y Fr(0)1089 1051 y Fp(h)1115 1058 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )1215 1063 y Fl(0)1234 1051 y Fz(\()p FA(y)1281 1032 y Fy(0)1292 1051 y Fz(\))496 1174 y Fq(\000)p Fp(h)557 1181 y Fs(\014)r(; )612 1186 y Fl(0)632 1174 y Fz(\()p FA(y)q Fz(\))e(+)h Fp(b)771 1181 y Fs(\013)795 1174 y Fp(h)821 1181 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )921 1186 y Fl(0)940 1174 y Fz(\()p FA(y)q Fz(\)\))420 1253 y(=)563 1210 y Fn(X)496 1314 y Fs(\013)519 1304 y Fr(0)530 1314 y Fs(;)p Fo(y)562 1304 y Fr(0)573 1314 y Fy(2)p Fm(Y)624 1304 y Fl(\(2)p Fk(;)p Fl(+\))702 1253 y Fp(P)737 1229 y Fv(\(2\))731 1268 y Fs(v)q(v)767 1258 y Fr(0)785 1253 y Fz([\()p Fq(r)p Fp(h)880 1260 y Fs(\014)r(; )935 1265 y Fl(0)953 1253 y Fz(\()p FA(y)q Fz(\))p Fp(;)e FA(y)1068 1234 y Fy(0)1087 1253 y Fq(\000)j FA(y)q Fz(\))e(+)h Fp(R)1269 1260 y Fv(1)1288 1253 y Fz(\()p Fp(h)1332 1260 y Fs(\014)r(; )1387 1265 y Fl(0)1406 1253 y Fp(;)e FA(y)q Fp(;)g FA(y)1506 1234 y Fy(0)1525 1253 y Fq(\000)i FA(y)q Fz(\))496 1379 y(+)p Fp(b)551 1386 y Fs(\013)574 1377 y Fr(0)587 1379 y Fz(\()p Fq(r)p Fp(h)669 1386 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )769 1391 y Fl(0)788 1379 y Fz(\()p FA(y)q Fz(\))p Fp(;)e FA(y)903 1360 y Fy(0)922 1379 y Fq(\000)i FA(y)q Fz(\))f(+)i Fp(b)1089 1386 y Fs(\013)1112 1377 y Fr(0)1124 1379 y Fp(R)1159 1386 y Fv(1)1178 1379 y Fz(\()p Fp(h)1222 1386 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )1322 1391 y Fl(0)1341 1379 y Fp(;)d FA(y)q Fp(;)g FA(y)1441 1360 y Fy(0)1460 1379 y Fq(\000)i FA(y)q Fz(\))496 1448 y(+\()p Fp(b)569 1455 y Fs(\013)592 1446 y Fr(0)615 1448 y Fq(\000)h Fp(b)681 1455 y Fs(\013)705 1448 y Fz(\))p Fp(h)749 1455 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )849 1460 y Fl(0)867 1448 y Fz(\()p FA(y)q Fz(\)])p Fp(;)24 1543 y Fz(where)16 b Fp(R)191 1550 y Fv(1)210 1543 y Fz(\()p Fp(f)r(;)8 b Fq(\001)p Fp(;)g Fq(\001)p Fz(\))13 b(is)k(the)f(\014rst)f(order)h(remainder)h(of)e(the)h(T)l(a)o(ylor)f (expansion)i(of)f Fp(f)21 b Fz(that)15 b(can)h(b)q(e)g(sho)o(wn)g(to)24 1599 y(b)q(e)f(subleading)i(along)f(the)f(same)g(lines)i(as)d(for)h (the)g(one)h(w)o(edge)f(problem.)20 b(Similarly)l(,)206 1700 y Fp(D)244 1707 y Fv(2)305 1700 