Content-Type: multipart/mixed; boundary="-------------0012291202823" This is a multi-part message in MIME format. ---------------0012291202823 Content-Type: text/plain; name="00-517.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-517.comments" AMS-Code: 37H99, 60H10 (primary), 34F05, 34E15 (secondary). ---------------0012291202823 Content-Type: text/plain; name="00-517.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-517.keywords" Stochastic resonance, noise-induced synchronization, double-well potential, additive noise, random dynamical systems, non-autonomous stochastic differential equations, singular perturbations, pathwise description, concentration of measure. ---------------0012291202823 Content-Type: application/postscript; name="Sres.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Sres.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58a Copyright 1986, 1994 Radical Eye Software %%Title: sres.dvi %%Pages: 46 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips sres -o %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2000.12.29:1737 %%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 /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin 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b(In)n(tro)r(d)n(u)n(ct)n(ion)59 2017 y FC(Since)24 b(it)o(s)f(in)o(tro)q(d)o(u)o(ct)o(ion)d(as)i(a)h (mo)q(d)o(el)g(for)f(t)n(h)o(e)g(p)q(er)q(io)q(dic)g(ap)o(p)q(earance)f (of)h(t)n(h)o(e)h(ice)g(age)q(s)f([BPSV)o(],)59 2073 y(st)o(o)q(c)o(h)o(ast)o(ic)16 b(re)q(son)o(ance)h(h)o(as)g(b)q(een)h (obs)q(erv)o(e)q(d)g(in)h(a)e(large)h(n)n(u)o(m)n(b)q(er)f(of)h(ph)o (ys)q(ical)f(an)o(d)h(biological)f(sys-)59 2130 y(t)o(ems,)c(includin)o (g)j(las)q(ers,)d(electronic)h(circuit)o(s)g(an)o(d)g(t)n(h)o(e)g(s)q (ensory)f(syst)o(em)f(of)i(cray\014sh)f(\(for)g(reviews)g(of)59 2186 y(ap)o(plica)o(t)o(ions,)g(s)q(ee)h(for)g(inst)o(ance)g([MW]\).) 130 2243 y(Th)o(e)g(m)o(ec)o(h)o(ani)q(sm)g(of)g(st)o(o)q(c)o(h)o(ast)o (ic)g(re)q(son)o(ance)g(can)h(b)q(e)g(illustra)o(t)o(e)q(d)g(in)h(a)f (s)q(imple)g(mo)q(d)o(el.)21 b(Cons)q(id)o(er)59 2299 y(t)n(h)o(e)d(o)o(v)o(erd)o(amp)q(e)q(d)f(mot)o(ion)g(of)h(a)g(part)o (icle)g(in)h(a)g(dou)n(ble-w)o(ell)f(p)q(ot)o(en)o(t)o(ial.)30 b(Th)o(e)18 b(t)o(w)o(o)e(p)q(ot)o(en)o(t)o(ial)i(w)o(ells)59 2356 y(d)o(e)q(scr)q(ib)q(e)e(t)o(w)o(o)f(m)o(acrosco)o(pically)g (di\013eren)o(t)h(st)o(a)o(t)o(e)q(s)f(of)h(t)n(h)o(e)g(u)o(np)q(ert)o (urb)q(e)q(d)f(syst)o(em,)g(for)g(inst)o(ance)h(cold)59 2412 y(an)o(d)j(w)o(arm)e(clim)o(a)o(t)o(e.)32 b(Th)o(e)19 b(part)o(icle)g(i)q(s)g(su)n(b)r(ject)g(t)o(o)g(t)o(w)o(o)f(di\013eren) o(t)h(kin)o(ds)g(of)g(p)q(ert)o(urba)o(t)o(ion:)25 b(a)19 b(d)o(e-)59 2469 y(t)o(ermini)q(st)o(ic)f(p)q(er)q(io)q(dic)h(dr)q (ivin)o(g)h(force)e(\(su)o(c)o(h)h(as)g(t)n(h)o(e)g(p)q(er)q(io)q(dic)g (v)n(ar)q(ia)o(t)o(ion)e(of)i(insula)o(t)o(ion)g(ca)n(us)q(e)q(d)g(b)o (y)59 2525 y(t)n(h)o(e)14 b(c)o(h)o(an)o(gin)o(g)g(eccen)o(tr)q(icit)o (y)e(of)i(t)n(h)o(e)g(eart)n(h's)e(orbit\),)h(an)o(d)h(an)f(addit)o(iv) o(e)h(noi)q(s)q(e)g(\(mo)q(d)o(elin)o(g)g(t)n(h)o(e)g(ran)o(dom)59 2581 y(in\015uence)k(of)e(t)n(h)o(e)h(w)o(ea)o(t)n(h)o(er\).)24 b(Eac)o(h)16 b(of)h(t)n(h)o(e)q(s)q(e)g(t)o(w)o(o)e(p)q(ert)o(urba)o(t) o(ions)o(,)g(t)o(ak)o(en)g(b)o(y)i(it)o(s)q(elf,)h(do)q(e)q(s)e(not)h (pro-)59 2638 y(d)o(u)o(ce)e(an)o(y)g(in)o(t)o(ere)q(st)o(in)o(g)g(dyn) o(amics)g(\(f)q(rom)e(t)n(h)o(e)i(p)q(oin)o(t)h(of)f(view)g(of)h(re)q (son)o(ance\).)k(In)o(d)o(ee)q(d,)c(t)n(h)o(e)f(p)q(er)q(io)q(dic)59 2694 y(dr)q(ivin)o(g)f(i)q(s)g(assu)o(m)o(e)q(d)g(t)o(o)f(h)o(a)o(v)o (e)g(t)o(o)q(o)g(sm)o(all)h(an)g(amplit)o(ud)o(e)f(t)o(o)h(allo)o(w)f (for)g(an)o(y)g(trans)q(it)o(ions)f(b)q(et)o(w)o(een)g(t)n(h)o(e)59 2751 y(p)q(ot)o(en)o(t)o(ial)e(w)o(ells)i(in)h(t)n(h)o(e)e(a)o(bs)q (ence)g(of)g(noi)q(s)q(e.)18 b(On)12 b(t)n(h)o(e)f(ot)n(h)o(er)g(h)o (an)o(d,)g(wit)n(h)o(ou)o(t)g(p)q(er)q(io)q(dic)g(forcin)o(g,)g(addit)o (iv)o(e)59 2807 y(noi)q(s)q(e)k(will)h(ca)n(us)q(e)e(t)n(h)o(e)h(part)o (icle)f(t)o(o)h(ju)o(mp)g(f)q(rom)e(on)o(e)h(p)q(ot)o(en)o(t)o(ial)g(w) o(ell)h(t)o(o)g(t)n(h)o(e)f(ot)n(h)o(er)g(a)o(t)g(ran)o(dom)f(t)o(im)o (e)q(s.)934 2936 y(1)p eop %%Page: 2 2 2 1 bop 59 164 a FC(Th)o(e)14 b(exp)q(ect)o(e)q(d)g(t)o(im)o(e)g(b)q (et)o(w)o(een)f(trans)q(it)o(ions)g(i)q(s)h(giv)o(en)h(asympt)o(ot)o (ically)l(,)d(in)k(t)n(h)o(e)e(sm)o(all)h(noi)q(s)q(e)f(limit,)h(b)o(y) 59 220 y(Kram)o(ers')h(t)o(im)o(e,)h(whic)o(h)h(i)q(s)g(pro)o(p)q(ort)o (ion)o(al)d(t)o(o)j(t)n(h)o(e)f(exp)q(on)o(en)o(t)o(ial)g(of)g(t)n(h)o (e)h(barr)q(ier)f(h)o(e)q(igh)o(t)g FB(H)22 b FC(o)o(v)o(er)16 b(t)n(h)o(e)59 277 y(noi)q(s)q(e)h(in)o(t)o(ens)q(it)o(y)e(square)q(d,) i(n)o(am)o(ely)f(e)721 260 y FA(H)q(=\033)790 249 y Fz(2)809 277 y FC(.)27 b(Wh)o(en)17 b(b)q(ot)n(h)g(p)q(ert)o(urba)o(t)o(ions)d (are)j(com)n(bin)o(e)q(d,)g(h)o(o)o(w)o(ev)o(er,)59 333 y(an)o(d)11 b(t)n(h)o(e)q(ir)g(amplit)o(ud)o(e)q(s)g(suit)o(a)o(bly)f (t)o(u)o(n)o(e)q(d,)i(t)n(h)o(e)f(part)o(icle)g(will)i(\015ip)f(bac)o (k)e(an)o(d)h(fort)n(h)g(b)q(et)o(w)o(een)e(t)n(h)o(e)i(w)o(ells)h(in) 59 390 y(a)h(clos)q(e)g(t)o(o)g(p)q(er)q(io)q(dic)g(w)o(ay)l(.)18 b(Th)n(us)12 b(t)n(h)o(e)h(in)o(t)o(er)q(n)o(al)f(noi)q(s)q(e)h(can)f (s)q(igni\014can)o(t)n(ly)h(enh)o(ance)f(t)n(h)o(e)h(w)o(eak)f(ext)o (er)q(n)o(al)59 446 y(p)q(er)q(io)q(dic)h(forcin)o(g,)g(b)o(y)g(pro)q (d)o(u)o(cin)o(g)f(large)h(amplit)o(ud)o(e)f(oscilla)o(t)o(ions)h(of)g (t)n(h)o(e)g(syst)o(em,)e(h)o(ence)i(t)n(h)o(e)g(n)o(am)o(e)f(of)59 503 y(re)q(son)o(ance.)130 559 y(Th)o(e)j(c)o(h)o(oice)h(of)g(t)n(h)o (e)g(t)o(erm)e(\\re)q(son)o(ance")g(h)o(as)i(b)q(een)g(que)q(st)o(ion)o (e)q(d,)f(as)h(\\it)g(w)o(ould)f(b)q(e)h(more)f(ap)o(pro-)59 615 y(pr)q(ia)o(t)o(e)j(t)o(o)h(refer)g(t)o(o)g Fy(noise-induc)n(e)n(d) h(signal-to-noise)h(r)n(atio)f(enhanc)n(ement)t FC(")e([F)l(o)o(x)o(].) 34 b(In)20 b(t)n(h)o(e)f(regim)o(e)59 672 y(of)d(a)g(p)q(er)q(io)q(dic) g(dr)q(ivin)o(g)h(wh)o(os)q(e)f(amplit)o(ud)o(e)f(i)q(s)i(not)f(a)g(sm) o(all)g(param)o(et)o(er,)d(on)o(e)j(also)g(sp)q(eaks)g(of)g Fy(noise-)59 728 y(induc)n(e)n(d)f(synchr)n(onization)k FC([SNAS)o(].)g(A)n(p)o(precia)o(ble,)13 b(t)n(h)o(ough)h(st)o(ill)i (su)n(b-t)n(hre)q(sh)o(old)e(amplit)o(ud)o(e)q(s)g(of)g(t)n(h)o(e)59 785 y(p)q(er)q(io)q(dic)e(dr)q(ivin)o(g)h(h)o(a)o(v)o(e)e(t)n(h)o(e)h (adv)n(an)o(t)o(a)o(ge)d(t)o(o)j(en)o(a)o(ble)g(trans)q(it)o(ions)e (for)h(sm)o(all)h(noi)q(s)q(e)g(in)o(t)o(ens)q(it)o(ie)q(s,)g(wit)n(h)o (ou)o(t)59 841 y(requir)q(in)o(g)j(astronom)o(ically)e(lon)o(g)i(dr)q (ivin)o(g)g(p)q(er)q(io)q(ds.)130 898 y(While)f(t)n(h)o(e)e(h)o(eur)q (i)q(st)o(ic)g(m)o(ec)o(h)o(ani)q(sm)f(of)h(st)o(o)q(c)o(h)o(ast)o(ic)f (re)q(son)o(ance)g(i)q(s)i(ra)o(t)n(h)o(er)e(w)o(ell)j(u)o(n)o(d)o (erst)o(o)q(o)q(d,)e(a)h(com-)59 954 y(plet)o(e)18 b(m)o(a)o(t)n(h)o (em)o(a)o(t)n(ical)d(d)o(e)q(scr)q(ipt)o(ion)j(i)q(s)g(st)o(ill)h(lac)o (kin)o(g,)f(t)n(h)o(ough)f(imp)q(ort)o(an)o(t)f(progre)q(ss)g(h)o(as)i (b)q(een)f(m)o(ad)o(e)59 1011 y(in)f(s)q(ev)o(eral)e(limit)o(in)o(g)i (cas)q(e)q(s.)k(Dep)q(en)o(din)o(g)15 b(on)g(t)n(h)o(e)g(regim)o(e)f (on)o(e)h(i)q(s)g(in)o(t)o(ere)q(st)o(e)q(d)f(in,)h(s)q(ev)o(eral)f(ap) 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e(mo)q(d)o(el,)i(whic)o(h)f(i)q(s)i(d)o(e)q(scr)q(ib)q(e)q(d)e(b)o(y)h (a)59 1349 y(Mar)o(k)o(o)o(vian)h(ju)o(mp)h(pro)q(ce)q(ss)h([ET)n(].)21 b(Th)o(e)15 b(t)o(w)o(o-st)o(a)o(t)n(e)d(mo)q(d)o(el)k(h)o(as)f(also)g (b)q(een)g(reali)q(s)q(e)q(d)h(exp)q(er)q(im)o(en)o(t)o(ally)59 1406 y(b)o(y)f(an)f(electronic)g(circuit,)h(calle)q(d)h(t)n(h)o(e)e(Sc) o(hmit)n(t)g(tr)q(igger)f([FH,)i(McNW].)130 1462 y(In)d(ph)o(ys)q(ical) g(exp)q(er)q(im)o(en)o(t)o(s,)f(on)o(e)g(h)o(as)h(oft)o(en)f(acce)q(ss) h(t)o(o)g(in)o(direct)g(c)o(h)o(aract)o(er)q(i)q(st)n(ics)e(of)i(t)n(h) o(e)f(dyn)o(amics,)59 1519 y(su)o(c)o(h)k(as)f(t)n(h)o(e)h(p)q(o)o(w)o (er)f(sp)q(ectru)o(m,)f(whic)o(h)i(di)q(splays)f(a)h(p)q(eak)f(a)o(t)h (t)n(h)o(e)f(dr)q(ivin)o(g)i(f)q(requency)l(.)j(Th)o(e)14 b(stren)o(gt)n(h)59 1575 y(of)e(t)n(h)o(e)g(re)q(son)o(ance)f(i)q(s)i (quan)o(t)o(i\014e)q(d)e(t)n(hrough)g(t)n(h)o(e)h(s)q(ign)o(al-t)o (o-noi)q(s)q(e)f(ra)o(t)o(io)g(\(SNR\),)g(whic)o(h)h(i)q(s)g(pro)o(p)q (ort)o(ion)o(al)59 1632 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741.6 l 553.4 738.8 l 553.5 740.3 l 553.6 738.8 l 553.7 741 l 553.8 740.4 l 553.9 735.7 l 554.1 732.8 l 554.2 735.4 l 554.3 739.3 l 554.4 733.7 l 554.5 737.2 l 554.6 735.3 l 554.8 735.5 l 554.9 736.4 l 555 738.4 l 555.1 735.3 l s n 555.1 735.3 m 555.2 737.5 l 555.3 740.7 l 555.4 737.9 l s n 0 0 0 setrgbcolor s n 1 setlinewidth s showpage %%EndDocument endTexFig 134 826 a Fv(Figure)11 b(2.)21 b FG(A)10 b(t)o(ypical)g(solu)o(t)o (ion)f(of)h(t)n(h)o(e)h(SDE)f(\(1.1\))g(in)f(t)n(h)o(e)i(cas)q(e)h(of)d (t)n(h)o(e)i(asymm)o(etr)q(ic)d(p)q(ot)o(en)o(t)o(ial)h(\(1.3\))o(.)134 876 y(Th)o(e)17 b(up)o(p)q(er)g(an)o(d)g(lo)o(w)o(er)f(h)o(ea)o(vy)h (curv)o(e)q(s)i(in)o(dica)o(t)o(e)d(t)n(h)o(e)h(p)q(os)q(it)o(ion)g(of) f(t)n(h)o(e)h(p)q(ot)o(en)o(t)o(ial)f(w)o(ells,)h(while)f(t)n(h)o(e)134 926 y(middle)g(curv)o(e)k(i)q(s)e(t)n(h)o(e)h(lo)q(ca)o(t)o(ion)e(of)h (t)n(h)o(e)h(saddle.)32 b(P)o(aram)o(et)o(er)18 b(v)n(alue)q(s)h FF(")g FG(=)h(0)p FF(:)p FG(005,)d FF(\033)k FG(=)e(0)p FF(:)p 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y(a)h(su)n(bst)o(an)o(t)o(ial)e(progre)q(ss)h(compare)q(d)f(t)o(o) i(t)n(h)o(e)g(\\quas)q(i)q(st)o(a)o(t)n(ic")d(ap)o(proac)o(h,)g(whic)o (h)j(cons)q(id)o(ers)g(p)q(ot)o(en)o(t)o(ials)59 1889 y(t)n(h)o(a)o(t)d(are)h(piecewi)q(s)q(e)h(const)o(an)o(t)d(in)j(t)o(im) o(e.)k(Not)o(e)13 b(t)n(h)o(a)o(t)g(som)o(e)g(of)i(our)e(re)q(sul)o(t)o (s)i(m)o(ay)f(com)o(e)f(as)i(a)f(surpr)q(i)q(s)q(e.)59 1946 y(In)22 b(part)o(icular,)e(n)o(e)q(it)n(h)o(er)h(t)n(h)o(e)g(wid)o (t)n(h)g(\(in)h(t)o(im)o(e\))e(of)g(t)n(h)o(e)h(trans)q(it)o(ion)e(zon) o(e)i(nor)f(t)n(h)o(e)h(asympt)o(ot)o(ics)d(of)59 2002 y(t)n(h)o(e)h(trans)q(it)o(ion)e(proba)o(bilit)o(y)i(d)o(ep)q(en)o(d)g (on)g(t)n(h)o(e)g(minim)o(al)g(barr)q(ier)g(h)o(e)q(igh)o(t)f FB(a)1381 2009 y Fw(0)1401 2002 y FC(.)33 b(In)20 b(f)q(act,)f(t)n(h)o (e)g(pict)o(ure)59 2059 y(i)q(s)g(in)o(d)o(ep)q(en)o(d)o(en)o(t)g(of)f FB(a)443 2066 y Fw(0)482 2059 y FC(as)h(so)q(on)f(as)g FB(a)733 2066 y Fw(0)772 2059 y FC(i)q(s)i(sm)o(aller)e(t)n(h)o(an)g FB(")1108 2042 y Fw(2)p FA(=)p Fw(3)1183 2059 y FC(\(in)h(t)n(h)o(e)g (symm)o(etr)q(ic)d(cas)q(e\))i(or)g FB(")i FC(\(in)59 2115 y(t)n(h)o(e)c(asymm)o(etr)q(ic)e(cas)q(e\),)i(ev)o(en)h(for)e FB(a)706 2122 y Fw(0)742 2115 y FC(=)h(0.)25 b(Thi)q(s)16 b(i)q(s)h(d)o(ue)g(t)o(o)g(t)n(h)o(e)f(f)q(act)f(t)n(h)o(a)o(t)h(wh)o (en)g FB(a)1555 2122 y Fw(0)1592 2115 y FC(i)q(s)h(sm)o(all,)f(t)n(h)o (e)59 2172 y(t)o(im)o(e)e(d)o(ur)q(in)o(g)g(whic)o(h)g(t)n(h)o(e)g(p)q (ot)o(en)o(t)o(ial)f(barr)q(ier)g(i)q(s)h(lo)o(w)g(i)q(s)h(t)o(o)q(o)e (sh)o(ort)g(t)o(o)h(con)o(tr)q(ibu)o(t)o(e)f(s)q(igni\014can)o(t)n(ly)g (t)o(o)h(t)n(h)o(e)59 2228 y(trans)q(it)o(ion)f(proba)o(bilit)o(y)l(.) 