y Fz(=)474 1657 y Fn(X)381 1761 y Fs(\013)404 1752 y Fr(0)415 1761 y Fv(=)p Fs(L;)p Fo(y)498 1752 y Fr(0)510 1761 y Fy(2)p Fm(Y)560 1751 y Fl(\(2)p Fk(;)p Fr(\000)p Fl(\))640 1700 y Fp(P)675 1676 y Fv(\(2\))669 1715 y Fs(v)q(v)705 1705 y Fr(0)722 1700 y Fz([)p Fp(c)p Fz(\()p Fp(\014)r Fz(\))p Fp(h)845 1707 y Fs(\014)r(; )900 1713 y Fk(L)924 1700 y Fz(\()p FA(y)971 1681 y Fy(0)981 1700 y Fz(\))10 b(+)h Fp(b)1075 1707 y Fs(L)1101 1700 y Fz(~)-24 b Fp(c)p Fz(\()p Fp(\014)r Fz(\))p Fp(h)1210 1707 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )1310 1713 y Fk(L)1334 1700 y Fz(\()p FA(y)1381 1681 y Fy(0)1392 1700 y Fz(\))381 1823 y Fq(\000)p Fp(h)442 1830 y Fs(\014)r(; )497 1835 y Fl(0)517 1823 y Fz(\()p FA(y)q Fz(\))8 b(+)j Fp(b)656 1830 y Fs(\013)680 1823 y Fp(h)706 1830 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )806 1835 y Fl(0)825 1823 y Fz(\()p FA(y)q Fz(\)])305 1902 y(=)474 1859 y Fn(X)381 1963 y Fs(\013)404 1954 y Fr(0)415 1963 y Fv(=)p Fs(L;)p Fo(y)498 1954 y Fr(0)510 1963 y Fy(2)p Fm(Y)560 1953 y Fl(\(2)p Fk(;)p Fr(\000)p Fl(\))640 1902 y Fp(P)675 1878 y Fv(\(2\))669 1917 y Fs(v)q(v)705 1908 y Fr(0)722 1902 y Fz([)p Fp(c)p Fz(\()p Fp(\014)r Fz(\)\(\()p Fp(h)881 1909 y Fs(\014)r(; )936 1915 y Fk(L)959 1902 y Fz(\()p FA(y)1006 1883 y Fy(0)1017 1902 y Fz(\))f Fq(\000)g Fp(h)1116 1909 y Fs(\014)r(; )1171 1915 y Fk(L)1196 1902 y Fz(\()p FA(y)q Fz(\)\))381 2025 y(+)p Fp(c)p Fz(\()p Fp(\014)r Fz(\))p Fp(h)526 2032 y Fs(\014)r(; )581 2038 y Fk(L)606 2025 y Fz(\()p FA(y)q Fz(\))e Fq(\000)j Fp(h)751 2032 y Fs(\014)r(; )806 2037 y Fl(0)825 2025 y Fz(\()p FA(y)q Fz(\)\))381 2094 y(+)p Fp(b)436 2101 y Fs(L)463 2094 y Fz(~)-24 b Fp(c)p Fz(\()p Fp(\014)r Fz(\)\()p Fp(h)590 2101 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )690 2107 y Fk(L)714 2094 y Fz(\()p FA(y)761 2075 y Fy(0)772 2094 y Fz(\))9 b Fq(\000)i Fp(h)871 2101 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )971 2107 y Fk(L)996 2094 y Fz(\()p FA(y)q Fz(\)\))381 2163 y(+)p Fp(b)436 2170 y Fs(L)463 2163 y Fz(~)-24 b Fp(c)p Fz(\()p Fp(\014)r Fz(\)\()p Fp(h)590 2170 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )690 2176 y Fk(L)714 2163 y Fz(\()p FA(y)q Fz(\))9 b Fq(\000)h Fp(h)859 