130 2284 y(Th)o(e)f(rem)o(ain)o(d)o(er)f(of)h(t)n(hi)q(s)h(pap)q(er)e (i)q(s)i(organize)q(d)f(as)g(follo)o(ws.)17 b(Th)o(e)12 b(re)q(sul)o(t)o(s)h(are)f(form)n(ula)o(t)o(e)q(d)e(in)k(d)o(et)o(ail) 59 2341 y(in)j(Sect)o(ion)e(2,)h(Su)n(bs)q(ect)o(ion)g(2.2)e(b)q(e)q (in)o(g)j(d)o(ev)o(ot)o(e)q(d)e(t)o(o)h(symm)o(etr)q(ic)d(p)q(ot)o(en)o (t)o(ials,)h(an)o(d)i(Su)n(bs)q(ect)o(ion)g(2.3)e(t)o(o)59 2397 y(asymm)o(etr)q(ic)9 b(p)q(ot)o(en)o(t)o(ials.)16 b(Sect)o(ion)c(3)f(con)o(t)o(ains)f(t)n(h)o(e)i(pro)q(ofs)e(for)h(t)n (h)o(e)g(symm)o(etr)q(ic)e(cas)q(e,)i(while)i(Sect)o(ion)e(4)59 2454 y(con)o(t)o(ains)i(t)n(h)o(e)i(pro)q(ofs)e(for)h(t)n(h)o(e)g (asymm)o(etr)q(ic)e(cas)q(e.)59 2574 y Fo(Ac)o(kno)o(wle)q(dgem)o(en)n (t)o(s:)59 2660 y FC(Th)o(e)j(f)q(ascin)o(a)o(t)o(in)o(g)f(problems)g (rela)o(t)o(e)q(d)g(t)o(o)h(st)o(o)q(c)o(h)o(ast)o(ic)f(re)q(son)o (ance)g(w)o(ere)h(brough)o(t)e(t)o(o)i(our)g(a)o(t)n(t)o(en)o(t)o(ion)e (b)o(y)59 2716 y(An)o(t)o(on)i(Bo)o(vier,)h(wh)o(om)f(w)o(e)h(w)o(ould) g(lik)o(e)h(t)o(o)f(t)n(h)o(ank)f(for)h(st)o(im)n(ula)o(t)o(in)o(g)f (di)q(scuss)q(ions.)24 b(N.)7 b(B.)15 b(t)n(h)o(anks)g(t)n(h)o(e)59 2773 y(WIAS)g(for)f(kin)o(d)h(h)o(ospit)o(alit)o(y)l(.)934 2936 y(5)p eop %%Page: 6 6 6 5 bop 59 164 a FD(2)66 b(Re)r(sul)o(t)o(s)59 267 y Fn(2.1)56 b(Prelimin)o(ar)q(i)o(e)q(s)59 353 y FC(W)l(e)19 b(cons)q(id)o(er)g(non-a)n(u)o(t)o(onomous)d(SDEs)j(of)f(t)n(h)o(e)g (form)g(\(1.1\))m(.)31 b(In)o(tro)q(d)o(u)o(cin)o(g)18 b(t)n(h)o(e)g(slo)o(w)g(t)o(im)o(e)g FB(t)i FC(=)f FB("s)59 409 y FC(allo)o(ws)g(t)o(o)f(st)o(udy)h(t)n(h)o(e)g(syst)o(em)e(on)i(a) g(t)o(im)o(e)f(in)o(t)o(erv)n(al)g(of)h(ord)o(er)f(on)o(e.)32 b(Wh)o(en)20 b(su)n(bst)o(it)o(u)o(t)o(in)o(g)f FB(t)g FC(for)g FB("s)p FC(,)59 466 y(Bro)o(wnian)13 b(mot)o(ion)h(i)q(s)g(re) q(scale)q(d)h(an)o(d)f(w)o(e)h(obt)o(ain)e(an)i(SDE)517 576 y(d)p FB(x)568 583 y FA(t)595 576 y FC(=)648 545 y(1)p 648 565 23 2 v 649 607 a FB(")675 576 y(f)5 b FC(\()p FB(x)746 583 y FA(t)761 576 y FB(;)j(t)p FC(\))g(d)n FB(t)j FC(+)940 545 y FB(\033)p 924 565 60 2 v 924 574 a Fs(p)p 962 574 22 2 v 33 x FB(")996 576 y FC(d)p FB(W)1064 583 y FA(t)1078 576 y FB(;)98 b(x)1215 583 y FA(t)1228 588 y Fz(0)1259 576 y FC(=)13 b FB(x)1333 583 y Fw(0)1353 576 y FB(;)372 b FC(\(2.1\))59 702 y(wh)o(ere)19 b FB(f)26 b FC(i)q(s)20 b(t)n(h)o(e)g(force,)f(d)o(er)q(iv)o(e)q(d)i(f)q(rom)d(t) n(h)o(e)i(p)q(ot)o(en)o(t)o(ial)e FB(V)10 b FC(,)21 b(an)o(d)f Fs(f)p FB(W)1277 709 y FA(t)1291 702 y Fs(g)1314 709 y FA(t)p Fm(>)p FA(t)1367 714 y Fz(0)1406 702 y FC(i)q(s)h(a)e(st)o(an) o(d)o(ard)f(Wien)o(er)59 758 y(pro)q(ce)q(ss)k(on)f(som)o(e)g(proba)o (bilit)o(y)g(space)g(\(\012)p FB(;)8 b Fs(F)d FB(;)j Fl(P)-8 b FC(\))o(.)38 b(Init)o(ial)23 b(con)o(dit)o(ions)e FB(x)1389 765 y Fw(0)1431 758 y FC(are)g(alw)o(ays)f(assu)o(m)o(e)q(d) 59 815 y(t)o(o)e(b)q(e)g(square-in)o(t)o(egra)n(ble)e(wit)n(h)i(re)q (sp)q(ect)f(t)o(o)h Fl(P)11 b FC(an)o(d)18 b(in)o(d)o(ep)q(en)o(d)o(en) o(t)g(of)f Fs(f)p FB(W)1363 822 y FA(t)1378 815 y Fs(g)1401 822 y FA(t)p Fm(>)p FA(t)1454 827 y Fz(0)1473 815 y FC(.)30 b(Wit)n(h)o(ou)o(t)18 b(furt)n(h)o(er)59 871 y(m)o(en)o(t)o(ionin)o(g)9 b(w)o(e)h(alw)o(ays)f(assu)o(m)o(e)g(t)n(h)o(a)o(t)g FB(f)15 b FC(sa)o(t)o(i)q(s\014e)q(s)9 b(t)n(h)o(e)h(usual)g(\(lo)q (cal\))g(Lipsc)o(hitz)h(an)o(d)f(b)q(ou)o(n)o(d)o(e)q(d-gro)o(wt)n(h)59 928 y(con)o(dit)o(ions)18 b(whic)o(h)g(guaran)o(t)o(ee)e(exi)q(st)o (ence)h(an)o(d)h(pa)o(t)n(h)o(wi)q(s)q(e)f(u)o(niquen)o(e)q(ss)j(of)e (a)g(stron)o(g)f(solu)o(t)o(ion)h Fs(f)p FB(x)1779 935 y FA(t)1793 928 y Fs(g)1816 935 y FA(t)59 984 y FC(of)h(\(2.1\))n(.)24 b(Un)o(d)o(er)16 b(t)n(h)o(e)q(s)q(e)g(con)o(dit)o(ions,)g(t)n(h)o(ere) f(exi)q(st)o(s)h(a)g(con)o(t)o(in)n(uous)f(v)o(ers)q(ion)h(of)g Fs(f)p FB(x)1486 991 y FA(t)1500 984 y Fs(g)1523 991 y FA(t)1538 984 y FC(.)24 b(Th)o(erefore)14 b(w)o(e)59 1040 y(m)o(ay)g(assu)o(m)o(e)g(t)n(h)o(a)o(t)f(t)n(h)o(e)i(pa)o(t)n(hs) e FB(!)i Fs(7!)e FB(x)725 1047 y FA(t)740 1040 y FC(\()p FB(!)r FC(\))h(are)g(con)o(t)o(in)n(uous)f(for)h Fl(P)-5 b FC(-a)o(lmost)12 b(all)j FB(!)g Fs(2)e FC(\012.)130 1097 y(W)l(e)18 b(in)o(tro)q(d)o(u)o(ce)e(t)n(h)o(e)g(not)o(a)o(t)o (ion)f Fl(P)699 1080 y FA(t)712 1085 y Fz(0)726 1080 y FA(;x)756 1085 y Fz(0)793 1097 y FC(for)h(t)n(h)o(e)h(law)g(of)f(t)n (h)o(e)h(pro)q(ce)q(ss)g Fs(f)p FB(x)1366 1104 y FA(t)1381 1097 y Fs(g)1404 1104 y FA(t)p Fm(>)p FA(t)1457 1109 y Fz(0)1476 1097 y FC(,)g(st)o(art)o(in)o(g)f(in)i FB(x)1755 1104 y Fw(0)1792 1097 y FC(a)o(t)59 1153 y(t)o(im)o(e)10 b FB(t)171 1160 y Fw(0)191 1153 y FC(,)i(an)o(d)e(us)q(e)h Fl(E)403 1137 y FA(t)416 1142 y Fz(0)436 1137 y FA(;x)466 1142 y Fz(0)496 1153 y FC(t)o(o)g(d)o(enot)o(e)f(exp)q(ect)o(a)o(t)o (ions)e(wit)n(h)j(re)q(sp)q(ect)f(t)o(o)g Fl(P)1270 1137 y FA(t)1283 1142 y Fz(0)1297 1137 y FA(;x)1327 1142 y Fz(0)1346 1153 y FC(.)18 b(Not)o(e)10 b(t)n(h)o(a)o(t)f(t)n(h)o(e)i(st) o(o)q(c)o(h)o(ast)o(ic)59 1210 y(pro)q(ce)q(ss)i Fs(f)p FB(x)265 1217 y FA(t)280 1210 y Fs(g)303 1217 y FA(t)p Fm(>)p FA(t)356 1222 y Fz(0)389 1210 y FC(i)q(s)g(an)g(inh)o(omogen)o (eous)f(Mar)o(k)o(o)o(v)f(pro)q(ce)q(ss.)19 b(W)l(e)14 b(are)f(in)o(t)o(ere)q(st)o(e)q(d)f(in)i(\014rst)f(exit)h(t)o(im)o(e)q (s)59 1266 y(of)j FB(x)139 1273 y FA(t)171 1266 y FC(f)q(rom)f(space{t) o(im)o(e)g(s)q(et)o(s.)26 b(Let)18 b Fs(A)f(\032)g Fl(R)e Fs(\002)d FC([)p FB(t)943 1273 y Fw(0)962 1266 y FB(;)c(t)999 1273 y Fw(1)1019 1266 y FC(])17 b(b)q(e)g(Borel-m)o(easura)o(ble.)25 b(Assu)o(min)o(g)17 b(t)n(h)o(a)o(t)f Fs(A)59 1323 y FC(con)o(t)o(ains)d(\()p FB(x)278 1330 y Fw(0)297 1323 y FB(;)8 b(t)334 1330 y Fw(0)354 1323 y FC(\),)14 b(w)o(e)g(d)o(e\014n) o(e)h(t)n(h)o(e)g(\014rst)f(exit)g(t)o(im)o(e)h(of)f(\()p FB(x)1045 1330 y FA(t)1059 1323 y FB(;)8 b(t)p FC(\))14 b(f)q(rom)f Fs(A)j FC(b)o(y)613 1425 y FB(\034)633 1432 y Fx(A)675 1425 y FC(=)d(inf)778 1388 y Fq(\010)804 1425 y FB(t)g Fs(2)g FC([)p FB(t)905 1432 y Fw(0)925 1425 y FB(;)8 b(t)962 1432 y Fw(1)981 1425 y FC(])d(:)14 b(\()p FB(x)1070 1432 y FA(t)1085 1425 y FB(;)8 b(t)p FC(\))j Fs(62)i(A)1230 1388 y Fq(\011)1257 1425 y FB(;)468 b FC(\(2.2\))59 1527 y(an)o(d)14 b(agree)g(t)o(o)h(s)q(et)f FB(\034)409 1534 y Fx(A)439 1527 y FC(\()p FB(!)r FC(\))e(=)h Fs(1)i FC(for)f(t)n(h)o(os)q(e)h FB(!)f Fs(2)g FC(\012)g(whic)o(h)h(sa) o(t)o(i)q(sfy)e(\()p FB(x)1256 1534 y FA(t)1270 1527 y FC(\()p FB(!)r FC(\))p FB(;)8 b(t)p FC(\))j Fs(2)i(A)j FC(for)e(all)h FB(t)f Fs(2)f FC([)p FB(t)1730 1534 y Fw(0)1749 1527 y FB(;)8 b(t)1786 1534 y Fw(1)1806 1527 y FC(].)59 1583 y(F)l(or)k(con)o(v)o(enience,)f(w)o(e)g(sh)o(all)i (call)f FB(\034)657 1590 y Fx(A)699 1583 y FC(t)n(h)o(e)g Fy(\014rst)i(exit)e(time)g(of)i FB(x)1121 1590 y FA(t)1149 1583 y Fy(fr)n(om)e Fs(A)p FC(.)19 b(T)o(ypically)l(,)12 b(w)o(e)f(will)i(cons)q(id)o(er)59 1640 y(s)q(et)o(s)k(of)g(t)n(h)o(e)g (form)f Fs(A)i FC(=)f Fs(f)p FC(\()p FB(x;)8 b(t)p FC(\))16 b Fs(2)h Fl(R)e Fs(\002)d FC([)p FB(t)810 1647 y Fw(0)830 1640 y FB(;)c(t)867 1647 y Fw(1)886 1640 y FC(])d(:)15 b FB(g)954 1647 y Fw(1)974 1640 y FC(\()p FB(t)p FC(\))h FB(<)i(x)f(<)h(g)1213 1647 y Fw(2)1232 1640 y FC(\()p FB(t)p FC(\))p Fs(g)f FC(wit)n(h)g(con)o(t)o(in)n(uous)f(fu)o(nct)o (ions)59 1696 y FB(g)81 1703 y Fw(1)113 1696 y FB(<)d(g)183 1703 y Fw(2)202 1696 y FC(.)19 b(Not)o(e)10 b(t)n(h)o(a)o(t)g(in)i(t)n (hi)q(s)f(cas)q(e,)g FB(\034)686 1703 y Fx(A)728 1696 y FC(i)q(s)g(a)g(st)o(o)o(p)o(pin)o(g)g(t)o(im)o(e)1066 1680 y Fw(1)1097 1696 y FC(wit)n(h)g(re)q(sp)q(ect)f(t)o(o)h(t)n(h)o(e) g(canonical)g(\014l)o(tra)o(t)o(ion)59 1753 y(of)j(\(\012)p FB(;)8 b Fs(F)d FB(;)j Fl(P)-8 b FC(\))11 b(gen)o(era)o(t)o(e)q(d)i(b)o (y)i Fs(f)p FB(x)613 1760 y FA(t)627 1753 y Fs(g)650 1760 y FA(t)p Fm(>)p FA(t)703 1765 y Fz(0)722 1753 y FC(.)130 1809 y(Before)e(t)o(ur)q(nin)o(g)g(t)o(o)h(t)n(h)o(e)f(preci)q (s)q(e)h(st)o(a)o(t)o(em)o(en)o(t)n(s)e(of)h(our)h(re)q(sul)o(t)o(s,)g (let)g(us)g(in)o(tro)q(d)o(u)o(ce)f(som)o(e)g(not)o(a)o(t)o(ions.)59 1866 y(W)l(e)j(sh)o(all)f(us)q(e)90 1934 y Fs(\017)23 b(d)p FB(y)r Fs(e)16 b FC(for)e FB(y)g Fr(>)f FC(0)i(t)o(o)f(d)o(enot)o (e)g(t)n(h)o(e)g(sm)o(alle)q(st)h(in)o(t)o(eger)e(whic)o(h)i(i)q(s)g (grea)o(t)o(er)e(t)n(h)o(an)h(or)g(equal)g(t)o(o)h FB(y)r FC(,)f(an)o(d)90 1990 y Fs(\017)23 b FB(y)12 b Fs(_)e FB(z)17 b FC(an)o(d)c FB(y)f Fs(^)e FB(z)17 b FC(t)o(o)d(d)o(enot)o(e)f (t)n(h)o(e)i(m)o(axim)n(u)o(m)e(or)h(minim)n(u)o(m,)g(re)q(sp)q(ect)o (iv)o(ely)l(,)f(of)h(t)o(w)o(o)f(real)i(n)n(u)o(m)n(b)q(ers)136 2047 y FB(y)i FC(an)o(d)e FB(z)r FC(.)90 2103 y Fs(\017)23 b FC(If)c FB(')p FC(\()p FB(t;)8 b(")p FC(\))17 b(an)o(d)h FB( )r FC(\()p FB(t;)8 b(")p FC(\))16 b(are)i(d)o(e\014n)o(e)q(d)g(for) g(sm)o(all)g FB(")h FC(an)o(d)e(for)h FB(t)g FC(in)h(a)f(giv)o(en)g(in) o(t)o(erv)n(al)f FB(I)t FC(,)i(w)o(e)e(wr)q(it)o(e)136 2160 y FB( )r FC(\()p FB(t;)8 b(")p FC(\))24 b Fs(\020)i FB(')p FC(\()p FB(t;)8 b(")p FC(\))22 b(if)h(t)n(h)o(ere)f(exi)q(st)g (str)q(ict)n(ly)f(p)q(os)q(it)o(iv)o(e)h(const)o(an)o(t)o(s)f FB(c)1337 2167 y Fx(\006)1389 2160 y FC(su)o(c)o(h)h(t)n(h)o(a)o(t)f FB(c)1618 2167 y Fx(\000)1647 2160 y FB(')p FC(\()p FB(t;)8 b(")p FC(\))24 b Fr(6)136 2216 y FB( )r FC(\()p FB(t;)8 b(")p FC(\))21 b Fr(6)i FB(c)361 2223 y Fw(+)391 2216 y FB(')p FC(\()p FB(t;)8 b(")p FC(\))19 b(for)h(all)i FB(t)h Fs(2)g FB(I)i FC(an)o(d)20 b(all)i(su\016cien)o(t)n(ly)e(sm)o (all)h FB(")p FC(.)38 b(Th)o(e)20 b(const)o(an)o(t)o(s)e FB(c)1720 2223 y Fx(\006)1771 2216 y FC(are)136 2273 y(u)o(n)o(d)o(erst)o(o)q(o)q(d)d(t)o(o)f(b)q(e)g(in)o(d)o(ep)q(en)o(d)o (en)o(t)h(of)f FB(t)h FC(an)o(d)f FB(")h FC(\(an)o(d)f(h)o(ence)g(also) h(in)o(d)o(ep)q(en)o(d)o(en)o(t)f(of)g(quan)o(t)o(it)o(ie)q(s)f(lik)o (e)136 2329 y FB(\033)k FC(an)o(d)d FB(a)289 2336 y Fw(0)324 2329 y FC(whic)o(h)h(w)o(e)f(cons)q(id)o(er)g(as)h(fu)o(nct)o(ions)f (of)g FB(")p FC(\).)90 2386 y Fs(\017)23 b FC(By)18 b FB(g)r FC(\()p FB(u)p FC(\))d(=)i Fs(O)q FC(\()p FB(u)p FC(\))g(w)o(e)g(in)o(dica)o(t)o(e)g(t)n(h)o(a)o(t)f(t)n(h)o(ere)h(exi)q (st)g FB(\016)i(>)e FC(0)h(an)o(d)e FB(K)k(>)e FC(0)f(su)o(c)o(h)g(t)n (h)o(a)o(t)f FB(g)r FC(\()p FB(u)p FC(\))f Fr(6)j FB(K)s(u)136 2442 y FC(for)f(all)h FB(u)g Fs(2)g FC([0)p FB(;)8 b(\016)r FC(],)15 b(wh)o(ere)i FB(\016)j FC(an)o(d)d FB(K)k FC(of)c(cours)q(e)g (do)g(not)g(d)o(ep)q(en)o(d)h(on)f FB(")h FC(or)f(on)g(t)n(h)o(e)h(ot)n (h)o(er)e(sm)o(all)136 2498 y(param)o(et)o(ers)c FB(a)391 2505 y Fw(0)426 2498 y FC(an)o(d)i FB(\033)r FC(.)19 b(Similarly)l(,)c FB(g)r FC(\()p FB(u)p FC(\))c(=)i Fx(O)s FC(\(1\))g(i)q(s)i(t)o(o)g(b)q(e)f(u)o(n)o(d)o(erst)o(o)q(o)q(d)h(as)f (lim)1532 2505 y FA(u)p Fx(!)p Fw(0)1615 2498 y FB(g)r FC(\()p FB(u)p FC(\))d(=)h(0.)59 2567 y(Fin)o(ally)l(,)18 b(let)e(us)h(p)q(oin)o(t)f(ou)o(t)g(t)n(h)o(a)o(t)f(most)g(e)q(st)o(im) o(a)o(t)o(e)q(s)g(h)o(old)h(for)f(sm)o(all)h(enough)g FB(")h FC(only)l(,)f(an)o(d)g(oft)o(en)f(only)59 2623 y(for)f Fl(P)-5 b FC(-a)o(lmost)12 b(all)j FB(!)g Fs(2)e FC(\012.)19 b(W)l(e)c(will)h(stre)q(ss)e(t)n(h)o(e)q(s)q(e)h(f)q(act)o (s)e(only)i(wh)o(ere)f(confus)q(ion)g(migh)o(t)g(ar)q(i)q(s)q(e.)p 59 2663 709 2 v 111 2690 a Fz(1)128 2706 y Fk(F)m(or)f(a)g(gen)o(eral)h (Borel-m)o(easura)o(ble)h(s)q(et)e Fj(A)p Fk(,)f(t)n(h)o(e)h(\014rst)g (exit)h(t)o(im)o(e)f Fi(\034)1095 2710 y Fh(A)1134 2706 y Fk(i)q(s)g(st)o(ill)i(a)e(st)o(o)o(p)o(pin)o(g)i(t)o(im)o(e)e(wit)n (h)h(re)q(sp)q(ect)f(t)o(o)g(t)n(h)o(e)59 2751 y(canonical)i(\014l)o (tra)o(t)o(ion,)h(complet)o(e)q(d)e(b)o(y)f(t)n(h)o(e)g(n)n(ull)j(s)q (ets.)934 2936 y FC(6)p eop %%Page: 7 7 7 6 bop 59 164 a Fn(2.2)56 b(Symm)n(etric)15 b(cas)q(e)59 250 y