2170 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )959 2175 y Fl(0)978 2163 y Fz(\()p FA(y)q Fz(\))381 2232 y(+\()p Fp(b)454 2239 y Fs(L)490 2232 y Fq(\000)h Fp(b)556 2239 y Fs(\013)580 2232 y Fz(\))p Fp(h)624 2239 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )724 2244 y Fl(0)743 2232 y Fz(\()p FA(y)q Fz(\)])305 2311 y(=)474 2268 y Fn(X)381 2372 y Fs(\013)404 2362 y Fr(0)415 2372 y Fv(=)p Fs(L;)p Fo(y)498 2362 y Fr(0)510 2372 y Fy(2)p Fm(Y)560 2362 y Fl(\(2)p Fk(;)p Fr(\000)p Fl(\))640 2311 y Fp(P)675 2287 y Fv(\(2\))669 2326 y Fs(v)q(v)705 2316 y Fr(0)722 2311 y Fz([)p Fp(c)p Fz(\()p Fp(\014)r Fz(\)\()p Fq(r)p Fp(h)901 2318 y Fs(\014)r(; )956 2324 y Fk(L)979 2311 y Fz(\()p FA(y)q Fz(\))p Fp(;)d FA(y)1094 2292 y Fy(0)1113 2311 y Fq(\000)j FA(y)q Fz(\))e(+)h Fp(c)p Fz(\()p Fp(\014)r Fz(\))p Fp(R)1379 2318 y Fv(1)1398 2311 y Fz(\()p Fp(h)1442 2318 y Fs(\014)r(; )1497 2324 y Fk(L)1521 2311 y Fp(;)e FA(y)q Fp(;)g FA(y)1621 2292 y Fy(0)1640 2311 y Fq(\000)i FA(y)q Fz(\))381 2434 y(+)p Fp(c)p Fz(\()p Fp(\014)r Fz(\))p Fp(h)526 2441 y Fs(\014)r(; )581 2447 y Fk(L)606 2434 y Fz(\()p FA(y)q Fz(\))e Fq(\000)j Fp(h)751 2441 y Fs(\014)r(; )806 2446 y Fl(0)825 2434 y Fz(\()p FA(y)q Fz(\))381 2503 y(+)p Fp(b)436 2510 y Fs(L)463 2503 y Fz(~)-24 b Fp(c)p Fz(\()p Fp(\014)r Fz(\)\(\()p Fq(r)p Fp(h)646 2510 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )746 2516 y Fk(L)770 2503 y Fz(\()p FA(y)q Fz(\))p Fp(;)8 b FA(y)885 2484 y Fy(0)904 2503 y Fq(\000)i FA(y)q Fz(\))f(+)j(~)-24 b Fp(c)o Fz(\()p Fp(\014)r Fz(\))p Fp(R)1169 2510 y Fv(1)1188 2503 y Fz(\()p Fp(h)1232 2510 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )1332 2516 y Fk(L)1356 2503 y Fp(;)8 b FA(y)q Fp(;)g FA(y)1456 2484 y Fy(0)1475 2503 y Fq(\000)j FA(y)q Fz(\))381 2571 y(+)p Fp(b)436 2578 y Fs(L)462 2571 y Fz(\()q(~)-24 b Fp(c)p Fz(\()p Fp(\014)r Fz(\))p Fp(h)590 2578 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )690 2584 y Fk(L)714 2571 y Fz(\()p FA(y)q Fz(\))9 b Fq(\000)h Fp(h)859 2578 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )959 2583 y Fl(0)978 2571 y Fz(\()p FA(y)q Fz(\)\))381 2640 y(+\()p Fp(b)454 2647 y Fs(L)490 2640 y Fq(\000)h Fp(b)556 2647 y Fs(\013)580 2640 y Fz(\))p