FC(W)l(e)j(cons)q(id)o(er)f(in)h(t)n(hi)q(s)e(su)n(bs)q(ect)o (ion)h(t)n(h)o(e)g(SDE)g(\(2.1\))e(in)j(t)n(h)o(e)f(cas)q(e)f(of)h FB(f)23 b FC(b)q(e)q(in)o(g)17 b(p)q(er)q(io)q(dic)g(in)h FB(t)p FC(,)g(o)q(dd)f(in)59 306 y FB(x)p FC(,)g(an)o(d)g(admit)n(t)o (in)o(g)f(t)o(w)o(o)f(st)o(a)o(ble)h(equilibr)q(iu)o(m)h(branc)o(h)o(e) q(s,)f(wit)n(h)h(a)g(\\barr)q(ier")d(b)q(et)o(w)o(een)i(t)n(h)o(e)g (branc)o(h)o(e)q(s)59 363 y(b)q(ecomin)o(g)e(sm)o(all)h(once)f(d)o(ur)q (in)o(g)h(ev)o(ery)f(t)o(im)o(e)g(p)q(er)q(io)q(d.)19 b(A)c(t)o(ypical)f(example)g(of)g(su)o(c)o(h)g(a)h(fu)o(nct)o(ion)f(i)q (s)388 464 y FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))j(=)i FB(a)p FC(\()p FB(t)p FC(\))p FB(x)d Fs(\000)g FB(x)756 445 y Fw(3)866 464 y FC(wit)n(h)90 b FB(a)p FC(\()p FB(t)p FC(\))12 b(=)g FB(a)1201 471 y Fw(0)1231 464 y FC(+)e(1)g Fs(\000)g FC(cos)e(2)p FB(\031)r(t:)235 b FC(\(2.3\))59 565 y(W)l(e)15 b(will)g(cons)q(id)o(er)f(a)f(more)g(gen)o(eral)g(class) i(of)e(fu)o(nct)o(ions)h FB(f)j FC(:)12 b Fl(R)1152 548 y Fw(2)1181 565 y Fs(!)h Fl(R)6 b FC(,)11 b(whic)o(h)i(w)o(e)h(assu)o (m)o(e)f(t)o(o)h(sa)o(t)o(i)q(sfy)59 621 y(t)n(h)o(e)g(follo)o(win)o(g) g(h)o(yp)q(ot)n(h)o(e)q(s)q(e)q(s:)90 689 y Fs(\017)23 b Fy(Smo)n(othness:)d FB(f)e Fs(2)13 b(C)506 673 y Fw(4)525 689 y FC(\()p Fs(M)p FB(;)8 b Fl(R)t FC(\),)j(wh)o(ere)j Fs(M)e FC(=)h([)p Fs(\000)p FB(d;)8 b(d)t FC(])g Fs(\002)i Fl(R)18 b FC(an)o(d)c FB(d)e(>)h FC(0)i(i)q(s)g(a)f(const)o(an)o(t;)90 746 y Fs(\017)23 b Fy(Perio)n(dicity:)c FB(f)5 b FC(\()p FB(x;)j(t)h FC(+)h(1\))i(=)g FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))14 b(for)g(all)i(\()p FB(x;)8 b(t)p FC(\))j Fs(2)i(M)p FC(;)90 802 y Fs(\017)23 b Fy(Symmetry:)c FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))k(=)g Fs(\000)p FB(f)5 b FC(\()p Fs(\000)p FB(x;)j(t)p FC(\))15 b(for)e(all)j(\()p FB(x;)8 b(t)p FC(\))j Fs(2)i(M)p FC(;)90 859 y Fs(\017)23 b Fy(Equilibrium)e(br)n (anches:)30 b FC(Th)o(ere)19 b(exi)q(st)o(s)h(a)g(con)o(t)o(in)n(uous)e (fu)o(nct)o(ion)i FB(x)1340 842 y FA(?)1382 859 y FC(:)h Fl(R)j Fs(!)e FC(\(0)p FB(;)8 b(d)t FC(])17 b(wit)n(h)j(t)n(h)o(e)136 915 y(pro)o(p)q(ert)o(y)13 b(t)n(h)o(a)o(t)g FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))j(=)i(0)h(in)i Fs(M)f FC(if)g(an)o(d)f(only)h (if)g FB(x)e FC(=)f(0)j(or)f FB(x)f FC(=)f Fs(\006)p FB(x)1350 899 y FA(?)1370 915 y FC(\()p FB(t)p FC(\);)90 972 y Fs(\017)23 b Fy(Stability:)g FC(Th)o(e)16 b(or)q(igin)g(i)q(s)h (u)o(nst)o(a)o(ble)f(an)o(d)g(t)n(h)o(e)g(equilibr)q(iu)o(m)h(branc)o (h)o(e)q(s)f Fs(\006)p FB(x)1437 955 y FA(?)1473 972 y FC(are)g(st)o(a)o(ble,)f(t)n(h)o(a)o(t)g(i)q(s,)136 1028 y(for)f(all)i FB(t)d Fs(2)g Fl(R)5 b FC(,)757 1125 y FB(a)p FC(\()p FB(t)p FC(\))840 1124 y(:)850 1125 y(=)i FB(@)916 1132 y FA(x)938 1125 y FB(f)e FC(\(0)p FB(;)j(t)p FC(\))j FB(>)i FC(0)737 1194 y FB(a)761 1176 y FA(?)781 1194 y FC(\()p FB(t)p FC(\))840 1193 y(:)850 1194 y(=)7 b FB(@)916 1201 y FA(x)938 1194 y FB(f)e FC(\()p FB(x)1009 1176 y FA(?)1029 1194 y FC(\()p FB(t)p FC(\))p FB(;)j(t)p FC(\))j FB(<)i FC(0)p FB(:)1738 1160 y FC(\(2.4\))90 1292 y Fs(\017)23 b Fy(Behaviour)14 b(ne)n(ar)g FB(t)f FC(=)g(0)p Fy(:)18 b FC(W)l(e)c(w)o(an)o(t)e(t)n(h)o(e)g(t)n(hree)h (equilibr)q(iu)o(m)h(branc)o(h)o(e)q(s)e(t)o(o)h(com)o(e)f(clos)q(e)i (a)o(t)e(in)o(t)o(eger)136 1348 y(t)o(im)o(e)q(s.)25 b(Giv)o(en)17 b(t)n(h)o(e)f(symm)o(etry)e(of)i FB(f)5 b FC(,)17 b(t)n(h)o(e)f(n)o(a)o(t)o(ural)f(assu)o(mpt)o(ion)h(i)q(s)h (t)n(h)o(a)o(t)e(w)o(e)h(h)o(a)o(v)o(e)g(an)g(\\a)o(v)o(oid)o(e)q(d)136 1405 y(pit)o(c)o(hfor)o(k)e(bifurca)o(t)o(ion",)e(t)n(h)o(a)o(t)i(i)q (s,)738 1502 y FB(@)762 1509 y FA(xxx)824 1502 y FB(f)5 b FC(\(0)p FB(;)j FC(0\))i FB(<)j FC(0)738 1575 y FB(a)p FC(\()p FB(t)p FC(\))f(=)h FB(a)898 1582 y Fw(0)928 1575 y FC(+)d FB(a)997 1582 y Fw(1)1016 1575 y FB(t)1032 1557 y Fw(2)1063 1575 y FC(+)g Fs(O)q FC(\()p FB(t)1179 1557 y Fw(3)1199 1575 y FC(\))p FB(;)1738 1539 y FC(\(2.5\))136 1675 y(wh)o(ere)17 b FB(a)292 1682 y Fw(1)329 1675 y FB(>)g FC(0)g(an)o(d)g FB(@)534 1682 y FA(xxx)596 1675 y FB(f)5 b FC(\(0)p FB(;)j FC(0\))14 b(are)j(\014xe)q(d)h(\(of)e(ord)o (er)g(on)o(e\),)h(while)h FB(a)1380 1682 y Fw(0)1417 1675 y FC(=)f FB(a)1493 1682 y Fw(0)1512 1675 y FC(\()p FB(")p FC(\))f(=)h Fx(O)1666 1682 y FA(")1685 1675 y FC(\(1\))f(i)q(s)h(a)136 1732 y(p)q(os)q(it)o(iv)o(e)f(sm)o(all)g (param)o(et)o(er)o(.)k(Is)c(i)q(s)g(easy)f(t)o(o)h(sh)o(o)o(w)e(t)n(h)o (a)o(t)g FB(x)1139 1715 y FA(?)1159 1732 y FC(\()p FB(t)p FC(\))h(b)q(e)o(h)o(a)o(v)o(e)q(s)g(lik)o(e)1477 1693 y Fq(p)p 1523 1693 76 2 v 1523 1732 a FB(a)p FC(\()p FB(t)p FC(\))g(for)g(sm)o(all)h FB(t)p FC(,)136 1788 y(an)o(d)f(admit)o(s)g(a)g(quadra)o(t)o(ic)f(minim)n(u)o(m)h(a)o(t)g(a) g(t)o(im)o(e)f FB(t)1018 1772 y FA(?)1052 1788 y FC(=)g Fs(O)q FC(\()p FB(a)1180 1795 y Fw(0)1199 1788 y FC(\).)21 b(Moreo)o(v)o(er,)13 b FB(a)1486 1772 y FA(?)1506 1788 y FC(\()p FB(t)p FC(\))g Fs(\020)h(\000)p FB(a)p FC(\()p FB(t)p FC(\))h(n)o(ear)136 1845 y FB(t)e FC(=)g(0.)136 1901 y(W)l(e)h(can)e(c)o(h)o(o)q(os)q(e)g(a)g(const)o(an)o(t)e FB(T)19 b Fs(2)13 b FC(\(0)p FB(;)8 b FC(1)p FB(=)p FC(2\))h(su)o(c)o (h)j(t)n(h)o(a)o(t)f(t)n(h)o(e)h(d)o(er)q(iv)n(a)o(t)o(iv)o(e)q(s)f(of) h FB(a)p FC(\()p FB(t)p FC(\))g(an)o(d)g FB(x)1622 1885 y FA(?)1642 1901 y FC(\()p FB(t)p FC(\))g(v)n(ani)q(sh)136 1958 y(only)21 b(once)g(in)h(t)n(h)o(e)e(in)o(t)o(erv)n(al)g([)p Fs(\000)p FB(T)t(;)8 b(T)e FC(].)35 b(W)l(e)22 b(\014n)o(ally)f (require)g(t)n(h)o(a)o(t)e FB(x)1359 1941 y FA(?)1379 1958 y FC(\()p FB(t)p FC(\),)i FB(a)p FC(\()p FB(t)p FC(\))g(an)o(d)f FB(a)1678 1941 y FA(?)1698 1958 y FC(\()p FB(t)p FC(\))h(are)136 2014 y(b)q(ou)o(n)o(d)o(e)q(d)16 b(aw)o(ay)d(f)q(rom)g(zero)h(ou)o(t)o(s)q(id)o(e)h(t)n(hi)q(s)g(in)o(t) 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154.6 672.3 l 155 668.4 l 155.2 662.1 l 155.4 668.3 l 155.4 675.2 l 155.6 668.8 l 156.1 668.3 l 156.5 675.3 l 157 677.2 l 157.3 684.6 l 157.4 678.1 l 157.9 679.8 l 158.1 686.7 l 158.3 693.5 l 158.5 686.9 l 158.6 680.6 l s n 158.6 680.6 m 158.8 674.2 l 159 682.2 l 159.2 688.7 l 159.4 682.1 l 159.9 682.4 l 160.4 678.8 l 160.5 685.6 l 160.7 679 l 160.8 685.3 l 161.3 678.7 l 161.7 685.3 l 162.1 691.5 l 162.3 684.8 l 162.8 684.9 l 163.1 691.1 l 163.4 698.2 l 163.7 692 l 163.9 685.6 l 164.4 685.8 l 164.7 679.6 l 165.1 673.2 l 165.3 666.9 l 165.4 673.2 l 165.8 668.5 l 166 675.1 l 166.2 682.1 l 166.4 675.1 l 166.7 681.3 l 166.9 674.1 l 167.4 681.4 l 167.4 688.1 l 167.9 686.8 l 167.9 694.3 l 168.4 687.3 l 168.5 681.1 l 168.7 674.7 l 169 668.6 l 169.5 666.1 l 169.8 672.5 l 169.8 665.9 l 170.1 672.5 l 170.4 679.4 l 170.6 686.1 l 170.8 679.7 l 171.1 673.3 l 171.2 667 l 171.4 660.7 l 171.8 654.4 l 172 660.6 l 172.5 659.4 l s n 172.5 659.4 m 172.9 666.1 l 173.4 672.3 l 173.5 678.6 l 174 679.4 l 174.2 673.2 l 174.6 680.3 l 174.7 686.5 l 175.2 684.7 l 175.5 678.5 l 175.9 672.4 l 176.4 673.1 l 176.4 666.8 l 176.6 660.5 l 177.1 666.7 l 177.2 660.5 l 177.3 666.7 l 177.5 672.9 l 178 673.2 l 178.1 667.1 l 178.3 674.1 l 178.6 680.9 l 179.1 686.8 l 179.5 679.2 l 179.9 673 l 180.2 666.7 l 180.2 673.2 l 180.5 679.6 l 180.6 673.5 l 180.8 680.3 l 181 673.7 l 181.5 669.3 l 181.8 661.3 l 182 668.1 l 182.1 660.9 l 182.4 668 l 182.6 661.7 l 182.7 655.2 l 182.9 661.6 l 183.4 661.9 l 183.8 669 l 183.9 675.4 l 184.1 681.5 l 184.2 675.3 l 184.5 668.5 l 184.7 675.4 l 184.9 668.6 l 185.2 675.7 l 185.5 681.9 l 185.7 688.1 l 185.8 695.2 l s n 185.8 695.2 m 186 688.1 l 186.4 694.5 l 186.6 688.2 l 186.8 681.7 l 187.1 688.7 l 187.2 682.4 l 187.5 675 l 188 673.2 l 188.3 666.6 l 188.7 669 l 189.2 668.1 l 189.4 661.8 l 189.5 655 l 189.9 648.6 l 190 654.7 l 190.5 652.9 l 190.9 655.4 l 191.4 647.7 l 191.7 641.6 l 192.1 647.9 l 192.5 641.2 l 192.7 648.2 l 193 655.4 l 193.3 649.1 l 193.8 647.7 l 194.1 641 l 194.6 640.5 l 194.7 647.5 l 194.7 654.6 l 195.1 647.9 l 195.2 641.4 l 195.4 647.7 l 195.5 640.7 l 195.7 646.9 l 195.9 639.7 l 196.4 637.7 l 196.7 631.3 l 196.7 637.7 l 196.9 629.8 l 197.4 631.2 l 197.7 624.9 l 197.8 618.2 l 198.2 624.8 l 198.3 618.2 l 198.7 626.2 l 199.2 632.4 l 199.3 639.5 l 199.5 632.2 l 199.6 638.8 l 200.1 643.9 l s n 200.1 643.9 m 200.2 637 l 200.3 643.3 l 200.8 641.8 l 200.9 635.5 l 201 629.3 l 201.1 636 l 201.3 642.5 l 201.8 647.4 l 202 640.8 l 202.2 647.9 l 202.5 640.6 l 202.6 633 l 203.1 630 l 203.3 623.4 l 203.6 629.6 l 203.7 636 l 204.2 629.4 l 204.4 635.8 l 204.9 630.8 l 205.4 626.3 l 205.6 632.5 l 205.8 625.2 l 205.9 631.6 l 206.2 637.9 l 206.6 640.3 l 207.1 636.7 l 207.6 635.7 l 207.9 629.5 l 208 622.4 l 208 629.1 l 208.3 622.9 l 208.6 629.2 l 208.7 635.4 l 209 642 l 209.2 648.4 l 209.3 642.1 l 209.8 643.9 l 210 651.5 l 210.5 650.1 l 210.6 643.7 l 210.7 649.9 l 210.8 656.7 l 211.2 664.1 l 211.3 656.9 l 211.7 650.2 l 211.9 658 l 212.2 664.2 l 212.4 657.9 l 212.9 657 l 213.1 664.2 l s n 213.1 664.2 m 213.2 657.8 l 213.4 651.4 l 213.6 657.6 l 214.1 655.5 l 214.3 648.7 l 214.8 655.5 l 215 661.8 l 215.5 658.9 l 215.8 652 l 216.1 644.6 l 216.3 638.1 l 216.7 631 l 216.8 637.7 l 217.2 630.9 l 217.3 637.2 l 217.4 643.3 l 217.5 649.6 l 217.7 643.5 l 217.9 650.6 l 218.4 647.9 l 218.5 655.1 l 218.7 648.5 l 219 641.9 l 219.4 641.8 l 219.9 636 l 220.3 642.7 l 220.3 650.3 l 220.7 643.3 l 220.8 650 l 221.2 642.8 l 221.4 635 l 221.7 641.7 l 221.8 648 l 222.2 641.6 l 222.4 648.3 l 222.4 654.9 l 222.7 648.1 l 222.9 641.5 l 223.1 634.7 l 223.2 641.4 l 223.6 648.2 l 223.8 654.3 l 224.1 647.7 l 224.2 653.9 l 224.6 660.2 l 224.6 666.5 l 224.8 659.9 l 225 666.9 l 225.4 666.8 l 225.7 659.8 l s n 225.7 659.8 m 226.1 666 l 226.4 672.5 l 226.7 666.2 l 227.2 665.8 l 227.5 658.9 l 227.8 651.3 l 227.9 657.6 l 228.2 651.4 l 228.3 645 l 228.8 651 l 229.3 648 l 229.4 641.8 l 229.6 648.3 l 230.1 645.8 l 230.6 643.3 l 230.7 637 l 231.1 639.4 l 231.2 633.3 l 231.5 626.8 l 231.7 619.8 l 231.7 626.1 l 232.2 626 l 232.5 632.7 l 233 640.3 l 233.3 633.7 l 233.4 627 l 233.6 619.8 l 233.8 626 l 234 619.6 l 234.5 616 l 234.9 609.5 l 235.2 602.5 l 235.2 608.7 l 235.5 615 l 235.7 608.8 l 236.1 602.6 l 236.4 596.3 l 236.7 589.8 l 237 596.9 l 237.3 590 l 237.5 583.5 l 237.6 590.1 l 237.7 583.6 l 238.2 583.5 l 238.7 584 l 239.2 590.7 l 239.2 584.4 l 239.7 584.4 l 239.9 577.1 l 240.3 583.5 l s n 240.3 583.5 m 240.4 577 l 240.5 583.3 l 241 582.1 l 241.1 588.6 l 241.2 582.2 l 241.7 584.8 l 242.2 584.5 l 242.4 590.7 l 242.9 590.1 l 243.4 591.1 l 243.6 597.6 l 244.1 600.4 l 244.2 606.6 l 244.4 600.5 l 244.4 607.7 l 244.5 614.3 l 244.5 620.7 l 245 615.9 l 245.4 608.2 l 245.5 615.2 l 245.8 621.5 l 245.9 628.4 l 246.4 630.9 l 246.8 624.7 l 247 618.3 l 247.3 611.9 l 247.8 612.4 l 247.9 618.5 l 248.4 612 l 248.5 618.8 l 248.7 612.5 l 249.2 613.8 l 249.4 606.9 l 249.5 613.4 l 249.9 619.8 l 250.2 613.1 l 250.6 606.4 l 251.1 604.9 l 251.6 602.4 l 251.8 608.6 l 252.3 613.4 l 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516.5 l 274.3 509.8 l 274.4 515.9 l 274.9 515.4 l 275.4 512.2 l 275.6 504.9 l 276.1 511.5 l 276.6 513.7 l 276.8 520.5 l 276.8 527.5 l 277.3 520.8 l 277.8 523.1 l 278.2 518.9 l 278.4 525.2 l 278.9 528.3 l 279.4 530.6 l 279.4 537.3 l 279.7 531 l 279.8 524.8 l 280.3 529.9 l 280.8 523.8 l 281.1 517.5 l 281.3 511.3 l 281.7 505.1 l 281.9 498.4 l 282.1 491.6 l 282.6 485.1 l 282.7 478.9 l 283.1 472.3 l 283.3 478.9 l 283.5 472.5 l 283.7 479.4 l 284.1 473.1 l 284.2 466.3 l 284.3 472.5 l 284.7 479.1 l 284.8 472.8 l 285.2 479.5 l s n 285.2 479.5 m 285.5 473.3 l 286 471.6 l 286.4 465.2 l 286.5 458.9 l 286.8 465.7 l 287.3 472 l 287.6 465.7 l 287.9 458.5 l 288 452.4 l 288.5 448.2 l 288.7 440.8 l 288.8 433.1 l 288.8 440.3 l 289.3 444.7 l 289.6 436.9 l 290 434.9 l 290.3 441.2 l 290.8 440.2 l 291.2 433.9 l 291.6 427 l 291.6 433.6 l 291.8 439.8 l 292.2 446.1 l 292.4 439.7 l 292.5 433.5 l 293 428.4 l 293.5 424.5 l 293.9 417.6 l 294.1 411.1 l 294.5 404.4 l 294.7 411.3 l 295.2 417.5 l 295.4 411.3 l 295.5 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426.7 l 315.7 420.3 l 315.9 426.6 l 316.1 434 l 316.5 440.7 l 316.7 434.4 l 316.8 440.6 l 317 447.3 l 317.2 454.1 l 317.2 460.5 l 317.5 466.6 l 317.9 466.9 l 318.4 467.4 l 318.9 469 l 319.4 466.2 l 319.9 466.1 l 320.4 462.5 l 320.5 468.9 l 320.9 475.3 l 321.3 482.3 l 321.4 488.6 l 321.8 490.6 l 322.3 491.2 l 322.8 490.1 l 323 483.2 l 323.1 476.7 l 323.6 470.5 l 323.8 464.2 l 323.9 458 l 324.2 451.4 l 324.4 445.2 l 324.6 437 l 324.9 430.3 l 325 436.7 l 325.2 442.9 l 325.4 436.6 l 325.4 430.4 l 325.6 423.7 l 326 417.5 l 326.4 410.5 l 326.7 403.9 l 327.2 403.2 l 327.5 409.8 l 328 412 l 328.5 417.4 l 329 419 l s n 329 419 m 329.4 425.7 l 329.9 427.6 l 330.4 427.5 l 330.8 421.3 l 330.9 414.8 l 331.4 421.1 l 331.5 427.3 l 331.9 420.6 l 332.1 413.5 l 332.5 411.4 l 332.7 405.1 l 333.1 411.7 l 333.5 405.2 l 333.8 412 l 334 405.8 l 334.2 399 l 334.7 398 l 335.2 392.7 l 335.4 399.3 l 335.7 406.2 l 335.8 399.8 l 336 393.7 l 336.5 395.5 l 336.7 402.2 l 337.2 402.1 l 337.6 395.9 l 337.8 389.6 l 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304.4 l 538 304 l 538.4 303.6 l 538.9 303.3 l 539.4 302.9 l 539.9 302.5 l 540.4 302.1 l 540.9 301.7 l 541.4 301.4 l 541.9 301 l 542.4 300.6 l 542.9 300.3 l 543.3 299.9 l 543.8 299.5 l 544.3 299.1 l 544.8 298.8 l 545.3 298.4 l 545.8 298 l 546.3 297.7 l 546.8 297.3 l 547.3 296.9 l 547.8 296.6 l 548.2 296.2 l 548.7 295.8 l 549.2 295.5 l 549.7 295.1 l 550.2 294.7 l 550.7 294.4 l 551.2 294 l 551.7 293.7 l 552.2 293.3 l 552.7 292.9 l 553.1 292.6 l 553.6 292.2 l 554.1 291.9 l 554.6 291.5 l 555.1 291.1 l 555.6 290.8 l 556.1 290.4 l 556.6 290.1 l 557.1 289.7 l 557.6 289.4 l 558 289 l 558.5 288.7 l 559 288.3 l 559.5 288 l 560 287.6 l s n 560 287.6 m s n 0 0 0 setrgbcolor s n 2 setlinewidth s showpage %%EndDocument endTexFig 134 786 a Fv(Figure)20 b(3.)h FG(Solu)o(t)o(ions)d(of)g(t)n(h)o(e)i (SDE)f(\(2.1\))f(wit)n(h)h(symm)o(etr)q(ic)d(dr)q(ift)j(t)o(erm)f (\(2.3\))o(,)i(sh)o(o)o(wn)f(for)f(t)o(w)o(o)134 835 y(di\013eren)o(t)d(noi)q(s)q(e)e(in)o(t)o(ens)q(it)o(ie)q(s,)h(bu)o(t)h (for)e(t)n(h)o(e)i(sam)o(e)e(realiza)o(t)o(ion)h(of)f(Bro)o(wnian)h (mot)o(ion.)i(Hea)o(vy)e(curv)o(e)q(s)134 885 y(repre)q(s)q(en)o(t)21 b(t)n(h)o(e)g(equilibr)q(iu)o(m)16 b(branc)o(h)o(e)q(s)21 b Ft(\006)p FF(x)854 870 y Fp(?)873 885 y FG(\()p FF(t)p FG(\),)g(an)o(d)e(t)n(h)o(e)h(straigh)o(t)g(lin)o(e)f(repre)q(s)q(en)o (t)o(s)j(t)n(h)o(e)e(saddle.)134 935 y(Smo)q(ot)n(h)11 b(ligh)o(t)h(curv)o(e)q(s)i(are)g(solu)o(t)o(ions)e(of)g(t)n(h)o(e)i(d) o(et)o(ermini)q(st)o(ic)e(equa)o(t)o(ion)g(\(2.9\))g(trac)o(kin)o(g)h (t)n(h)o(e)h(p)q(ot)o(en)o(t)o(ial)134 985 y(w)o(ells)j(wit)n(h)g(a)f (sm)o(all)f(lag,)i(while)f(rugge)q(d)i(curv)o(e)q(s)h(are)e(pa)o(t)n (hs)h(of)e(t)n(h)o(e)i(SDE.)f(P)o(aram)o(et)o(er)g(v)n(alue)q(s)g(are) 134 1035 y FF(")12 b FG(=)f(0)p FF(:)p FG(01,)i FF(a)330 1041 y Fu(0)360 1035 y FG(=)f(0)p FF(:)p FG(02,)g FF(\033)g FG(=)g(0)p FF(:)p FG(02)h(\(left\))h(an)o(d)f FF(\033)g FG(=)f(0)p FF(:)p FG(08)h(\(r)q(igh)o(t\).)130 1173 y 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1354 y Fs(_)11 b FB(")334 1338 y Fw(2)p FA(=)p Fw(3)389 1354 y FC(\))407 1338 y Fw(2)426 1354 y FC(\).)22 b(On)16 b(t)n(h)o(e)f(ot)n(h)o(er)g(h)o(an)o (d,)g(if)h FB(\033)h FC(i)q(s)f(not)f(so)g(sm)o(all,)h(\(2.18\))d(can)i (st)o(ill)i(b)q(e)e(ap)o(plie)q(d)h(t)o(o)59 1410 y(sh)o(o)o(w)d(t)n(h) o(a)o(t)g(a)h(trans)q(it)o(ion)e(i)q(s)i(u)o(nlik)o(ely)i(t)o(o)d(o)q (ccur)h(b)q(efore)f(a)h(t)o(im)o(e)f(of)h(ord)o(er)f Fs(\000)1362 1378 y(p)p 1400 1378 28 2 v 32 x FB(\033)r FC(.)19 b(Figure)c(3)f(illustra)o(t)o(e)q(s)59 1467 y(t)n(hi)q(s)h(ph)o (enom)o(enon)e(b)o(y)h(sh)o(o)o(win)o(g)g(t)o(ypical)g(pa)o(t)n(hs)g (for)g(t)o(w)o(o)f(di\013eren)o(t)i(noi)q(s)q(e)f(in)o(t)o(ens)q(it)o (ie)q(s.)130 1523 y(Let)f(us)h(no)o(w)f(assu)o(m)o(e)g(t)n(h)o(a)o(t)f FB(\033)k FC(i)q(s)e(su\016cien)o(t)n(ly)f(large)g(t)o(o)h(allo)o(w)f (for)g(a)g(trans)q(it)o(ion,)f(an)o(d)h(examin)o(e)g(t)n(h)o(e)59 1580 y(trans)q(it)o(ion)h(regim)o(e)h(in)i(more)d(d)o(et)o(ail.)23 b(W)l(e)17 b(will)g(pro)q(cee)q(d)f(in)h(t)o(w)o(o)d(st)o(eps.)22 b(First)17 b(w)o(e)e(will)i(e)q(st)o(im)o(a)o(t)o(e)d(t)n(h)o(e)59 1636 y(proba)o(bilit)o(y)d(of)h Fy(not)17 b FC(reac)o(hin)o(g)12 b(t)n(h)o(e)g(saddle)g(a)o(t)f FB(x)i FC(=)f(0)g(d)o(ur)q(in)o(g)g(a)g (t)o(im)o(e)f(in)o(t)o(erv)n(al)g([)p FB(t)1428 1643 y Fw(0)1448 1636 y FB(;)d(t)1485 1643 y Fw(1)1504 1636 y FC(].)18 b(Th)o(e)12 b(symm)o(etry)59 1693 y(of)i FB(f)20 b FC(implie)q(s)c(t)n(h)o(a)o(t)d(for)h(an)o(y)g FB(t)f Fr(>)g FB(t)646 1700 y Fw(1)681 1693 y FC(an)o(d)i FB(x)794 1700 y Fw(0)826 1693 y FB(>)e FC(0,)441 1811 y Fl(P)475 1793 y FA(t)488 1798 y Fz(0)502 1793 y FA(;x)532 1798 y Fz(0)551 1775 y Fq(\010)578 1811 y FB(x)604 1818 y FA(t)631 1811 y FB(<)g FC(0)702 1775 y Fq(\011)741 1811 y FC(=)793 1781 y(1)p 793 1801 23 2 v 793 1843 a(2)825 1811 y Fl(P)858 1793 y FA(t)871 1798 y Fz(0)885 1793 y FA(;x)915 1798 y Fz(0)935 1775 y Fq(\010)961 1811 y Fs(9)p FB(s)g Fs(2)g FC(\()p FB(t)1097 1818 y Fw(0)1117 1811 y FB(;)8 b(t)p FC(\))d(:)14 b FB(x)1230 1818 y FA(s)1261 1811 y FC(=)e(0)1331 1775 y Fq(\011)741 1919 y FC(=)793 1889 y(1)p 793 1909 V 793 1950 a(2)831 1919 y Fs(\000)881 1889 y FC(1)p 881 1909 V 881 1950 a(2)913 1919 y Fl(P)946 1901 y FA(t)959 1906 y Fz(0)974 1901 y FA(;x)1004 1906 y Fz(0)1023 1882 y Fq(\010)1049 1919 y FB(x)1075 1926 y FA(s)1106 1919 y FB(>)h FC(0)f Fs(8)p FB(s)h Fs(2)g FC([)p FB(t)1320 1926 y Fw(0)1340 1919 y FB(;)8 b(t)p FC(])1390 1882 y Fq(\011)741 2027 y Fr(>)794 1996 y FC(1)p 794 2017 V 794 2058 a(2)831 2027 y Fs(\000)882 1996 y FC(1)p 882 2017 V 882 2058 a(2)913 2027 y Fl(P)947 2008 y FA(t)960 2013 y Fz(0)974 2008 y FA(;x)1004 2013 y Fz(0)1023 1990 y Fq(\010)1049 2027 y FB(x)1075 2034 y FA(s)1106 2027 y FB(>)13 b FC(0)g Fs(8)p FB(s)g Fs(2)g FC([)p FB(t)1321 2034 y Fw(0)1340 2027 y FB(;)8 b(t)1377 2034 y Fw(1)1397 2027 y FC(])1410 1990 y Fq(\011)1436 2027 y FB(:)267 b FC(\(2.22\))59 2140 y(In)15 b(t)n(h)o(e)g(s)q(econ)o(d)f(st)o(ep,)g (w)o(e)g(will)i(sh)o(o)o(w,)d(in)o(d)o(ep)q(en)o(d)o(en)o(t)n(ly)l(,)i (t)n(h)o(a)o(t)e(pa)o(t)n(hs)h(are)g(lik)o(ely)i(t)o(o)e(lea)o(v)o(e)h (a)g(n)o(e)q(igh)o(b)q(our-)59 2196 y(h)o(o)q(o)q(d)i(of)f FB(x)h FC(=)f(0)h(aft)o(er)e(t)o(im)o(e)569 2163 y Fs(p)p 607 2163 28 2 v 33 x FB(\033)r FC(.)26 b(Th)n(us)17 b(if)g(t)n(h)o(e)g (proba)o(bilit)o(y)f(of)g(not)h(reac)o(hin)o(g)g FB(x)f FC(=)g(0)h(i)q(s)g(sm)o(all,)g(t)n(h)o(e)59 2253 y(proba)o(bilit)o(y)12 b(of)h(m)o(akin)o(g)g(a)f(trans)q(it)o(ion)g(f)q(rom)f(t)n(h)o(e)i(p)q (os)q(it)o(iv)o(e)g(w)o(ell)g(t)o(o)g(t)n(h)o(e)g(n)o(ega)o(t)o(iv)o(e) e(on)o(e)i(will)i(b)q(e)e(clos)q(e)g(t)o(o)59 2309 y(1)p FB(=)p FC(2)i(\(it)g(can)h(n)o(ev)o(er)f(excee)q(d)h(1)p FB(=)p FC(2)e(b)q(eca)n(us)q(e)h(of)g(t)n(h)o(e)h(symm)o(etry\))o(.)k (Thi)q(s)15 b(do)q(e)q(s)g(not)g(exclud)o(e,)h(of)f(cours)q(e,)59 2365 y(t)n(h)o(a)o(t)d(pa)o(t)n(hs)g(f)q(requen)o(t)n(ly)g(swit)o(c)o (h)g(bac)o(k)h(an)o(d)f(fort)n(h)g(b)q(et)o(w)o(een)g(t)n(h)o(e)h(t)o (w)o(o)e(p)q(ot)o(en)o(t)o(ial)h(w)o(ells)i(d)o(ur)q(in)o(g)f(t)n(h)o (e)g(t)o(im)o(e)59 2422 y(in)o(t)o(erv)n(al)i([)p Fs(\000)270 2389 y(p)p 308 2389 V 33 x FB(\033)r(;)357 2389 y Fs(p)p 394 2389 V 394 2422 a FB(\033)5 b FC(].)25 b(Bu)o(t)16 b(it)h(sh)o(o)o(ws)e(t)n(h)o(a)o(t)g(\(2.22\))f(can)j(in)o(d)o(ee)q(d)g (b)q(e)g(in)o(t)o(erpret)o(e)q(d)e(as)h(a)g(lo)o(w)o(er)g(b)q(ou)o(n)o (d)59 2478 y(on)f(t)n(h)o(e)f(trans)q(it)o(ion)f(proba)o(bilit)o(y)l(.) 130 2535 y(Let)h FB(\016)h(>)e FC(0)h(b)q(e)h(a)g(const)o(an)o(t)d(su)o (c)o(h)i(t)n(h)o(a)o(t)524 2637 y FB(x@)574 2644 y FA(xx)616 2637 y FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))k Fr(6)g FC(0)90 b(for)14 b Fs(j)p FB(x)p Fs(j)e Fr(6)h FB(\016)k FC(an)o(d)d Fs(j)p FB(t)p Fs(j)e Fr(6)h FB(T)6 b FC(.)350 b(\(2.23\))59 2739 y(Our)14 b(h)o(yp)q(ot)n(h)o(e)q(s)q(e)q(s)f(on)g FB(f)20 b FC(imply)14 b(t)n(h)o(a)o(t)e(su)o(c)o(h)i(a)g FB(\016)i FC(of)d(ord)o(er)h(on)o(e)f(alw)o(ays)g(exi)q(st)o(s.)19 b(In)14 b(som)o(e)f(sp)q(ecial)h(cas)q(e)q(s,)59 2796 y(for)g(inst)o(ance)g(if)h FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))k(=)g FB(a)p FC(\()p FB(t)p FC(\))p FB(x)e Fs(\000)g FB(x)711 2779 y Fw(3)731 2796 y FC(,)15 b FB(\016)i FC(m)o(ay)c(b)q(e)i (c)o(h)o(os)q(en)f(arbitrar)q(ily)f(large.)922 2936 y(10)p eop %%Page: 11 11 11 10 bop 59 164 a Fo(Th)o(eorem)15 b(2.3)i(\(T)l(rans)q(it)o(ion)g (regim)o(e\).)k Fy(L)n(et)12 b FB(c)928 171 y Fw(1)960 164 y FB(>)h FC(0)e Fy(b)n(e)g(a)g(c)n(onstant)h(and)g(assume)g FB(c)1558 147 y Fw(2)1558 176 y(1)1577 164 y FB(\033)j Fr(>)e FB(a)1690 171 y Fw(0)1710 164 y Fs(_)q FB(")1762 147 y Fw(2)p FA(=)p Fw(3)1817 164 y Fy(.)59 220 y(Cho)n(ose)22 b(times)g Fs(\000)p FB(T)30 b Fr(6)24 b FB(t)513 227 y Fw(0)557 220 y Fr(6)g FB(t)632 227 y Fw(1)676 220 y Fr(6)g FB(T)k Fy(with)22 b FB(t)910 227 y Fw(1)954 220 y Fs(2)i FC([)p Fs(\000)p FB(c)1076 227 y Fw(1)1095 188 y Fs(p)p 1133 188 28 2 v 32 x FB(\033)q(;)8 b(c)1201 227 y Fw(1)1220 188 y Fs(p)p 1258 188 V 32 x FB(\033)d FC(])p Fy(,)23 b(and)g(let)f FB(h)i(>)g FC(2)p FB(\033)f Fy(b)n(e)f(such)59 277 y(that)f FB(x)182 260 y Fw(d)o(et)182 288 y FA(s)247 277 y FC(+)14 b FB(h)322 238 y Fq(p)p 367 238 80 2 v 39 x FB(\020)s FC(\()p FB(s)p FC(\))22 b FB(<)h(\016)h Fy(for)d(al)r(l)h FB(s)h Fs(2)g FC([)p FB(t)843 284 y Fw(0)862 277 y FB(;)8 b(t)899 284 y Fw(1)919 277 y FC(])p Fy(.)36 b(Then,)22 b(if)f FB(c)1188 284 y Fw(1)1229 277 y Fy(is)h(su\016ciently)f(smal)r(l)i(and)e FB(x)1758 284 y Fw(0)1800 277 y Fs(2)59 333 y FC(\(0)p FB(;)8 b(x)147 317 y Fw(d)o(et)147 344 y FA(t)160 349 y Fz(0)206 333 y FC(+)256 315 y Fw(1)p 256 322 18 2 v 256 349 a(2)279 333 y FB(h)305 294 y Fq(p)p 350 294 95 2 v 39 x FB(\020)s FC(\()p FB(t)407 340 y Fw(0)427 333 y FC(\)])p Fy(,)192 432 y Fl(P)226 413 y FA(t)239 418 y Fz(0)253 413 y FA(;x)283 418 y Fz(0)302 395 y Fq(\010)329 432 y FB(x)355 439 y FA(s)386 432 y FB(>)13 b FC(0)f Fs(8)p FB(s)h Fs(2)g FC([)p FB(t)600 439 y Fw(0)620 432 y FB(;)8 b(t)657 439 y Fw(1)676 432 y FC(])689 395 y Fq(\011)268 532 y Fr(6)321 501 y FC(5)p 321 522 23 2 v 321 563 a(2)349 482 y Fq(\020)381 501 y Fs(j)p 399 476 22 2 v FB(\013)p FC(\()p FB(t)457 508 y Fw(1)477 501 y FB(;)g(t)514 508 y Fw(0)533 501 y FC(\))p Fs(j)p 381 522 182 2 v 461 563 a FB(")578 532 y FC(+)i(1)646 482 y Fq(\021)680 532 y FC(e)700 513 y Fx(\000)q Fw(\026)-19 b FA(\024)q(h)768 502 y Fz(2)786 513 y FA(=\033)825 502 y Fz(2)852 532 y FC(+)t(2)8 b(exp)989 482 y Fq(n)1020 532 y Fs(\000)r FC(\026)-25 b FB(\024)1160 501 y FC(1)p 1086 522 170 2 v 1086 563 a(log\()p FB(h=\033)r FC(\))1266 501 y FB(\013)p FC(\()p FB(t)1329 508 y Fw(1)1349 501 y FB(;)8 