Fp(h)624 2647 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )724 2652 y Fl(0)743 2640 y Fz(\()p FA(y)q Fz(\)])p Fp(:)24 2749 y Fz(No)o(w)f(c)o(ho)q(ose)h Fp(c)p Fz(\()p Fp(\014)r Fz(\))h(=)415 2722 y Fs(h)435 2728 y Fk(\014)q(; )482 2735 y Fl(0)503 2722 y Fv(\()p Fo(y)q Fv(\))p 412 2739 144 2 v 412 2766 a Fs(h)432 2772 y Fk(\014)q(; )479 2780 y(L)506 2766 y Fv(\()p Fo(y)q Fv(\))571 2749 y Fz(for)f FA(y)i Fq(2)g Fp(@)745 2756 y Fv(2)764 2749 y Ft(Y)-9 b Fz(.)16 b(Hence,)c Fp(c)p Fz(\()p Fp(\014)r Fz(\))g(=)1125 2727 y Fv(cos)o(\()p Fs(\014)r( )1231 2731 y Fl(+)1256 2727 y Fy(\000)p Fs( )1306 2732 y Fl(0)1324 2727 y Fv(\))p 1122 2739 219 2 v 1122 2766 a(cos)o(\()p Fs(\014)r( )1228 2770 y Fr(\000)1253 2766 y Fy(\000)p Fs( )1303 2772 y Fk(L)1327 2766 y Fv(\))1346 2749 y Fz(.)18 b(This)12 b(c)o(hoice)g(guaran)o(tees)e(that)24 2837 y(the)15 b(second)h(line)h (in)g(the)e(last)h(expression)g(for)f Fp(D)881 2844 y Fv(2)916 2837 y Fz(v)m(anishes.)22 b(Similarly)l(,)c(w)o(e)d(c)o(ho)q (ose)h(~)-24 b Fp(c)p Fz(\()p Fp(\014)r Fz(\))13 b(=)1680 2809 y Fs(h)1700 2815 y Fk(\014)q Fr(\000)p Fl(1)p Fk(; )1786 2822 y Fl(0)1807 2809 y Fv(\()p Fo(y)q Fv(\))p 1677 2826 183 2 v 1677 2853 a Fs(h)1697 2859 y Fk(\014)q Fr(\000)p Fl(1)p Fk(; )1783 2867 y(L)1810 2853 y Fv(\()p Fo(y)q Fv(\))1878 2837 y Fz(=)p eop %%Page: 25 25 25 24 bop 24 74 a FC(Markov)15 b(chains)g(in)g(a)g(we)n(dge)g(with)f (excitable)h(b)n(oundaries)912 b FB(25)32 153 y Fv(cos)o(\(\()p Fs(\014)r Fy(\000)p Fv(1\))p Fs( )211 157 y Fl(+)235 153 y Fy(\000)p Fs( )285 158 y Fl(0)303 153 y Fv(\))p 29 165 292 2 v 29 191 a(cos)o(\(\()p Fs(\014)r Fy(\000)p Fv(1\))p Fs( )208 195 y Fr(\000)233 191 y Fy(\000)p Fs( )283 197 y Fk(L)306 191 y Fv(\))336 175 y Fz(that)11 b(guaran)o(tees)f(v)m (anshing)j(of)d(the)i(p)q(en)o(ultimate)g(line.)20 b(Therefore,)12 b Fp(D)1564 182 y Fv(2)1594 175 y Fz(reads,)g(in)g(leading)24 243 y(order)j Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)273 227 y Fs(\014)r Fy(\000)p Fv(1)340 243 y Fz(\),)127 352 y Fp(D)165 359 y Fv(2)197 352 y Fz(=)338 309 y Fn(X)245 413 y Fs(\013)268 404 y Fr(0)279 413 y Fv(=)p Fs(L;)p Fo(y)362 404 y Fr(0)374 413 y Fy(2)p Fm(Y)424 404 y Fl(\(2)p Fk(;)p Fr(\000)p Fl(\))503 352 y Fp(P)538 328 y Fv(\(2\))532 367 y Fs(v)q(v)568 358 y Fr(0)586 352 y Fz([)p Fp(c)p Fz(\()p Fp(\014)r Fz(\)\()p Fq(r)p Fp(h)765 359 y Fs(\014)r(; )820 365 y Fk(L)843 352 y Fz(\()p FA(y)q Fz(\))p Fp(;)8 b FA(y)958 333 y Fy(0)977 352 y Fq(\000)j FA(y)q Fz(\))e(+)h(\()p Fp(b)1162 359 y Fs(L)1198 352 y Fq(\000)g Fp(b)1263 359 y Fs(\013)1287 352 y Fz(\))p Fp(h)1331 359 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )1431 364 y Fl(0)1450 352 y Fz(\()p FA(y)q Fz(\)])e(+)j Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)1712 333 y Fs(\014)r Fy(\000)p Fv(2)1779 352 y Fz(\))p Fp(:)24 505 y Fz(Recollecting)17 b(terms,)e(w)o(e)f(get)321 606 y Fp(D)q Fz(\()p Fp(v)r Fz(\))41 b(=)h Fp(D)576 613 y Fv(1)605 606 y Fz(+)11 b Fp(D)689 613 y Fv(2)461 674 y Fz(=)42 b(\()p Fq(r)p Fp(h)620 681 y Fs(\014)r(; )675 686 y Fl(0)693 674 y Fz(\()p FA(y)q Fz(\))p Fp(;)8 b FA(n)808 656 y Fv(\(2\))853 674 y Fz(\()p Fp(\013)p Fz(;)g Fq(\025)p Fz(\)\))h(+)i Fp(c)p Fz(\()p Fp(\014)r Fz(\)\()p Fq(r)p Fp(h)1213 681 y Fs(\014)r(; )1268 687 y Fk(L)1291 674 y Fz(\()p FA(y)q Fz(\))p Fp(;)d(\025)p FA(n)1433 656 y Fv(\(2\))1478 674 y Fz(\()p Fp(\013)p Fz(;)g Fp(<)p Fz(\)\))538 754 y(+)p Fp(h)599 761 y Fs(\014)r Fy(\000)p Fv(1)p Fs(; )699 766 y Fl(0)718 754 y Fz(\()p FA(y)q Fz(\))797 711 y Fn(X)791 810 y Fs(\013)814 800 y Fr(0)823 810 y Fy(2)p Fm(A)878 754 y Fp(Q)914 730 y Fv(\(2\))914 769 y Fs(\013;\013)970 759 y Fr(0)983 754 y Fz(\()p Fp(b)1021 761 y Fs(\013)1044 752 y Fr(0)1066 754 y Fq(\000)i Fp(b)1131 761 y Fs(\013)1155 754 y Fz(\))g(+)h Fq(O)q Fz(\()p Fq(k)p FA(y)q Fq(k)1359 735 y Fs(\014)r Fy(\000)p Fv(2)1426 754 y Fz(\))p Fp(:)24 905 y Fz(No)o(w)k(w)o(e)g(can)h(determine)h (parametres)e(\()p Fp(b)766 912 y Fs(\013)790 905 y Fz(\))h(so)f(that)g Fp(D)q Fz(\()p Fp(v)r Fz(\))e Fq(\024)h(\000)p Fp(K)s(\017)p Fq(k)p FA(y)q Fq(k)1310 888 y Fs(\014)r Fy(\000)p Fv(1)1394 905 y Fz(for)h(ev)o(ery)h Fp(\013)p Fz(,)f(pro)o(vided)i(that)24 961 y(the)e(compatibilit)o(y)i(condition)f(holds,)g(i.e.)501 1062 y Fp(\025c)p Fz(\()p Fp(\014)r Fz(\)\()p FA(g)656 1069 y Fy(\000)684 1062 y Fz(\()p Fp(\014)r Fz(\))p Fp(;)p 769 1036 30 2 v 8 w FA(n)796 1043 y Fv(\(2\))844 1062 y Fz(\()p Fp(<)p Fz(\)\))9 b(+)i(\()p