b Fs(\000)p FB(c)1425 508 y Fw(1)1444 469 y Fs(p)p 1482 469 28 2 v 32 x FB(\033)q FC(\))p 1266 522 261 2 v 1386 563 a FB(")1532 482 y Fq(o)1562 532 y FB(;)141 b FC(\(2.24\))59 652 y Fy(wher)n(e)17 b FC(\026)-25 b FB(\024)16 b Fy(is)g(a)g(p)n(ositive)f(c)n(onstant,)h(and)h FB(\013)p FC(\()p FB(t;)8 b(s)p FC(\))j(=)940 616 y Fq(R)970 629 y FA(t)962 668 y(s)993 652 y FB(a)p FC(\()p FB(u)p FC(\))d(d)n FB(u)p Fy(.)130 732 y FC(Th)o(e)14 b(pro)q(of)f(i)q(s)h (giv)o(en)h(in)g(Su)n(bs)q(ect)o(ion)f(3.3.)k(Th)o(e)c(\014rst)g(t)o (erm)f(in)i(\(2.24\))d(i)q(s)i(an)h(up)o(p)q(er)e(b)q(ou)o(n)o(d)i(on)f (t)n(h)o(e)59 789 y(proba)o(bilit)o(y)k(t)n(h)o(a)o(t)f FB(x)415 796 y FA(t)448 789 y FC(e)q(scap)q(e)q(s)h(\\up)o(w)o(ard".)27 b(In)o(d)o(ee)q(d,)20 b(our)e(h)o(yp)q(ot)n(h)o(e)q(s)q(e)q(s)f(on)h FB(f)24 b FC(do)18 b(not)g(exclud)o(e)h(t)n(h)o(a)o(t)59 845 y(ot)n(h)o(er)13 b(st)o(a)o(ble)g(equilibr)q(ia)j(exi)q(st)e(for)f (su\016cien)o(t)n(ly)h(large)g FB(x)p FC(,)g(whic)o(h)g(migh)o(t)g (trap)f(e)q(scapin)o(g)h(tra)r(ject)o(or)q(ie)q(s.)59 902 y(Th)o(e)h(s)q(econ)o(d)f(t)o(erm)f(b)q(ou)o(n)o(ds)i(t)n(h)o(e)g (proba)o(bilit)o(y)f(of)h FB(x)939 909 y FA(s)972 902 y FC(rem)o(ainin)o(g)g(b)q(et)o(w)o(een)e(0)i(an)o(d)f FB(x)1504 885 y Fw(d)o(et)1504 913 y FA(s)1565 902 y FC(+)c FB(h)1636 863 y Fq(p)p 1682 863 80 2 v 1682 902 a FB(\020)s FC(\()p FB(s)p FC(\))15 b(for)59 958 y Fs(\000)p FB(c)114 965 y Fw(1)134 926 y Fs(p)p 172 926 28 2 v 32 x FB(\033)j Fr(6)e FB(s)h Fr(6)f FB(t)372 965 y Fw(1)392 958 y FC(.)26 b(Thi)q(s)17 b(e)q(st)o(im)o(a)o(t)o(e)e(lie)q(s)j(a)o(t) e(t)n(h)o(e)h(core)f(of)h(our)f(argu)o(m)o(en)o(t,)e(an)o(d)j(can)f(b)q (e)h(u)o(n)o(d)o(erst)o(o)q(o)q(d)59 1015 y(as)f(follo)o(ws.)24 b(Assu)o(m)o(e)16 b FB(x)480 1022 y FA(s)515 1015 y FC(st)o(art)o(s)f (n)o(ear)h FB(x)768 998 y Fw(d)o(et)768 1026 y FA(s)818 1015 y FC(.)26 b(It)16 b(will)i(p)q(erform)d(a)h(cert)o(ain)g(n)n(u)o (m)n(b)q(er)g(of)g(excurs)q(ions)f(t)o(o)59 1071 y(a)o(t)n(t)o(empt)e (reac)o(hin)o(g)i(t)n(h)o(e)g(saddle)h(a)o(t)e FB(x)g FC(=)f(0.)21 b(Eac)o(h)14 b(excurs)q(ion)h(require)q(s)f(a)i(t)o (ypical)e(t)o(im)o(e)h(of)g(ord)o(er)f(\001)p FB(s)p FC(,)59 1128 y(su)o(c)o(h)d(t)n(h)o(a)o(t)g FB(\013)p FC(\()p FB(s)t FC(+)t(\001)p FB(s;)d(s)p FC(\))j Fs(\020)i FB(")f FC(\(t)n(h)o(a)o(t)e(i)q(s,)i FB(a)p FC(\()p FB(s)p FC(\)\001)p FB(s)g Fs(\020)h FB(")p FC(\),)e(in)i(t)n(h)o(e)e(s)q(ens)q 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2079 l 2475 2716 l 2512 2424 l 2550 2775 l 2595 2266 l 2640 2604 l 2700 2175 l 2722 2416 l 2767 2244 l 2775 2700 l 2835 2469 l 2865 2791 l 2925 2250 l 2970 2821 l 3022 2529 l 3075 3300 l 3105 2963 l 3135 3316 l 3202 2544 l 3225 2700 l 3270 2581 l 3315 3233 l 3390 2888 l 3450 3750 l 3480 3428 l 3525 3676 l 3600 3000 l 3675 3525 l 3727 2903 l 3780 3083 l 3825 2700 l 3900 3248 l 3952 3083 l 3975 3825 l 4035 3218 l 4087 3526 l 4125 3300 l 4215 4276 l 4268 3983 l 4275 4650 l 4328 4081 l 4373 4209 l 4425 3450 l 4470 4013 l 4508 3871 l 4545 4779 l 4575 4381 l 4635 4989 l 4665 3518 l 4710 3818 l 4725 3300 l 4747 3601 l 4770 3143 l 4800 3675 l gs col0 s gr $F2psEnd rs endTexFig 3172 2544 a Fi(t)975 1965 y Fh(B)s Fg(\()p Fi(h)p Fg(\))1247 2532 y Fh(\000)1312 2462 y(p)p 1380 2462 48 4 v 1380 2532 a Fi(h)2983 2048 y Fh(D)2203 1954 y Fi(x)2250 1923 y Ff(?)2289 1954 y Fg(\()p Fi(t)p Fg(\))2133 2969 y Fh(\000)p Fi(x)2245 2939 y Ff(?)2283 2969 y Fg(\()p Fi(t)p Fg(\))1778 1339 y Fi(x)3148 1316 y(x)3195 1286 y Fe(det)3195 1336 y Ff(t)1258 3111 y Fh(\000)p Fi(x)1370 3081 y Fe(det)1370 3132 y Ff(t)3154 1835 y Fg(~)-48 b Fi(x)q Fg(\()p Fi(t)p Fg(\))2038 1812 y Fi(x)2085 1824 y Ff(t)2014 3111 y Fh(\000)p Fi(x)2126 3123 y Ff(t)1920 1481 y Fi(x)1967 1451 y Fe(det)1967 1502 y Ff(t)2082 1481 y Fg(+)18 b Fi(h)2213 1410 y Fd(p)p 2296 1410 138 4 v 71 x Fi(\020)6 b Fg(\()p Fi(t)p Fg(\))268 4559 y Fc(Figure)32 b(4.)40 b Fb(Nils)28 b(Berglund)f(and)g(Barbara)e (Gen)n(tz)268 4658 y(Sto)r(c)n(hastic)i(resonance)e(in)j(a)f(double-w)n (ell)g(p)r(oten)n(tial)1867 5871 y Fa(1)p eop end userdict /end-hook known{end-hook}if %%EndDocument endTexFig 134 1246 a Fv(Figure)11 b(4.)21 b FG(A)11 b(t)o(ypical)f(pa)o(t)n(h)g FF(x)634 1252 y Fp(t)659 1246 y FG(of)g(t)n(h)o(e)h(SDE)g(\(2.1\))f(in) g(t)n(h)o(e)h(symm)o(etr)q(ic)e(cas)q(e,)i(sh)o(o)o(wn)g(in)f(a)g(n)o (e)q(igh)o(b)q(our-)134 1296 y(h)o(o)q(o)q(d)15 b(of)g(t)n(h)o(e)h(or)q (igin,)d(wh)o(ere)k(t)n(h)o(e)f(p)q(ot)o(en)o(t)o(ial)e(barr)q(ier)i (reac)o(h)o(e)q(s)h(it)o(s)f(minim)n(al)c(h)o(e)q(igh)o(t.)22 b(W)m(e)14 b(sh)o(o)o(w)i(h)o(ere)134 1346 y(a)f(s)q(it)o(ua)o(t)o(ion) f(wh)o(ere)j(t)n(h)o(e)f(noi)q(s)q(e)g(in)o(t)o(ens)q(it)o(y)e(i)q(s)h (large)h(enough)g(t)o(o)f(allo)o(w)f(a)i(trans)q(it)o(ion.)22 b(Th)o(e)16 b(p)q(ot)o(en)o(t)o(ial)134 1396 y(w)o(ells)d(a)o(t)h Ft(\006)p FF(x)340 1381 y Fp(?)359 1396 y FG(\()p FF(t)p FG(\))g(b)q(e)o(h)o(a)o(v)o(e)g(lik)o(e)f Ft(\006)p FG(\()680 1369 y Ft(p)p 715 1369 41 2 v 27 x FF(a)737 1402 y Fu(0)765 1396 y Ft(_)c(j)p FF(t)p Ft(j)p FG(\).)17 b(Th)o(e)d(d)o(et)o(ermini)q (st)o(ic)f(solu)o(t)o(ion)g FF(x)1396 1381 y Fu(d)o(et)1396 1406 y Fp(t)1457 1396 y FG(st)o(art)o(in)o(g)i(n)o(ear)f(t)n(h)o(e)134 1445 y(r)q(igh)o(t-h)o(an)o(d)c(p)q(ot)o(en)o(t)o(ial)g(w)o(ell)g(trac) o(ks)i FF(x)729 1430 y Fp(?)748 1445 y FG(\()p FF(t)p FG(\))f(a)o(t)g(a)g(di)q(st)o(ance)h(a)o(t)f(most)f(of)h(ord)o(er)h(\() p FF("=a)1418 1451 y Fu(0)1437 1445 y FG(\))t Ft(_)t FF(")1508 1430 y Fu(1)p Fp(=)p Fu(3)1560 1445 y FG(,)f(an)o(d)g(n)o(ev) o(er)134 1495 y(ap)o(proac)o(h)o(e)q(s)j(t)n(h)o(e)g(saddle)g(a)o(t)f FF(x)e FG(=)h(0)h(clos)q(er)i(t)n(h)o(an)913 1468 y Ft(p)p 948 1468 V 27 x FF(a)970 1501 y Fu(0)998 1495 y Ft(_)9 b FF(")1054 1480 y Fu(1)p Fp(=)p Fu(3)1106 1495 y FG(.)18 b(Th)o(e)c(pa)o(t)n(h)f FF(x)1337 1501 y Fp(t)1365 1495 y FG(i)q(s)g(lik)o(ely)f(t)o(o)i(st)o(ay)f(in)g(t)n(h)o(e)134 1545 y(s)q(et)i Ft(B)q FG(\()p FF(h)p FG(\))f(up)g(t)o(o)h(t)o(im)o(e)d Ft(\000)533 1510 y(p)p 568 1510 24 2 v 35 x FF(h)i FG(wh)o(en)g FF(h)e Ft(\035)f FF(\033)q FG(.)19 b(Bet)o(w)o(een)c(t)o(im)o(e)q(s)e Ft(\000)1165 1515 y(p)p 1200 1515 26 2 v 30 x FF(\033)i FG(an)o(d)1319 1515 y Ft(p)p 1354 1515 V 30 x FF(\033)q FG(,)f(t)n(h)o(e)g(pa)o(t)n(h)h(i)q(s)e(lik)o(ely)g(t)o(o)134 1595 y(reac)o(h)k(t)n(h)o(e)f(or)q(igin.)21 b(It)16 b(m)o(ay)e(con)o(t) o(in)n(ue)h(t)o(o)h(ju)o(mp)e(bac)o(k)i(an)o(d)f(fort)n(h)h(b)q(et)o(w) o(een)g(t)n(h)o(e)g(p)q(ot)o(en)o(t)o(ial)f(w)o(ells)g(up)134 1645 y(t)o(o)e(t)o(im)o(e)274 1615 y Ft(p)p 309 1615 V 30 x FF(\033)q FG(,)g(bu)o(t)h(i)q(s)e(lik)o(ely)g(t)o(o)h(lea)o(v)o (e)g(a)g(n)o(e)q(igh)o(b)q(ourh)o(o)q(o)q(d)g Ft(D)h FG(of)e(t)n(h)o(e)i(saddle)f(for)g(t)o(im)o(e)q(s)f(sligh)o(t)n(ly)g (larger)134 1694 y(t)n(h)o(an)228 1665 y Ft(p)p 263 1665 V 29 x FF(\033)q FG(.)19 b(F)m(or)13 b(eac)o(h)i(realiza)o(t)o(ion)f FF(!)i FG(su)o(c)o(h)e(t)n(h)o(a)o(t)h FF(x)934 1700 y Fp(t)948 1694 y FG(\()p FF(!)q FG(\))g(reac)o(h)o(e)q(s)h(t)n(h)o(e)f (saddle)f(a)o(t)g(a)g(t)o(im)o(e)e FF(\034)5 b FG(,)14 b(t)n(h)o(ere)i(i)q(s)d(a)134 1744 y(realiza)o(t)o(ion)e FF(!)360 1729 y Ff(0)384 1744 y FG(su)o(c)o(h)i(t)n(h)o(a)o(t)f FF(x)583 1750 y Fp(t)597 1744 y FG(\()p FF(!)640 1729 y Ff(0)653 1744 y FG(\))f(=)h Ft(\000)p FF(x)780 1750 y Fp(t)795 1744 y FG(\()p FF(!)q FG(\))g(i)q(s)g(t)n(h)o(e)h(mirror)d (im)o(age)g(of)i FF(x)1288 1750 y Fp(t)1314 1744 y FG(for)g FF(t)f Fe(>)h FF(\034)5 b FG(,)12 b(whic)o(h)g(explains)134 1792 y(wh)o(y)j(t)n(h)o(e)i(proba)o(bilit)o(y)d(t)o(o)i(c)o(h)o(o)q(os) q(e)h(on)o(e)e(w)o(ell)g(or)h(t)n(h)o(e)h(ot)n(h)o(er)f(aft)o(er)g(t)n (h)o(e)h(trans)q(it)o(ion)d(region)i(i)q(s)f(clos)q(e)i(t)o(o)134 1848 y(1)p FF(=)p FG(2.)130 1987 y FC(It)i(rem)o(ains)e(t)o(o)i(sh)o(o) o(w)e(t)n(h)o(a)o(t)h(t)n(h)o(e)g(pa)o(t)n(hs)g(are)g(lik)o(ely)i(t)o (o)f(ap)o(proac)o(h)e(e)q(it)n(h)o(er)i FB(x)1440 1971 y FA(?)1459 1987 y FC(\()p FB(t)p FC(\))g(or)f Fs(\000)p FB(x)1650 1971 y FA(?)1670 1987 y FC(\()p FB(t)p FC(\))h(aft)o(er)59 2044 y(lea)o(vin)o(g)e Fs(D)q FC(\()p FB(\024)p FC(\).)23 b(Let)16 b(us)g(\014rst)g(cons)q(id)o(er)g(t)n(h)o(e)f(solu)o(t)o(ion)i Fq(b)-26 b FB(x)1035 2027 y Fw(d)o(et)1031 2055 y FA(t)1101 2044 y FC(of)16 b(t)n(h)o(e)g(d)o(et)o(ermini)q(st)o(ic)f(di\013eren)o (t)o(ial)h(equa-)59 2100 y(t)o(ion)d(\(2.9\))e(wit)n(h)i(init)o(ial)h (t)o(im)o(e)f FB(t)596 2107 y Fw(2)629 2100 y Fr(>)g FB(c)697 2107 y Fw(2)716 2067 y Fs(p)p 754 2067 28 2 v 33 x FB(\033)i FC(an)o(d)e(init)o(ial)h(con)o(dit)o(ion)f Fs(j)p FB(x)1240 2107 y Fw(2)1266 2100 y Fs(\000)7 b FB(x)1334 2084 y FA(?)1354 2100 y FC(\()p FB(t)p FC(\))p Fs(j)12 b Fr(6)h FB(c)t(t)1519 2107 y Fw(2)1552 2100 y FC(for)f(som)o(e)g(sm)o(all)59 2157 y(const)o(an)o(t)e FB(c)i(>)h FC(0.)18 b(Here)12 b(w)o(e)g(n)o(ee)q(d)g(t)o(o)g(c)o(h)o(o) q(os)q(e)g FB(c)g FC(sm)o(all)g(in)h(ord)o(er)e(t)o(o)h(arran)o(ge)f (for)f Fq(b)-24 b FB(a)p FC(\()p FB(t)p FC(\))12 b(=)g FB(@)1581 2164 y FA(x)1604 2157 y FB(f)5 b FC(\()r Fq(b)-27 b FB(x)1679 2140 y Fw(d)o(et)1675 2168 y FA(t)1729 2157 y FB(;)8 b(t)p FC(\))j Fs(\020)59 2213 y(\000)p FB(t)110 2197 y Fw(2)144 2213 y FC(whic)o(h)j(allo)o(ws)f(us)g(t)o(o)h(pro)q (cee)q(d)f(as)g(in)h(our)f(in)o(v)o(e)q(st)o(iga)o(t)o(ion)f(of)h(t)n (h)o(e)g(mot)o(ion)g(for)f FB(t)h Fr(6)g Fs(\000)p FC(\()1597 2180 y Fs(p)p 1635 2180 25 2 v 33 x FB(a)1659 2226 y Fw(0)1686 2213 y Fs(_)8 b FB(")1745 2197 y Fw(1)p FA(=)p Fw(3)1801 2213 y FC(\),)59 2270 y(cf.)18 b(Su)n(bs)q(ect)o(ions)g(3.1)f (an)o(d)h(3.2.)30 b(Un)o(d)o(er)18 b(t)n(h)o(e)q(s)q(e)g(assu)o(mpt)o (ions,)h Fq(b)-26 b FB(x)1195 2253 y Fw(d)o(et)1191 2281 y FA(t)1264 2270 y FC(ap)o(proac)o(h)o(e)q(s)16 b(a)i(n)o(e)q(igh)o(b)q (ourh)o(o)q(o)q(d)59 2326 y(of)f Fs(\006)p FB(x)174 2310 y FA(?)194 2326 y FC(\()p FB(t)p FC(\))g(exp)q(on)o(en)o(t)o(ially)f(f) q(ast)g(an)o(d)h(t)n(h)o(en)g(trac)o(ks)e(t)n(h)o(e)i(equilibr)q(iu)o (m)h(branc)o(h)e(a)o(t)h(di)q(st)o(ance)f FB("=t)1719 2310 y Fw(2)1739 2326 y FC(.)27 b(As)59 2382 y(b)q(efore,)13 b(on)o(e)h(can)g(sh)o(o)o(w)f(t)n(h)o(a)o(t)g(t)n(h)o(e)h(pa)o(t)n(h)f FB(x)773 2389 y FA(t)803 2382 y FC(of)g(t)n(h)o(e)h(solu)o(t)o(ion)g (of)g(t)n(h)o(e)g(SDE)g(\(2.1\))f(wit)n(h)h(t)n(h)o(e)g(sam)o(e)f(init) o(ial)59 2439 y(con)o(dit)o(ion)j(i)q(s)h(lik)o(ely)g(t)o(o)f(rem)o (ain)g(in)i(a)e(str)q(ip)g(arou)o(n)o(d)h Fq(b)-27 b FB(x)1015 2422 y Fw(d)o(et)1011 2450 y FA(t)1082 2439 y FC(of)17 b(wid)o(t)n(h)f(scalin)o(g)h(wit)n(h)g FB(\033)r(=)1565 2400 y Fq(p)p 1609 2400 76 2 v 1608 2439 a(b)-24 b FB(a)p FC(\()p FB(t)p FC(\))15 b Fs(\020)h FB(\033)r(=t)p FC(.)59 2495 y(So)d(if)h(a)f(pa)o(t)n(h)f FB(x)322 2502 y FA(t)351 2495 y FC(lea)o(v)o(e)q(s)h Fs(D)q FC(\()p FB(\024)p FC(\))g(a)o(t)f(t)o(im)o(e)h FB(\034)762 2505 y Fx(D)q Fw(\()p FA(\024)p Fw(\))840 2495 y FC(,)g(t)n(h)o(en)g(t)n(hi)q(s)g(pa) o(t)n(h)f(i)q(s)i(lik)o(ely)g(t)o(o)f(ap)o(proac)o(h)e FB(x)1581 2479 y FA(?)1601 2495 y FC(\()p FB(t)p FC(\),)h(if)i FB(x)1744 2502 y FA(\034)1760 2509 y Fh(D)q Fz(\()p Fg(\024)p Fz(\))59 2552 y FC(i)q(s)19 b(p)q(os)q(it)o(iv)o(e,)g(an)o(d)f Fs(\000)p FB(x)444 2535 y FA(?)464 2552 y FC(\()p FB(t)p FC(\),)g(ot)n(h)o(erwi)q(s)q(e.)29 b(Not)o(e)18 b(t)n(h)o(a)o(t)f Fs(j)p FB(x)1023 2535 y FA(?)1042 2552 y FC(\()p FB(\034)1080 2561 y Fx(D)q Fw(\()p FA(\024)p Fw(\))1158 2552 y FC(\))12 b Fs(\000)h FB(x)1262 2559 y FA(\034)1278 2566 y Fh(D)q Fz(\()p Fg(\024)p Fz(\))1348 2552 y Fs(j)19 b FC(h)o(as)f(t)o(o)g(b)q (e)h(sm)o(aller)f(t)n(h)o(an)59 2608 y FB(c)t(\034)103 2618 y Fx(D)q Fw(\()p FA(\024)p Fw(\))181 2608 y FC(,)f(whic)o(h)f(re)q (str)q(ict)o(s)g(t)n(h)o(e)g(p)q(oss)q(ible)i(v)n(alue)q(s)e(for)g FB(\024)p FC(.)26 b(Th)o(erefore,)15 b(w)o(e)i(c)o(h)o(o)q(os)q(e)f FB(\024)h FC(sm)o(all)g(enough)f(t)o(o)59 2665 y(guaran)o(t)o(ee)e(t)n (h)o(a)o(t)i FB(x)388 2648 y FA(?)407 2665 y FC(\()p FB(t)p FC(\))11 b Fs(\000)k FC(~)-26 b FB(x)p FC(\()p FB(t)p FC(\))16 b Fr(6)h FB(c)t(t)g FC(for)f(all)i FB(t)f Fr(>)g FB(c)961 2672 y Fw(1)980 2632 y Fs(p)p 1018 2632 28 2 v 33 x FB(\033)q FC(.)27 b(Fin)o(ally)18 b(not)o(e)e(t)n(h)o(a)o (t)g(pa)o(t)n(hs)g(whic)o(h)g(are)h(not)59 2721 y(in)f Fs(D)q FC(\()p FB(\024)p FC(\))g(a)o(t)e(t)o(im)o(e)h FB(c)401 2728 y Fw(2)421 2688 y Fs(p)p 459 2688 V 33 x FB(\033)i FC(bu)o(t)f(are)f(not)f(furt)n(h)o(er)h(aw)o(ay)e(f)q(rom)h Fs(\006)p FB(x)1175 2705 y FA(?)1211 2721 y FC(t)n(h)o(an)h FB(c)t(t)g FC(a)o(t)g(som)o(e)f(t)o(im)o(e)h FB(t)i FC(will)f(also)59 2778 y(ap)o(proac)o(h)d(t)n(h)o(e)h(corre)q(sp)q(on)o(din)o(g)g (equilibr)q(iu)o(m)h(branc)o(h.)922 2936 y(12)p eop %%Page: 13 13 13 12 bop 130 164 a FC(Let)20 b(us)g(no)o(w)g(di)q(scuss)g(t)n(h)o(e)g (c)o(h)o(oice)h(of)f(t)n(h)o(e)g(param)o(et)o(er)o(s)e(in)j(\(2.24\))d (givin)o(g)j(an)f(o)o(pt)o(im)o(al)f(b)q(ou)o(n)o(d.)59 220 y(F)l(or)e FB(\033)h Fr(>)f FC(\()p FB(a)280 227 y Fw(0)311 220 y Fs(_)12 b FB(")374 204 y Fw(2)p FA(=)p Fw(3)429 220 y FC(\))p FB(=c)490 204 y Fw(2)490 232 y(1)508 220 y FC(,)18 b(Th)o(eorem)d(2.2)h(sh)o(o)o(ws)f(t)n(h)o(a)o(t)h(up)h (t)o(o)g(t)o(im)o(e)f FB(t)i FC(sligh)o(t)n(ly)f(le)q(ss)h(t)n(h)o(an)e Fs(\000)1675 188 y(p)p 1713 188 28 2 v 32 x FB(\033)r FC(,)h(t)n(h)o(e)59 277 y(pa)o(t)n(hs)h(are)g(concen)o(tra)o(t)o(e)q(d) e(arou)o(n)o(d)i FB(x)712 260 y Fw(d)o(et)712 288 y FA(t)762 277 y FC(.)32 b(Th)o(erefore,)17 b(h)o(ere)i(w)o(e)f(sh)o(ould)h(c)o(h) o(o)q(os)q(e)f(an)h(init)o(ial)g(t)o(im)o(e)g FB(t)1811 284 y Fw(0)59 333 y FC(sligh)o(t)n(ly)c(b)q(efore)e Fs(\000)384 300 y(p)p 422 300 V 33 x FB(\033)r FC(.)19 b(In)c(ord)o(er)f(for)f(t)n (h)o(e)h(s)q(econ)o(d)g(t)o(erm)f(in)i(\(2.24\))d(t)o(o)i(b)q(e)g(sm)o (all,)g(w)o(e)g(w)o(an)o(t)f(t)o(o)h(c)o(h)o(o)q(os)q(e)59 390 y FB(t)75 397 y Fw(1)109 390 y Fs(\000)f FC(\()p Fs(\000)p FB(c)230 397 y Fw(1)250 357 y Fs(p)p 287 357 V 287 390 a FB(\033)r FC(\))19 b(as)h(large)e(as)i(p)q(oss)q(ible.)34 b(Not)o(e,)19 b(h)o(o)o(w)o(ev)o(er,)g(t)n(h)o(a)o(t)f FB(h)1231 351 y Fq(p)p 1276 351 80 2 v 39 x FB(\020)s FC(\()p FB(s)p FC(\))i(h)o(as)f(t)o(o)g(b)q(e)g(sm)o(aller)h(t)n(h)o (an)59 446 y FB(\016)9 b Fs(\000)e FB(x)156 430 y Fw(d)o(et)156 457 y FA(s)221 446 y FC(for)12 b(all)j FB(s)p FC(.)k(Since)14 b(t)n(h)o(e)f(ord)o(er)g(of)g FB(\020)s FC(\()p FB(s)p FC(\))g(i)q(s)g(increas)q(in)o(g)h(for)e FB(s)h Fr(6)g Fs(\000)p FC(\()1307 413 y Fs(p)p 1345 413 25 2 v 33 x FB(a)1369 459 y Fw(0)1396 446 y Fs(_)7 b FB(")1454 430 y Fw(1)p FA(=)p Fw(3)1510 446 y FC(\),)13 b(it)g(t)o(ur)q(ns)f(ou)o (t)h(t)o(o)59 503 y(b)q(e)i(more)g(adv)n(an)o(t)o(a)o(geo)o(us)e(t)o(o) i(c)o(h)o(o)q(os)q(e)g FB(t)726 510 y Fw(1)761 503 y FC(n)o(ega)o(t)o(iv)o(e)f(an)o(d)h(of)g(ord)o(er)f Fs(\000)1227 470 y(p)p 1266 470 28 2 v 1266 503 a FB(\033)q FC(,)i(say)e FB(t)1417 510 y Fw(1)1451 503 y FC(=)g Fs(\000)1540 485 y Fw(1)p 1540 492 18 2 v 1540 518 a(2)1563 503 y FB(c)1583 510 y Fw(1)1602 470 y Fs(p)p 1640 470 28 2 v 33 x FB(\033)r FC(.)21 b(In)16 b(t)n(hi)q(s)59 559 y(cas)q(e,)j Fs(j)p FB(\013)p FC(\()p FB(t)249 566 y Fw(1)268 559 y FB(;)8 b Fs(\000)p FB(c)344 566 y Fw(1)363 526 y Fs(p)p 401 526 V 33 x FB(\033)q FC(\))p Fs(j)18 b FC(i)q(s)h(larger)f(t)n(h)o(an)f (a)i(const)o(an)o(t)d(t)o(im)o(e)q(s)i FB(\033)1141 543 y Fw(3)p FA(=)p Fw(2)1214 559 y FC(\(in)o(d)o(ep)q(en)o(d)o(en)o(t)n (ly)g(of)h FB(a)1600 566 y Fw(0)1619 559 y FC(\),)g(an)o(d)f(t)n(hi)q (s)59 615 y(do)q(e)q(s)13 b(not)g(impro)o(v)o(e)f(s)q(igni\014can)o(t)n (ly)g(for)h(larger)f(admi)q(ss)q(ible)h FB(t)1084 622 y Fw(1)1105 615 y FC(.)19 b(A)o(t)12 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y Fw(2)378 1282 y(1)411 1270 y FC(wit)n(h)f(\()p FB(")p Fs(j)p FC(log)7 b FB(")p Fs(j)p FC(\))678 1253 y Fw(2)p FA(=)p Fw(3)745 1270 y Fs(\034)13 b FB(\033)i Fs(\034)e FC(1)p FB(=)p Fs(j)p FC(log)6 b FB(")p Fs(j)p FC(,)13 b(t)n(h)o(e)g(proba)o(bilit)o(y)f(of)g(a)h(trans)q(it)o(ion)e (b)q(et)o(w)o(een)136 1326 y(t)n(h)o(e)j(w)o(ells)h(i)q(s)f(exp)q(on)o (en)o(t)o(ially)f(clos)q(e)i(t)o(o)f(1)p FB(=)p FC(2,)e(wit)n(h)j(an)e (exp)q(on)o(en)o(t)g(giv)o(en)h(e)q(ss)q(en)o(t)o(ially)g(\(up)g(t)o(o) g(logar-)136 1383 y(it)n(hmic)h(correct)o(ions\))d(b)o(y)888 1485 y FB(\033)916 1469 y Fw(3)p FA(=)p Fw(2)p 888 1506 V 919 1547 a FB(")986 1516 y Fs(^)1034 1485 y FC(1)p 1031 1506 28 2 v 1031 1547 a FB(\033)1064 1516 y FC(;)639 b(\(2.29\))90 1631 y Fs(\017)23 b FC(for)15 b FB(\033)i Fr(>)e FC(1)p FB(=)p Fs(j)p FC(log)6 b FB(")p Fs(j)p FC(,)16 b(t)n(h)o(e)g(pa)o(t)n(hs)f(b)q(ecom)o(e)g(so)g(p)q(o)q(orly)h (lo)q(calize)q(d)h(t)n(h)o(a)o(t)e(it)h(i)q(s)g(no)g(lon)o(ger)g(m)o (eanin)o(gful)136 1688 y(t)o(o)f(sp)q(eak)f(of)g(a)h(trans)q(it)o(ion)e (proba)o(bilit)o(y)l(.)59 1756 y(\(2.29\))19 b(sh)o(o)o(ws)h(t)n(h)o(a) o(t)g(t)n(h)o(e)i(trans)q(it)o(ion)d(proba)o(bilit)o(y)i(b)q(ecom)o(e)q (s)f(o)o(pt)o(im)o(al)h(for)g FB(\033)k Fs(\020)g FB(")1522 1739 y Fw(2)p FA(=)p Fw(5)1577 1756 y FC(.)40 b(F)l(or)22 b(larger)59 1812 y(v)n(alue)q(s)12 b(of)g(t)n(h)o(e)f(noi)q(s)q(e)h(in) o(t)o(ens)q(it)o(y)l(,)f(t)n(h)o(e)h(p)q(oss)q(ibilit)o(y)i(of)d(pa)o (t)n(hs)h(e)q(scapin)o(g)g(\\up)o(w)o(ard")d(b)q(ecom)o(e)q(s)i (su\016cien)o(t)n(ly)59 1869 y(imp)q(ort)o(an)o(t)i(t)o(o)j(d)o(ecreas) q(e)f(t)n(h)o(e)h(trans)q(it)o(ion)e(proba)o(bilit)o(y)l(.)22 b(Ho)o(w)o(ev)o(er,)13 b(if)k(t)n(h)o(e)e(fu)o(nct)o(ion)h FB(f)21 b FC(i)q(s)16 b(su)o(c)o(h)f(t)n(h)o(a)o(t)g FB(\016)59 1925 y FC(can)e(b)q(e)g(c)o(h)o(os)q(en)g(arbitrar)q(ily)e (large,)h(t)n(h)o(e)h(s)q(econ)o(d)g(t)o(erm)e(in)j(\(2.29\))d(can)i(b) q(e)g(remo)o(v)o(e)q(d)e(wit)n(h)o(ou)o(t)h(c)o(h)o(an)o(gin)o(g)59 1982 y(t)n(h)o(e)j(\014rst)f(on)o(e)g(\(up)h(t)o(o)f(logar)q(it)n(hmic) f(correct)o(ions\))g(b)o(y)h(t)o(akin)o(g)g FB(\016)1147 1965 y Fw(2)1180 1982 y FC(=)e FB(\033)r(=")p FC(,)j(for)e(inst)o (ance.)19 b(In)d(t)n(h)o(a)o(t)d(cas)q(e,)59 2038 y(trans)q(it)o(ions)g (b)q(et)o(w)o(een)g(t)n(h)o(e)h(w)o(ells)h(b)q(ecom)o(e)f(t)n(h)o(e)g (more)g(lik)o(ely)h(t)n(h)o(e)g(larger)f(t)n(h)o(e)g(ra)o(t)o(io)f FB(\033)r(=")1557 2022 y Fw(2)p FA(=)p Fw(3)1627 2038 y FC(i)q(s.)130 2095 y(On)o(e)j(sh)o(ould)g(not)o(e)f(t)n(h)o(a)o(t)f (a)i(t)o(ypical)g(pa)o(t)n(h)f(will)j(reac)o(h)d(m)o(axim)o(al)g(v)n (alue)q(s)h(of)g(t)n(h)o(e)f(ord)o(er)h FB(\033)1655 2056 y Fq(p)p 1700 2056 81 2 v 39 x FB(\020)s FC(\(0\))d Fs(\020)59 2151 y FB(\033)r(=)p FC(\()128 2122 y Fs(p)p 165 2122 44 2 v 165 2151 a FB(a)189 2158 y Fw(0)221 2151 y Fs(_)f FB(")284 2135 y Fw(1)p FA(=)p Fw(3)339 2151 y FC(\).)28 b(Th)n(us,)17 b(d)o(ue)h(t)o(o)f(t)n(h)o(e)g(\015a)o(tn)o (e)q(ss)g(of)g(t)n(h)o(e)h(p)q(ot)o(en)o(t)o(ial)e(n)o(ear)h FB(t)h FC(=)f(0,)h(if)g FB(\033)h FC(i)q(s)f(larger)f(t)n(h)o(an)59 2178 y Fs(p)p 97 2178 V 30 x FB(a)121 2215 y Fw(0)155 2208 y Fs(_)e FB(")221 2191 y Fw(1)p FA(=)p Fw(3)276 2208 y FC(,)23 b(t)n(h)o(e)e(spreadin)o(g)g(of)g(t)n(h)o(e)g(pa)o(t)n (hs)g(d)o(ur)q(in)o(g)g(t)n(h)o(e)g(trans)q(it)o(ion)e(in)o(t)o(erv)n (al)h(i)q(s)h(larger)g(t)n(h)o(an)f(t)n(h)o(e)59 2264 y(m)o(axim)o(al)11 b(di)q(st)o(ance)h(b)q(et)o(w)o(een)e(t)n(h)o(e)i(w) o(ells)h(aw)o(ay)e(f)q(rom)f(t)n(h)o(e)i(trans)q(it)o(ion.)17 b(In)c(gen)o(eral)e(w)o(e)h(cannot)f(exclud)o(e)59 2321 y(t)n(h)o(a)o(t)i(pa)o(t)n(hs)h(e)q(scap)q(e)g(t)o(o)h(ot)n(h)o(er)e(a) o(t)n(tract)o(ors,)e(if)16 b(t)n(h)o(e)e(p)q(ot)o(en)o(t)o(ial)f(h)o (as)h(more)g(t)n(h)o(an)g(t)o(w)o(o)f(w)o(ells.)130 2377 y(It)h(m)o(ay)f(b)q(e)i(surpr)q(i)q(s)q(in)o(g)e(t)n(h)o(a)o(t)g(t)n(h) o(e)h(ord)o(er)f(of)h(t)n(h)o(e)g(trans)q(it)o(ion)e(proba)o(bilit)o(y) i(i)q(s)g(in)o(d)o(ep)q(en)o(d)o(en)o(t)g(of)g FB(a)1756 2384 y Fw(0)1790 2377 y FC(as)59 2433 y(so)q(on)f(as)g FB(\033)i(>)e(a)329 2440 y Fw(0)348 2433 y FC(.)19 b(In)o(t)o(uit)o(iv) o(ely)l(,)14 b(on)o(e)f(w)o(ould)g(ra)o(t)n(h)o(er)f(exp)q(ect)h(t)n (hi)q(s)h(proba)o(bilit)o(y)f(t)o(o)g(d)o(ep)q(en)o(d)h(on)f(t)n(h)o(e) h(ra)o(t)o(io)59 2490 y FB(a)83 2473 y Fw(2)83 2502 y(0)103 2490 y FB(=\033)154 2473 y Fw(2)173 2490 y FC(,)e(b)q(eca)n(us)q(e)f (of)h(Kram)o(ers')d(law.)18 b(Th)o(e)11 b(f)q(act)g(t)n(h)o(a)o(t)f(t)n (hi)q(s)i(i)q(s)g(not)f(t)n(h)o(e)g(cas)q(e)h(illustra)o(t)o(e)q(s)f(t) n(h)o(e)g(n)o(ece)q(ss)q(it)o(y)g(of)59 2546 y(a)k(go)q(o)q(d)f(u)o(n)o (d)o(erst)o(an)o(din)o(g)h(of)f(dyn)o(amical)h(e\013ect)o(s)g(\(as)f(o) o(p)o(p)q(os)q(e)q(d)h(t)o(o)f(a)h(quas)q(i)q(st)o(a)o(t)o(ic)d(pict)o (ure\).)19 b(Al)o(t)n(h)o(ough)59 2603 y(t)n(h)o(e)f(p)q(ot)o(en)o(t)o (ial)f(barr)q(ier)h(i)q(s)g(sm)o(alle)q(st)h(b)q(et)o(w)o(een)d(t)n(h)o (e)j(t)o(im)o(e)q(s)e Fs(\000)1120 2573 y(p)p 1159 2573 V 1159 2603 a FB(a)1183 2610 y Fw(0)1221 2603 y FC(an)o(d)1311 2573 y Fs(p)p 1349 2573 V 30 x FB(a)1373 2610 y Fw(0)1393 2603 y FC(,)i(t)n(h)o(e)f(pa)o(t)n(hs)g(h)o(a)o(v)o(e)f(more)59 2659 y(o)o(p)o(p)q(ort)o(u)o(nit)o(ie)q(s)g(t)o(o)h(reac)o(h)f(t)n(h)o (e)h(saddle)h(d)o(ur)q(in)o(g)e(larger)h(t)o(im)o(e)f(in)o(t)o(erv)n (als.)28 b(Th)o(e)18 b(o)o(pt)o(im)o(al)f(t)o(im)o(e)g(in)o(t)o(erv)n (al)59 2716 y(t)o(ur)q(ns)11 b(ou)o(t)i(t)o(o)f(h)o(a)o(v)o(e)g(a)h (len)o(gt)n(h)g(of)f(t)n(h)o(e)h(ord)o(er)811 2683 y Fs(p)p 849 2683 28 2 v 33 x FB(\033)r FC(,)g(whic)o(h)f(corre)q(sp)q (on)o(ds)f(t)o(o)i(t)n(h)o(e)f(regim)o(e)g(wh)o(ere)g(di\013us)q(iv)o (e)59 2772 y(b)q(e)o(h)o(a)o(viour)h(prev)n(ails)h(o)o(v)o(er)g(t)n(h)o (e)g(in\015uence)i(of)e(t)n(h)o(e)h(dr)q(ift.)922 2936 y(13)p eop %%Page: 14 14 14 13 bop 59 164 a Fn(2.3)56 b(Asymm)n(etr)q(i)o(c)16 b(cas)q(e)59 250 y FC(W)l(e)22 b(cons)q(id)o(er)f(in)h(t)n(hi)q(s)f(su) n(bs)q(ect)o(ion)g(t)n(h)o(e)g(SDE)g(\(2.1\))e(in)j(t)n(h)o(e)f(cas)q (e)g(of)g FB(f)27 b FC(b)q(e)q(in)o(g)21 b(p)q(er)q(io)q(dic)g(in)i FB(t)e FC(an)o(d)59 306 y(admit)n(t)o(in)o(g)12 b(t)o(w)o(o)e(st)o(a)o (ble)h(equilibr)q(iu)o(m)j(branc)o(h)o(e)q(s,)d(bu)o(t)h(wit)n(h)o(ou)o (t)g(t)n(h)o(e)g(symm)o(etry)d(assu)o(mpt)o(ion.)18 b(Inst)o(ead,)59 363 y(w)o(e)d(w)o(an)o(t)g(eac)o(h)g(of)h(t)n(h)o(e)f(p)q(ot)o(en)o(t)o (ial)g(w)o(ells)h(t)o(o)g(ap)o(proac)o(h)d(t)n(h)o(e)j(saddle)g(once)g (in)g(ev)o(ery)f(t)o(im)o(e)h(p)q(er)q(io)q(d,)f(bu)o(t)59 419 y(a)o(t)f(di\013eren)o(t)h(t)o(im)o(e)q(s)f(for)h(t)n(h)o(e)f (left-h)o(an)o(d)g(an)o(d)h(t)n(h)o(e)g(r)q(igh)o(t-h)o(an)o(d)d(p)q (ot)o(en)o(t)o(ial)i(w)o(ell.)21 b(A)15 b(t)o(ypical)f(example)g(of)59 475 y(su)o(c)o(h)g(a)h(fu)o(nct)o(ion)f(i)q(s)336 578 y FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))j(=)i FB(x)d Fs(\000)g FB(x)628 559 y Fw(3)658 578 y FC(+)g FB(\025)p FC(\()p FB(t)p FC(\))89 b(wit)n(h)h FB(\025)p FC(\()p FB(t)p FC(\))11 b(=)i Fs(\000)p FC(\()p FB(\025)1265 585 y Fw(c)1292 578 y Fs(\000)e FB(a)1362 585 y Fw(0)1381 578 y FC(\))d(cos)f(2)p FB(\031)r(t:)161 b FC(\(2.30\))59 686 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y Fs(\017)23 b Fy(Perio)n(dicity:)c