FA(g)1032 1069 y Fv(+)1061 1062 y Fz(\()p Fp(\014)r Fz(\))p Fp(;)p 1146 1036 V 8 w FA(n)1174 1043 y Fv(\(2\))1221 1062 y Fz(\()p Fq(\025)p Fz(\)\))h Fp(<)h Fq(\000)p Fp(\017;)24 1173 y Fz(since)j Fq(j)165 1145 y Fs(h)185 1151 y Fk(\014)q Fr(\000)p Fl(1)p Fk(; )271 1158 y Fl(0)p 153 1162 153 2 v 153 1189 a Fy(kr)p Fs(h)220 1195 y Fk(\014)q(; )267 1202 y Fl(0)288 1189 y Fy(k)310 1173 y Fq(j)d Fz(=)g Fq(O)q Fz(\(1\).)61 1285 y(The)20 b(b)q(eha)o(viour)h(on)f(the)g(other) g(b)q(oundaries)h(is)f(treated)g(in)h(exactly)f(the)g(same)g(lines)h (as)f(for)f(the)h(one)24 1342 y(w)o(edge)c(case.)24 b(As)17 b(for)f(the)h(\\almost)f(martingale")g(prop)q(ert)o(y)l(,)h(it)g(is)g (enough)g(to)f(remark)g(that)g(b)q(oth)h Fp(h)1818 1349 y Fs(\014)r Fy(\000)p Fv(1)p Fs(;\016)24 1398 y Fz(and)e Fp(h)138 1405 y Fs(\014)r(;\016)204 1398 y Fz(are)f(harmonic)i (functions.)61 1584 y Fu(Pr)n(o)n(of)g(of)g(the)n(or)n(em)h(5.1.)79 1724 y Fz(1.)22 b(The)15 b(existence)g(of)e(suc)o(h)i(a)f Fp(\014)637 1731 y Fv(0)670 1724 y Fz(is)h(obtained)f(b)o(y)g(the)h (con)o(tin)o(uit)o(y)f(of)p 1304 1688 39 2 v 14 w Fp(D)q Fz(\()p Fp(\014)r Fz(\))f(and)h(b)o(y)g(explicit)j(estimates)137 1781 y(of)p 189 1744 V 15 w Fp(D)q Fz(\()p Fp(\014)r Fz(\))e(near)g(the)g(edges)g(of)g(the)g(in)o(terv)m(al)i(\()p Fp(\014)946 1788 y Fv(min)1006 1781 y Fp(;)8 b(\014)1053 1788 y Fv(max)1120 1781 y Fz(\).)79 1874 y(2.)22 b(If)d Fp(\014)212 1881 y Fv(0)251 1874 y Fp(<)g Fz(0,)g(then)g(for)f(ev)o (ery)h Fp(\014)690 1881 y Fv(1)728 1874 y Fz(with)g Fp(\014)861 1881 y Fv(min)941 1874 y Fp(<)g(\014)1021 1881 y Fv(1)1059 1874 y Fp(<)g(\014)1139 1881 y Fv(0)1158 1874 y Fz(,)h(w)o(e)e(can)h (construct)g(a)f(sup)q(ermartingale)137 1930 y(on)f(the)h(whole)g (space.)26 b(Moreo)o(v)o(er,)16 b(since)i Fp(\014)913 1937 y Fv(0)949 1930 y Fp(<)e Fz(0,)i(the)f(leading)i(con)o(tribution)f (of)e(the)i(Ly)o(apuno)o(v's)137 1987 y(function)13 b(is)g(in)h Fq(k)p FA(y)q Fq(k)481 1970 y Fs(\014)501 1975 y Fl(1)531 1987 y Fz(that)e(remains)g(b)q(ounded)i(outside)f(a)f(\014nite)i(set)e (near)g(the)h(origin.)19 b(W)l(e)13 b(conclude)137 2043 y(therefore)i(transience.)79 2137 y(3.)22 b(If)d Fp(\014)212 2144 y Fv(0)249 2137 y Fp(>)g Fz(0,)f(then)h(for)f(ev)o(ery)g