FB(f)5 b FC(\()p FB(x;)j(t)h FC(+)h(1\))i(=)g FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))14 b(for)g(all)i(\()p FB(x;)8 b(t)p FC(\))j Fs(2)i(M)p FC(;)90 1500 y Fs(\017)23 b Fy(Equilibrium)f(br)n(anches:)31 b FC(Th)o(ere)20 b(exi)q(st)g(con)o (t)o(in)n(uous)g(fu)o(nct)o(ions)g FB(x)1303 1484 y FA(?)1303 1512 y Fx(\000)1355 1500 y FB(<)j(x)1439 1484 y FA(?)1439 1513 y Fw(0)1482 1500 y FB(<)g(x)1566 1484 y FA(?)1566 1512 y Fw(+)1617 1500 y FC(f)q(rom)c Fl(R)k FC(t)o(o)136 1557 y([)p Fs(\000)p FB(d;)8 b(d)t FC(])14 b(wit)n(h)i(t)n(h)o(e)f(pro) o(p)q(ert)o(y)f(t)n(h)o(a)o(t)g FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))13 b(=)i(0)g(in)i Fs(M)e FC(if)i(an)o(d)e(only)h(if)g FB(x)f FC(=)f FB(x)1478 1540 y FA(?)1478 1568 y Fx(\006)1508 1557 y FC(\()p FB(t)p FC(\))h(or)g FB(x)g FC(=)f FB(x)1747 1540 y FA(?)1747 1569 y Fw(0)1767 1557 y FC(\()p FB(t)p FC(\);)136 1613 y(t)n(h)o(e)i(zero)q(e)q(s)e(of)h FB(f)21 b FC(sh)o(ould)16 b(b)q(e)f(i)q(sola)o(t)o(e)q(d)f(in)i(t)n(h)o(e)g 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Fk(,)f(w)o(e)g(w)o(an)o(t)g(t)o(o)h(a)o(v)o(oid)h(t)n(h)o(a)o(t)f Fi(f)t Fk(\()p Fi(x;)5 b(t)p Fk(\))12 b(ap)o(proac)o(h)o(e)q(s)i(zero)f (els)q(ewh)o(ere)h(bu)o(t)59 2757 y(n)o(ear)g(t)n(h)o(e)f(t)n(hree)h (equilibr)q(ium)i(branc)o(h)o(e)q(s,)d(ev)o(en)h(wh)o(en)f Fi(a)884 2761 y Fz(0)915 2757 y Fk(b)q(ecom)o(e)q(s)g(sm)o(all.)922 2936 y FC(14)p eop %%Page: 15 15 15 14 bop 136 164 a FB(t)152 147 y FA(?)152 176 y Fw(0)193 164 y FC(=)21 b Fs(O)q FC(\()p FB(a)328 171 y Fw(0)347 164 y FC(\).)34 b(W)l(e)20 b(can)f(assu)o(m)o(e)g(t)n(h)o(a)o(t)f(for)g (a)i(su\016cien)o(t)n(ly)f(sm)o(all)g(const)o(an)o(t)e FB(T)27 b(>)21 b FC(0,)e(t)n(h)o(e)h(t)n(hree)136 220 y(equilibr)q(iu)o(m)c(branc)o(h)o(e)q(s)e(an)o(d)g(t)n(h)o(e)g(lin)o (ear)q(iza)o(t)o(ion)h(of)f FB(f)20 b FC(arou)o(n)o(d)14 b(t)n(h)o(em)g(sa)o(t)o(i)q(sfy)164 354 y FB(x)190 335 y FA(?)190 365 y Fw(+)220 354 y FC(\()p FB(t)p FC(\))9 b Fs(\000)i FB(x)353 361 y Fw(c)383 354 y Fs(\020)431 276 y Fq(\()503 293 y Fs(p)p 541 293 44 2 v 29 x 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Fs(\000)p FB(f)5 b FC(\()p Fs(\000)p FB(x;)j(t)p FC(\))15 b(for)e(all)j(\()p FB(x;)8 b(t)p FC(\).)130 1243 y(W)l(e)15 b(st)o(art)e(b)o(y)h(cons)q(id)o(er)q(in)o (g)h(t)n(h)o(e)f(d)o(et)o(ermini)q(st)o(ic)g(equa)o(t)o(ion)754 1368 y FB(")780 1337 y FC(d)q FB(x)832 1321 y Fw(d)o(et)832 1348 y FA(t)p 780 1357 102 2 v 811 1399 a FC(d)p FB(t)900 1368 y FC(=)e FB(f)5 b FC(\()p FB(x)1018 1349 y Fw(d)o(et)1018 1379 y FA(t)1069 1368 y FB(;)j(t)p FC(\))p FB(:)579 b FC(\(2.35\))59 1476 y(As)21 b(in)h(t)n(h)o(e)e(symm)o(etr)q(ic)f(cas)q (e,)j(it)f(i)q(s)g(su\016cien)o(t)f(t)o(o)h(cons)q(id)o(er)g(t)n(h)o(e) f(dyn)o(amics)h(in)g(t)n(h)o(e)g(t)o(im)o(e)f(in)o(t)o(erv)n(al)59 1532 y([)p Fs(\000)p FB(T)t(;)8 b(T)e FC(],)14 b(wit)n(h)h(an)h(init)o (ial)h(con)o(dit)o(ion)e(sa)o(t)o(i)q(sfyin)o(g)f FB(x)948 1516 y Fw(d)o(et)948 1546 y Fx(\000)p FA(T)1013 1532 y Fs(\000)e FB(x)1086 1516 y FA(?)1086 1544 y Fw(+)1115 1532 y FC(\()p Fs(\000)p FB(T)6 b FC(\))14 b Fs(\020)g FB(")p FC(.)23 b(Th)o(e)15 b(s)q(it)o(ua)o(t)o(ion)f(in)i(t)n(h)o(e)g (t)o(im)o(e)59 1589 y(in)o(t)o(erv)n(al)d([)p FB(t)249 1596 y Fw(c)277 1589 y Fs(\000)e FB(T)t(;)d(t)391 1596 y Fw(c)417 1589 y FC(+)j FB(T)6 b FC(])14 b(can)h(b)q(e)f(d)o(e)q(scr)q (ib)q(e)q(d)h(in)g(exact)n(ly)g(t)n(h)o(e)f(sam)o(e)g(w)o(ay)l(.)59 1679 y Fo(Th)o(eorem)h(2.5)i(\(Det)n(ermini)q(st)o(ic)g(cas)q(e\).)23 b Fy(The)18 b(solution)h FB(x)1144 1662 y Fw(d)o(et)1144 1690 y FA(t)1213 1679 y Fy(and)f(the)g(curve)g FB(x)1529 1662 y FA(?)1529 1690 y Fw(+)1558 1679 y FC(\()p FB(t)p FC(\))g Fy(cr)n(oss)g(onc)n(e)59 1735 y(and)e(only)g(onc)n(e)g(during)f (the)h(time)f(interval)g FC([)p Fs(\000)p FB(T)t(;)8 b(T)e FC(])p Fy(.)18 b(This)f(cr)n(ossing)e(o)n(c)n(curs)g(at)h(a)g (time)1615 1727 y FC(~)1615 1735 y FB(t)g Fy(satisfying)60 1784 y FC(~)59 1792 y FB(t)11 b Fs(\000)f FB(t)147 1775 y FA(?)147 1803 y Fw(+)190 1792 y Fs(\020)j FC(\()p FB("=)300 1763 y Fs(p)p 337 1763 44 2 v 337 1792 a FB(a)361 1799 y Fw(0)384 1792 y FC(\))d Fs(^)452 1759 y(p)p 490 1759 22 2 v 33 x FB(")q Fy(.)20 b(Ther)n(e)15 b(exists)h(a)g(c)n(onstant)g FB(c)1036 1799 y Fw(0)1068 1792 y FB(>)d FC(0)j Fy(such)g(that)411 1975 y FB(x)437 1957 y Fw(d)o(et)437 1987 y FA(t)498 1975 y Fs(\000)10 b FB(x)569 1957 y FA(?)569 1987 y Fw(+)598 1975 y FC(\()p FB(t)p FC(\))j Fs(\020)711 1855 y Fq(8)711 1896 y(>)711 1910 y(>)711 1923 y(<)711 2005 y(>)711 2019 y(>)711 2032 y(:)802 1879 y FB(")p 791 1900 42 2 v 791 1941 a Fs(j)p FB(t)p Fs(j)884 1910 y Fy(for)j Fs(\000)p FB(T)i Fr(6)13 b FB(t)h Fr(6)e Fs(\000)p FB(c)1216 1917 y Fw(0)1236 1910 y FC(\()1254 1881 y Fs(p)p 1292 1881 44 2 v 29 x FB(a)1316 1917 y Fw(0)1345 1910 y Fs(_)1386 1877 y(p)p 1424 1877 22 2 v 33 x FB(")t FC(\))751 2026 y Fs(\000)802 1996 y FB(")p 791 2016 42 2 v 791 2058 a Fs(j)p FB(t)p Fs(j)884 2026 y Fy(for)k FB(c)976 2033 y Fw(0)995 2026 y FC(\()1013 1997 y Fs(p)p 1050 1997 44 2 v 1050 2026 a FB(a)1074 2033 y Fw(0)1104 2026 y Fs(_)1145 1994 y(p)p 1183 1994 22 2 v 32 x FB(")t FC(\))c Fr(6)h FB(t)g Fr(6)g FB(T)6 b Fy(,)1716 1975 y FC(\(2.36\))59 2161 y Fy(and)16 b(thus)h FB(x)269 2145 y Fw(d)o(et)269 2172 y FA(t)329 2161 y Fs(\000)11 b FB(x)401 2168 y Fw(c)431 2161 y Fs(\020)i(j)p FB(t)p Fs(j)j Fy(in)g(these)g(time)f(intervals.)20 b(F)m(or)c Fs(j)p FB(t)p Fs(j)d Fr(6)g FB(c)1216 2168 y Fw(0)1235 2161 y FC(\()1253 2132 y Fs(p)p 1290 2132 44 2 v 1290 2161 a FB(a)1314 2168 y Fw(0)1344 2161 y Fs(_)1385 2128 y(p)p 1423 2128 22 2 v 33 x FB(")t FC(\))p Fy(,)647 2300 y FB(x)673 2282 y Fw(d)o(et)673 2312 y FA(t)733 2300 y Fs(\000)e FB(x)805 2307 y Fw(c)835 2300 y Fs(\020)883 2223 y Fq(\()920 2239 y Fs(p)p 958 2239 44 2 v 30 x FB(a)982 2276 y Fw(0)1047 2269 y Fy(if)16 b FB(a)1115 2276 y Fw(0)1147 2269 y Fr(>)d FB(")920 2304 y Fs(p)p 958 2304 22 2 v 32 x FB(")68 b Fy(if)16 b FB(a)1115 2343 y Fw(0)1147 2336 y Fr(6)d FB(")p Fy(.)1716 2300 y FC(\(2.37\))59 2442 y Fy(The)j(line)n(arization)g(of)g FB(f)21 b Fy(at)16 b FB(x)586 2426 y Fw(d)o(et)586 2453 y FA(t)652 2442 y Fy(satis\014es)552 2540 y FC(\026)-24 b FB(a)p FC(\()p FB(t)p FC(\))635 2539 y(:)644 2540 y(=)8 b FB(@)711 2547 y FA(x)733 2540 y FB(f)d FC(\()p FB(x)804 2521 y Fw(d)o(et)804 2551 y FA(t)854 2540 y FB(;)j(t)p FC(\))k Fs(\020)h(\000)p FC(\()p Fs(j)p FB(t)p Fs(j)c(_)1114 2508 y(p)p 1151 2508 44 2 v 1151 2540 a FB(a)1175 2547 y Fw(0)1205 2540 y Fs(_)1246 2505 y(p)p 1284 2505 22 2 v 35 x FB(")t FC(\))p FB(:)376 b FC(\(2.38\))59 2642 y Fy(Mor)n(e)n(over,)13 b FC(\(2.35\))f Fy(admits)j(a)g(p)n(articular)f (solution)j Fq(b)-26 b FB(x)985 2625 y Fw(d)o(et)981 2653 y FA(t)1050 2642 y Fy(tr)n(acking)13 b(the)i FC(u)o(nst)o(a)o(ble) e Fy(e)n(quilibrium)h(br)n(anch)59 2698 y FB(x)85 2682 y FA(?)85 2710 y Fw(0)105 2698 y FC(\()p FB(t)p FC(\))p Fy(.)21 b(It)16 b(satis\014es)h(analo)n(gous)g(r)n(elations,)g(namely,) h Fq(b)-27 b FB(x)1012 2682 y Fw(d)o(et)1008 2709 y FA(t)1079 2698 y Fy(and)16 b FB(x)1192 2682 y FA(?)1192 2710 y Fw(0)1212 2698 y FC(\()p FB(t)p FC(\))g Fy(cr)n(oss)g(onc)n(e)g(at)g(a) h(time)1694 2690 y FC(^)1693 2698 y FB(t)g Fy(satis-)59 2755 y(fying)173 2746 y FC(^)173 2755 y FB(t)12 b Fs(\000)g FB(t)264 2738 y FA(?)264 2767 y Fw(0)301 2755 y Fs(\020)17 b(\000)p FC(\()407 2746 y(~)406 2755 y FB(t)12 b Fs(\000)g FB(t)497 2738 y FA(?)497 2766 y Fw(+)527 2755 y FC(\))p Fy(,)18 b(and)24 b FC(\(2.36\))m Fy(,)18 b FC(\(2.37\))e Fy(and)23 b FC(\(2.38\))15 b Fy(hold)k(for)h Fq(b)-27 b FB(x)1383 2738 y Fw(d)o(et)1452 2755 y Fy(and)18 b FB(x)1567 2738 y FA(?)1567 2767 y Fw(0)1587 2755 y FC(\()p FB(t)p FC(\))p Fy(,)g(but)g(with)59 2811 y(opp)n(osite)d(signs.)922 2936 y FC(15)p eop %%Page: 16 16 16 15 bop 130 164 a FC(Th)o(e)19 b(pro)q(of)g(i)q(s)h(s)q(imilar)g(t)o (o)f(t)n(h)o(e)h(pro)q(of)f(of)g(Th)o(eorem)f(2.1,)i(an)o(d)f(w)o(e)h (comm)o(en)o(t)e(on)h(a)h(few)g(minor)59 220 y(di\013erence)q(s)13 b(in)g(Su)n(bs)q(ect)o(ion)f(4.1.)17 b(Not)o(e)11 b(t)n(h)o(a)o(t)g (\(2.37\))f(implie)q(s)j(t)n(h)o(a)o(t)e FB(x)1229 204 y Fw(d)o(et)1229 232 y FA(t)1292 220 y FC(n)o(ev)o(er)g(ap)o(proac)o(h) 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Fc(Figure)32 b(5.)40 b Fb(Nils)28 b(Berglund)f(and)g(Barbara)e(Gen)n (tz)268 4843 y(Sto)r(c)n(hastic)i(resonance)e(in)j(a)f(double-w)n(ell)g (p)r(oten)n(tial)1867 5871 y Fa(1)p eop end userdict /end-hook known{end-hook}if %%EndDocument endTexFig 134 1388 a Fv(Figure)15 b(5.)21 b FG(A)15 b(t)o(ypical)e(pa)o(t)n(h)i FF(x)650 1394 y Fp(t)678 1388 y FG(of)f(t)n(h)o(e)h(SDE)g(\(2.1\))e(in) h(t)n(h)o(e)h(asymm)o(etr)q(ic)d(cas)q(e,)j(sh)o(o)o(wn)f(n)o(ear)i FF(t)c FG(=)h(0,)134 1438 y(wh)o(ere)j(t)n(h)o(e)h(r)q(igh)o(t-h)o(an)o (d)d(w)o(ell)h(a)o(t)g FF(x)690 1423 y Fp(?)690 1448 y Fu(+)733 1438 y FG(ap)o(proac)o(h)o(e)q(s)h(t)n(h)o(e)g(saddle)g(a)o (t)f FF(x)1221 1423 y Fp(?)1221 1448 y Fu(0)1240 1438 y FG(.)22 b(W)m(e)15 b(sh)o(o)o(w)h(again)e(a)h(s)q(it)o(ua)o(t)o(ion) 134 1487 y(wh)o(ere)g(t)n(h)o(e)g(noi)q(s)q(e)f(in)o(t)o(ens)q(it)o(y)f (i)q(s)g(large)h(enough)h(t)o(o)f(allo)o(w)f(a)g(trans)q(it)o(ion.)18 b(Th)o(e)d(d)o(et)o(ermini)q(st)o(ic)e(solu)o(t)o(ion)134 1537 y FF(x)158 1522 y Fu(d)o(et)158 1548 y Fp(t)218 1537 y FG(st)o(art)o(in)o(g)h(n)o(ear)g(t)n(h)o(e)g(r)q(igh)o(t-h)o(an) o(d)e(p)q(ot)o(en)o(t)o(ial)g(w)o(ell)h(trac)o(ks)h FF(x)1133 1522 y Fp(?)1133 1548 y Fu(+)1160 1537 y FG(\()p FF(t)p FG(\))g(a)o(t)f(a)g(di)q(st)o(ance)g(a)o(t)g(most)g(of)g(ord)o(er)134 1587 y(\()p FF("=)190 1560 y Ft(p)p 225 1560 41 2 v 27 x FF(a)247 1593 y Fu(0)265 1587 y FG(\))d Ft(^)327 1557 y(p)p 362 1557 20 2 v 30 x FF(")p FG(,)15 b(an)o(d)f(n)o(ev)o(er)i(ap)o (proac)o(h)o(e)q(s)f(t)n(h)o(e)h(saddle)e(a)o(t)h FF(x)1084 1572 y Fp(?)1084 1597 y Fu(0)1117 1587 y FG(clos)q(er)h(t)n(h)o(an)1330 1560 y Ft(p)p 1365 1560 41 2 v 27 x FF(a)1387 1593 y Fu(0)1414 1587 y Ft(_)1451 1557 y(p)p 1486 1557 20 2 v 30 x FF(")p FG(.)21 b(Th)o(e)15 b(pa)o(t)n(h)g FF(x)1742 1593 y Fp(t)134 1637 y FG(i)q(s)f(lik)o(ely)f(t)o(o)h(st)o(ay)h(in)f(t) n(h)o(e)h(s)q(et)g Ft(B)q FG(\()p FF(h)p FG(\))g(up)g(t)o(o)g(t)o(im)o (e)d Ft(\000)p FF(h)969 1622 y Fu(2)p Fp(=)p Fu(3)1036 1637 y FG(wh)o(en)j FF(h)d Ft(\035)g FF(\033)q FG(.)20 b(Bet)o(w)o(een)c(t)o(im)o(e)q(s)d Ft(\000)p FF(h)1623 1622 y Fu(2)p Fp(=)p Fu(3)1690 1637 y FG(an)o(d)134 1687 y FF(h)158 1672 y Fu(2)p Fp(=)p Fu(3)210 1687 y FG(,)h(t)n(h)o(e)i(pa)o (t)n(h)e(i)q(s)g(lik)o(ely)g(t)o(o)h(reac)o(h)h(t)n(h)o(e)f(saddle.)20 b(Al)o(t)n(h)o(ough)15 b(it)f(m)o(ay)f(f)q(all)g(bac)o(k)i(in)o(t)o(o)f (t)n(h)o(e)h(r)q(igh)o(t-h)o(an)o(d)134 1737 y(p)q(ot)o(en)o(t)o(ial)f (w)o(ell,)h(it)f(i)q(s)h(lik)o(ely)f(t)o(o)i(\014n)o(ally)e(o)o(v)o (ercom)o(e)h(t)n(h)o(e)h(p)q(ot)o(en)o(t)o(ial)e(barr)q(ier)i(an)o(d)f (reac)o(h)h(a)f(lev)o(el)g FF(\016)1688 1743 y Fu(1)1723 1737 y FG(of)134 1786 y(ord)o(er)k(1)f(b)q(elo)o(w)f(t)n(h)o(e)i (saddle,)g(aft)o(er)g(whic)o(h)e(it)h(quic)o(kly)f(reac)o(h)o(e)q(s)k (a)c(lo)o(w)o(er)h(lev)o(el)g FF(\016)1458 1792 y Fu(0)1477 1786 y FG(.)31 b(Th)o(e)19 b(di)q(st)o(ance)134 1836 y(b)q(et)o(w)o(een)14 b FF(\016)311 1842 y Fu(1)344 1836 y FG(an)o(d)f FF(\016)441 1842 y Fu(0)473 1836 y FG(can)h(b)q(e)g(m)n (u)o(c)o(h)e(larger)i(t)n(h)o(an)f(in)h(t)n(hi)q(s)f(pict)o(ure.)18 b(Fin)o(ally)m(,)11 