Fp(\014)686 2144 y Fv(1)724 2137 y Fz(with)h(0)f Fp(<)g(\014)951 2144 y Fv(1)989 2137 y Fp(<)g(\014)1068 2144 y Fv(0)1088 2137 y Fz(,)h(w)o(e)f(can)g(construct)g(a)h(sup)q(ermartingale)g(for) 137 2193 y(large)e Fq(k)p FA(y)q Fq(k)p Fz(.)24 b(On)18 b(the)f(other)f(hand,)i(the)f(leading)h(con)o(tribution)g(to)f(the)g (Ly)o(apuno)o(v's)f(function)i(is)g(in)137 2250 y Fq(k)p FA(y)q Fq(k)212 2233 y Fs(\014)232 2238 y Fl(0)250 2250 y Fz(.)i(Therefore,)15 b(its)g(lev)o(el)i(sets)e(are)f(compact)h(and)h (w)o(e)f(conclude)h(recurrence.)79 2350 y(4.)22 b(If)i Fp(\014)217 2357 y Fv(0)262 2350 y Fp(>)i Fz(0,)e(then)f(for)g(ev)o (ery)g Fp(p)f Fz(with)i(0)h Fp(<)h(p)g(<)g(p)1099 2357 y Fv(0)1144 2350 y Fz(=)1210 2327 y Fv(min)o(\()p Fs(\014)r(;\015)r Fv(\))p 1210 2339 138 2 v 1270 2365 a(2)1353 2350 y Fz(,)e(w)o(e)f(can) g(construct)g(a)f(strict)137 2406 y(sup)q(ermartingale,)16 b(as)f(in)h(the)f(one)g(w)o(edge)h(problem,)f(v)o(erifying)h(condition) 719 2507 y Ft(E)r Fz(\()p Fp(X)806 2485 y Fv(2)p Fs(p)802 2520 y(n)p Fv(+1)883 2507 y Fq(\000)11 b Fp(X)971 2488 y Fv(2)p Fs(p)967 2518 y(n)1007 2507 y Fq(jF)1053 2514 y Fs(n)1076 2507 y Fz(\))h Fq(\024)h(\000)p Fp(\017X)1249 2488 y Fv(2)p Fs(p)p Fy(\000)p Fv(2)1245 2518 y Fs(n)137 2608 y Fz(whic)o(h)j(pro)o(v)o(es)f(\014niteness)h(of)f(the)g Fp(p)p Fz(-th)g(momen)o(t)g(of)g(passage)f(time.)79 2701 y(5.)22 b(Finally)l(,)16 b(for)d Fp(p)f(>)h(p)474 2708 y Fv(0)494 2701 y Fz(,)h(if)g Fp(p)584 2708 y Fv(0)616 2701 y Fp(<)f(\015)s(=)p Fz(2,)f(w)o(e)i(can)g(construct)f(pro)q (cesses)h Fp(X)1341 2708 y Fs(n)1378 2701 y Fz(and)g Fp(Y)1491 2708 y Fs(n)1529 2701 y Fz(as)f(in)i(the)f(one)f(w)o(edge)137 2758 y(problem,)j(v)o(erifying)g(the)f(criterion)h(of)f(the)g (non-existence)i(of)e(momen)o(ts.)p eop %%Page: 26 26 26 25 bop 24 74 a FB(26)1306 b(M.)13 b(Menshik)o(o)o(v)h(and)g(D.)f(P)o (etritis)24 169 y FA(6.)23 b(Conclusion)18 b(and)g(op)q(en)g(problems) 61 315 y Fz(W)l(e)24 b(ha)o(v)o(e)g(established)i(that)d(the)i(ergo)q (dic)g(prop)q(erties)f(of)g(the)g(c)o(hains)h(are)f(determined)i(b)o(y) e(simple)24 372 y(geometrical)18 b(c)o(haracteristics)g(that)f(are)g (deriv)o(ed)i(form)e(the)h(Ly)o(apuno)o(v's)g(function)g(metho)q(d)g (and)g(this)h(in)24 428 y(spite)c(of)g(the)h(complication)g(steming)g (from)e(the)h(in)o(ternal)h(degrees)g(of)f(freedom.)61 532 y(The)j(results)g(of)g(the)g(second)g(mo)q(del)h(can)f(b)q(e)g (extended)h(with)f(only)h(computational)f(complications)h(to)24 588 y(w)o(edges)d(with)g(an)g(arbitrary)f(n)o(um)o(b)q(er)i(of)e (sectors)h(pro)o(vided)h(that)e(the)h(gluing)i(of)d(sectors)h(do)q(es)g (not)g(result)24 645 y(to)22 b(a)g(problem)h(without)g(external)g(b)q (oundaries.)43 b(The)23 b(treatmen)o(t)f(for)g(geometries)g(without)h (external)24 701 y(b)q(oundaries)16 b(remains)g(op)q(en)f(for)g(the)g (momen)o(t.)24 876 y FA(References)46 993 y Fz([1])22 b(F.)15 b(Asm)o(ussen,)f Fu(Applie)n(d)i(pr)n(ob)n(ability)g(in)g (queues)p Fz(,)f(Wiley)l(,)h(Chic)o(hester)g(\(1987\).)46 1087 y([2])22 b(S.)15 b(Aspandiiaro)o(v,)g(R.)g(Iasnogoro)q(dski,)f(M.) g(Menshik)o(o)o(v,)h(P)o(assage-time)f(momen)o(ts)g(for)g(non-negativ)o (e)117 1143 y(sto)q(c)o(hastic)e(pro)q(cesses)g(and)g(an)f(application) j(to)d(re\015ected)h(random)f(w)o(alks)h(in)g(a)g(quadran)o(t,)f(Ann.)h (Prob.)117 1200 y(24:932{960)g(\(1996\).)46 1294 y([3])22 b(I.)e(M.)g(Asymon)o(t,)g(G.)g(F)l(a)o(y)o(olle,)h(M.)f(Menshik)o(o)o (v,)h(Random)g(w)o(alks)f(in)h(a)f(quarter)g(plane)h(with)f(zero)117 1350 y(drifts:)g(transience)c(and)f(recurrence,)h Fu(J.)g(Appl.)g(Pr)n (ob.)p Fz(,)f(32:941{955)d(\(1995\).)46 1444 y([4])22 b(G.)f(F)l(a)o(y)o(olle,)i(V.)e(Malyshev,)i(M.)d(Menshik)o(o)o(v,)j Fu(T)m(opics)e(in)h(the)g(c)n(onstructive)f(the)n(ory)i(of)f(c)n (ountable)117 1500 y(Markov)17 b(chains)p Fz(,)d(Cam)o(bridge)h(Univ)o (ersit)o(y)h(Press,)e(Cam)o(bridge)i(\(1994\).)46 1594 y([5])22 b(M.)11 b(Menshik)o(o)o(v,)h(S.)g(P)o(op)q(o)o(v,)f(Exact)g(p) q(o)o(w)o(er)g(estimates)h(for)f(coun)o(table)h(Mark)o(o)o(v)e(c)o (hains,)j Fu(Markov)g(Pr)n(o)n(c.)117 1651 y(R)n(el.)j(Fields)p Fz(,)d(1:57{78)g(\(1995\).)46 1744 y([6])22 b(M.)14 b(Menshik)o(o)o(v,) 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(Williams,)k(Bro)o(wnian)c(motion)f(in)i(a)f(w)o(edge)f(with)h(oblique) i(re\015ection,)117 2308 y Fu(Commun.)16 b(Pur)n(e)h(Appl.)f(Math.)p Fz(,)f(38:405{443)d(\(1985\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9910250341145--