b(t)n(h)o(e)k(pa)o(t)n(h)e(will)f(trac)o(k)i(t)n(h) o(e)134 1886 y(d)o(et)o(ermini)q(st)o(ic)f(solu)o(t)o(ion)h(st)o(art)o (in)o(g)g(in)g FF(x)e FG(=)h FF(\016)837 1892 y Fu(0)856 1886 y FG(,)h(whic)o(h)g(ap)o(proac)o(h)o(e)q(s)h(a)f(n)o(e)q(igh)o(b)q (ourh)o(o)q(o)q(d)g(of)g(ord)o(er)i FF(")f FG(of)134 1936 y(t)n(h)o(e)f(left-h)o(an)o(d)f(p)q(ot)o(en)o(t)o(ial)g(w)o(ell)g (a)o(t)h FF(x)707 1921 y Fp(?)707 1946 y Ff(\000)735 1936 y FG(.)130 2074 y FC(Here)9 b FB(\016)251 2081 y Fw(0)282 2074 y FC(an)o(d)g FB(\016)383 2081 y Fw(1)413 2074 y FC(are)h(alw)o(ays)e(of)i(ord)o(er)f(1)h(\(in)g(f)q(act,)f(w)o (e)h(m)n(ust)f(h)o(a)o(v)o(e)g FB(\016)1265 2081 y Fw(0)1298 2074 y FB(>)k(x)1372 2058 y FA(?)1372 2086 y Fx(\000)1401 2074 y FC(\()p FB(t)p FC(\))d(for)f(all)i FB(t)i Fs(2)g FC([)p Fs(\000)p FB(T)t(;)8 b(T)e FC(]\),)59 2131 y(an)o(d)17 b(w)o(e)g(t)n(hink)h(of)f FB(\016)411 2138 y Fw(0)448 2131 y FC(as)g(b)q(e)q(in)o(g)h(in)g(t)n(h)o(e)g(bas)q(in)f(of)g(a)o(t) n(tract)o(ion)d(of)j FB(x)1226 2114 y FA(?)1226 2142 y Fx(\000)1256 2131 y FC(.)27 b(Our)18 b(h)o(yp)q(ot)n(h)o(e)q(s)q(e)q (s)e(imply)h(t)n(h)o(a)o(t)59 2187 y(a)e FB(\016)117 2194 y Fw(2)153 2187 y FC(of)g(ord)o(er)g(on)o(e)g(sa)o(t)o(i)q(sfyin)o (g)g(\(2.46\))e(alw)o(ays)h(exi)q(st)o(s,)h(bu)o(t)h FB(\016)1123 2194 y Fw(2)1159 2187 y FC(m)o(ay)f(b)q(e)g(c)o(h)o(os)q (en)g(arbitrar)q(ily)f(large)h(in)59 2244 y(part)o(icular)e(cas)q(e)q (s)h(su)o(c)o(h)h(as)f FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))j(=)i FB(x)d Fs(\000)g FB(x)828 2227 y Fw(3)858 2244 y FC(+)g FB(\025)p FC(\()p FB(t)p FC(\).)130 2300 y(Th)o(e)k(non-trans)q(it)o (ion)e(proba)o(bilit)o(y)i(can)h(b)q(e)f(e)q(st)o(im)o(a)o(t)o(e)q(d)f (b)o(y)i(di)q(st)o(in)o(gui)q(shin)o(g)g(t)n(hree)f(cas)q(e)q(s:)90 2369 y Fs(\017)23 b FC(Eit)n(h)o(er)15 b FB(x)298 2376 y FA(t)313 2369 y FC(,)f(st)o(art)o(in)o(g)f(in)j FB(x)584 2376 y Fw(0)617 2369 y FB(>)d(x)691 2376 y Fw(c)724 2369 y FC(a)o(t)h(som)o(e)g FB(t)906 2376 y Fw(0)939 2369 y FB(<)f FC(0,)i(n)o(ev)o(er)f(reac)o(h)o(e)q(s)g FB(\016)1334 2376 y Fw(1)1354 2369 y FC(.)20 b(Th)o(e)15 b(proba)o(bilit)o(y)f(of)g (t)n(hi)q(s)136 2425 y(ev)o(en)o(t)f(can)g(b)q(e)h(sh)o(o)o(wn)e(t)o(o) h(b)q(e)h(sm)o(all)f(in)i(a)e(s)q(imilar)g(w)o(ay)g(as)g(in)h(Th)o (eorem)e(2.3,)g(t)n(h)o(e)h(m)o(ain)g(di\013erence)136 2481 y(b)q(e)q(in)o(g)j(t)n(h)o(a)o(t)d(d)o(ue)h(t)o(o)h(t)n(h)o(e)f (asymm)o(etry)l(,)d(w)o(e)j(can)h(do)f(b)q(et)n(t)o(er)g(t)n(h)o(an)f (e)q(st)o(im)o(a)o(t)o(in)o(g)h(t)n(h)o(e)g(proba)o(bilit)o(y)g(not)136 2538 y(t)o(o)h(reac)o(h)f(t)n(h)o(e)h(saddle.)90 2594 y Fs(\017)23 b FC(If)16 b FB(x)208 2601 y FA(t)238 2594 y FC(reac)o(h)o(e)q(s)f FB(\016)416 2601 y Fw(1)436 2594 y FC(,)g(on)o(e)g(can)g(e)q(st)o(im)o(a)o(t)o(e)e(in)j(a)g(v)o(ery)e(s) q(imple)h(w)o(ay)f(t)n(h)o(e)h(proba)o(bilit)o(y)f(not)h(t)o(o)g(reac)o (h)g FB(\016)1811 2601 y Fw(0)136 2651 y FC(as)g(w)o(ell,)g(us)q(in)o 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b(pro)q(of)g(i)q(s)g(giv)o(en)h(in)h(Su)n (bs)q(ect)o(ion)e(4.2.)21 b(Th)o(e)16 b(t)n(hree)f(t)o(erms)f(on)i(t)n (h)o(e)f(r)q(igh)o(t-h)o(an)o(d)f(s)q(id)o(e)i(of)j(\(2.47\))59 950 y(b)q(ou)o(n)o(d,)i(re)q(sp)q(ect)o(iv)o(ely)l(,)f(t)n(h)o(e)g (proba)o(bilit)o(y)f(t)n(h)o(a)o(t)f FB(x)913 957 y FA(t)949 950 y FC(e)q(scap)q(e)q(s)h(t)n(hrough)f(t)n(h)o(e)i(up)o(p)q(er)f(b)q (ou)o(n)o(d)o(ary)h FB(x)1732 934 y Fw(d)o(et)1732 961 y FA(s)1796 950 y FC(+)59 1007 y FB(h)85 968 y Fq(p)p 131 968 80 2 v 131 1007 a FB(\020)s FC(\()p FB(s)p FC(\))o(,)13 b(t)n(h)o(e)g(proba)o(bilit)o(y)e(t)n(h)o(a)o(t)h FB(x)653 1014 y FA(t)681 1007 y FC(reac)o(h)o(e)q(s)g(n)o(e)q(it)n(h)o(er)g(t)n (h)o(e)h(up)o(p)q(er)f(b)q(ou)o(n)o(d)o(ary)g(nor)g FB(\016)1477 1014 y Fw(1)1497 1007 y FC(,)h(an)o(d)f(t)n(h)o(e)g(proba)o(b-)59 1063 y(ilit)o(y)k(t)n(h)o(a)o(t)d FB(x)271 1070 y FA(t)301 1063 y FC(do)q(e)q(s)i(not)f(reac)o(h)g FB(\016)623 1070 y Fw(0)658 1063 y FC(wh)o(en)h(st)o(art)o(in)o(g)e(on)i FB(\016)1023 1070 y Fw(1)1058 1063 y FC(\(Fig.)d(5\).)19 b(Th)o(e)14 b(cru)o(cial)h(t)o(erm)f(i)q(s)h(t)n(h)o(e)g(s)q(econ)o(d) 59 1120 y(on)o(e.)130 1176 y(Let)h(us)h(no)o(w)e(di)q(scuss)i(t)n(h)o (e)f(o)o(pt)o(im)o(al)g(c)o(h)o(oice)g(of)g(param)o(et)o(ers.)22 b(If)17 b(w)o(e)f(c)o(h)o(o)q(os)q(e)g FB(t)1478 1183 y Fw(1)1514 1176 y FC(=)g Fs(\000)1605 1158 y Fw(1)p 1605 1165 18 2 v 1605 1192 a(2)1628 1176 y FB(c)1648 1183 y Fw(1)1667 1176 y FB(\033)1695 1160 y Fw(2)p FA(=)p Fw(3)1750 1176 y FC(,)g(w)o(e)59 1238 y(can)f(t)o(ak)o(e)e FB(h)g Fs(\020)327 1226 y FC(~)325 1238 y FB(\016)345 1245 y Fw(2)365 1238 y FB(\033)393 1222 y Fw(1)p FA(=)p Fw(3)447 1238 y FC(,)i(wh)o(ere)606 1226 y(~)604 1238 y FB(\016)624 1245 y Fw(2)656 1238 y FC(=)e FB(\016)724 1245 y Fw(2)754 1238 y Fs(\000)d FB(x)825 1245 y Fw(c)843 1238 y FC(,)15 b(an)o(d)f(w)o(e)g(get)g(t)n(h)o(e)h(e)q(st)o(im)o(a)o (t)o(e)191 1362 y Fl(P)224 1343 y FA(t)237 1348 y Fz(0)251 1343 y FA(;x)281 1348 y Fz(0)300 1311 y Fq(n)331 1362 y FB(x)357 1369 y 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y(n)o(en)o(t)f(i)q(s)h(in)o(d)o(ep)q(en)o(d)o(en)o(t)f(of)h FB(a)531 2246 y Fw(0)550 2239 y FC(.)130 2296 y(If)d(t)n(h)o(e)f(fu)o (nct)o(ion)h FB(f)17 b FC(i)q(s)12 b(su)o(c)o(h)f(t)n(h)o(a)o(t)f FB(\016)704 2303 y Fw(2)736 2296 y FC(can)i(b)q(e)g(c)o(h)o(os)q(en)f (arbitrar)q(ily)f(large,)h(t)n(h)o(e)h(s)q(econ)o(d)f(t)o(erm)f(in)j (\(2.49\))59 2352 y(can)18 b(b)q(e)h(remo)o(v)o(e)q(d)d(wit)n(h)o(ou)o (t)i(c)o(h)o(an)o(gin)o(g)g(t)n(h)o(e)g(\014rst)g(on)o(e)g(\(up)g(t)o (o)g(logar)q(it)n(hmic)f(correct)o(ions\))f(b)o(y)i(t)o(akin)o(g)61 2396 y(~)59 2408 y FB(\016)81 2392 y Fw(2)79 2421 y(2)116 2408 y FC(=)d FB(\033)194 2392 y Fw(8)p FA(=)p Fw(3)249 2408 y FB(=")h FC(for)g(inst)o(ance.)23 b(If)17 b FB(\033)g(>)e(a)740 2385 y Fw(1)p FA(=)p Fw(4)740 2422 y(0)805 2408 y Fs(_)c FB(")867 2392 y Fw(1)p FA(=)p Fw(4)922 2408 y FC(,)17 b(t)n(h)o(e)f(pa)o(t)n(hs)f(m)o(ay)g(b)q(ecom)o(e)g(extrem)o(ely)g(d)o (elo)q(cali)q(s)q(e)q(d)59 2465 y(in)h(t)n(h)o(e)e(trans)q(it)o(ion)f 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FA(?)1220 2697 y FC(an)o(d)1307 2688 y(~)1306 2697 y FB(t)1322 2704 y Fw(2)1356 2697 y FB(>)g(T)6 b FC(.)20 b(Th)o(erefore,)13 b(t)n(h)o(ere)i(i)q(s)59 2753 y(a)o(t)g(most)f(on)o(e)h(cross)q(in)o(g.)21 b(W)l(e)16 b(sh)o(all)g(s)q(ee)f(b)q(elo)o(w)g(t)n(h)o(a)o(t)f FB(x)1003 2760 y FA(t)1033 2753 y FC(an)o(d)h FB(x)1146 2737 y FA(?)1166 2753 y FC(\()p FB(t)p FC(\))g(act)o(ually)g(cross)g(an)o(d)g (w)o(e)g(will)h(also)59 2810 y(d)o(et)o(ermin)o(e)e(t)n(h)o(e)g(ord)o (er)g(of)h(t)n(h)o(a)o(t)e(t)o(im)o(e)703 2801 y(~)702 2810 y FB(t)p FC(.)922 2936 y(19)p eop %%Page: 20 20 20 19 bop 130 164 a FC(W)l(e)15 b(cons)q(id)o(er)g(no)o(w)e(t)n(h)o(e)i (di\013erence)g FB(y)774 171 y FA(t)802 164 y FC(=)e FB(x)876 171 y FA(t)901 164 y Fs(\000)d FB(x)972 147 y FA(?)992 164 y FC(\()p FB(t)p FC(\).)19 b(It)14 b(sa)o(t)o(i)q (s\014e)q(s)g(t)n(h)o(e)g(equa)o(t)o(ion)631 288 y FB(")657 257 y FC(d)q FB(y)p 657 277 49 2 v 661 319 a FC(d)p FB(t)724 288 y FC(=)f FB(a)796 269 y FA(?)815 288 y FC(\()p FB(t)p 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Fy(,)i(and)h(de\014ne)21 b FC(~)-28 b FB(\013)p FC(\()p FB(t;)8 b(s)p FC(\))k(=)1116 1258 y Fq(R)1146 1271 y FA(t)1137 1310 y(s)1169 1295 y FC(~)-24 b FB(a)p FC(\()p FB(u)p FC(\))8 b(d)o FB(u)p Fy(.)20 b(Then)185 1501 y FB(\037)213 1508 y Fw(0)241 1501 y FC(e)264 1483 y Fw(~)-21 b FA(\013)o Fw(\()p FA(t;)p Fx(\000)p FA(T)5 b Fw(\))p FA(=")430 1501 y FC(+)470 1471 y(1)p 470 1491 23 2 v 471 1532 a FB(")505 1439 y Fq(Z)550 1453 y FA(t)530 1543 y Fx(\000)p FA(T)593 1501 y FC(e)616 1483 y Fw(~)-21 b FA(\013)o Fw(\()p FA(t;s)p Fw(\))p FA(=")745 1501 y FC(d)p FB(s)13 b Fs(\020)852 1367 y Fq(8)852 1408 y(>)852 1422 y(>)852 1435 y(>)852 1449 y(<)852 1531 y(>)852 1544 y(>)852 1558 y(>)852 1572 y(:)964 1390 y FC(1)p 897 1410 155 2 v 897 1453 a FB(\014)g Fs(_)d FB(")997 1440 y Fw(2)p FA(=)p Fw(3)1103 1420 y Fy(for)16 b Fs(j)p FB(t)p Fs(j)c Fr(6)1277 1386 y Fs(p)p 1315 1386 29 2 v 34 x FB(\014)g Fs(_)f FB(")1415 1404 y Fw(1)p FA(=)p Fw(3)904 1531 y FC(1)p 897 1551 37 2 v 897 1593 a FB(t)913 1580 y Fw(2)1103 1561 y Fy(for)1175 1527 y Fs(p)p 1213 1527 29 2 v 34 x FB(\014)h Fs(_)e FB(")1312 1545 y Fw(1)p FA(=)p Fw(3)1380 1561 y Fr(6)j Fs(j)p FB(t)p Fs(j)g Fr(6)g FB(T)6 b Fy(.)1716 1501 y FC(\(3.11\))59 1701 y Fd(Pr)o(oof:)23 b FC(T)l(o)13 b(pro)o(v)o(e)g(t)n (h)o(e)g(lemm)o(a,)g(w)o(e)g(t)o(ak)o(e)f(adv)n(an)o(t)o(ag)o(e)f(of)i (t)n(h)o(e)g(f)q(act)g(t)n(h)o(a)o(t)f(t)n(h)o(e)h(expre)q(ss)q(ion)g (on)h(t)n(h)o(e)f(left-)59 1757 y(h)o(an)o(d)i(s)q(id)o(e)i(of)i (\(3.11\))13 b(i)q(s)j(t)n(h)o(e)g(solu)o(t)o(ion)f(of)h(an)f(ordin)o (ary)g(di\013eren)o(t)o(ial)h(equa)o(t)o(ion.)22 b(By)16 b(t)n(h)o(e)g(s)q(emi-group)59 1814 y(pro)o(p)q(ert)o(y)l(,)f(w)o(e)h (m)o(ay)g(cons)q(id)o(er)g(s)q(epara)o(t)o(ely)f(t)n(h)o(e)h(follo)o (win)o(g)g(regim)o(e)q(s:)23 b(F)l(or)17 b FB(\014)h Fr(>)e FB(")1464 1797 y Fw(2)p FA(=)p Fw(3)1520 1814 y FC(,)h(w)o(e)f(di)q(st)o(in)o(gui)q(sh)59 1870 y(t)n(h)o(e)g(cas)q(e) q(s)f FB(t)g Fs(2)g FC([)p Fs(\000)p FB(T)t(;)8 b Fs(\000)p FB(T)d(=)p FC(2],)14 b([)p Fs(\000)p FB(T)5 b(=)p FC(2)p FB(;)j Fs(\000)762 1836 y(p)p 798 1836 V 798 1870 a FB(\014)e FC(],)15 b([)p Fs(\000)919 1836 y(p)p 957 1836 V 34 x FB(\014)r(;)8 b Fs(\000)p FB("=a)1109 1877 y Fw(0)1128 1870 y FC(],)16 b([)p Fs(\000)p FB("=a)1286 1877 y Fw(0)1305 1870 y FB(;)8 b("=a)1394 1877 y Fw(0)1413 1870 y FC(],)16 b([)p FB("=a)1536 1877 y Fw(0)1555 1870 y FB(;)1576 1836 y Fs(p)p 1613 1836 V 1613 1870 a FB(\014)6 b FC(],)16 b([)1700 1836 y Fs(p)p 1737 1836 V 1737 1870 a FB(\014)r(;)8 b(T)e FC(])59 1927 y(an)o(d)18 b(for)f FB(\014)j(<)f(")342 1910 y Fw(2)p FA(=)p Fw(3)397 1927 y FC(,)g(w)o(e)f(d)o(eal)h(s)q (epara)o(t)o(ely)d(wit)n(h)i FB(t)h Fs(2)g FC([)p Fs(\000)p FB(T)t(;)8 b Fs(\000)p FB(T)d(=)p FC(2],)16 b([)p Fs(\000)p FB(T)5 b(=)p FC(2)p FB(;)j Fs(\000)p FB(")1459 1910 y Fw(1)p FA(=)p Fw(3)1512 1927 y FC(],)18 b([)p Fs(\000)p FB(")1625 1910 y Fw(1)p FA(=)p Fw(3)1680 1927 y FB(;)8 b Fs(\000)1736 1892 y(p)p 1774 1892 V 35 x FB(\014)e FC(],)59 1983 y([)p Fs(\000)107 1949 y(p)p 145 1949 V 34 x FB(\014)r(;)194 1949 y Fs(p)p 231 1949 V 231 1983 a FB(\014)g FC(],)21 b([)323 1949 y Fs(p)p 360 1949 V 360 1983 a FB(\014)r(;)8 b(")430 1967 y Fw(1)p FA(=)p Fw(3)485 1983 y FC(],)21 b([)p FB(")566 1967 y Fw(1)p FA(=)p Fw(3)620 1983 y FB(;)8 b(T)e FC(].)35 b(On)20 b(eac)o(h)h(of)e(t)n(h)o(e)q(s)q(e)h(t)o(im)o(e)g(in)o(t)o(erv)n(als)e (t)n(h)o(e)i(claim)o(e)q(d)h(b)q(e)o(h)o(a)o(viour)59 2040 y(follo)o(ws)13 b(eas)q(ily)j(b)o(y)e(elem)o(en)o(t)o(ary)f (calculus,)h(s)q(ee)h(also)f(t)n(h)o(e)h(s)q(imilar)f(re)q(sul)o(t)h ([BG,)f(Lemm)o(a)g(4.2].)p 1798 2040 2 31 v 1800 2011 28 2 v 1800 2040 V 1827 2040 2 31 v 59 2134 a Fo(Pro)o(p)q(o)q(s)q(it)o (ion)j(3.3.)23 b Fy(Ther)n(e)11 b(exists)i(a)g(c)n(onstant)f FB(c)916 2141 y Fw(0)948 2134 y FB(>)h FC(0)f Fy(such)h(that)g(the)f (solution)h(of)22 b FC(\(3.7\))11 b Fy(with)h(initial)59 2190 y(c)n(ondition)j FB(y)277 2197 y Fx(\000)p FA(T)345 2190 y Fs(\020)e FB(")k Fy(satis\014es)522 2301 y FB(y)544 2308 y FA(t)572 2301 y Fs(\020)632 2270 y FB(")p 625 2290 37 2 v 625 2332 a(t)641 2319 y Fw(2)758 2301 y Fy(for)f Fs(\000)p FB(T)j Fr(6)13 b FB(t)g Fr(6)g Fs(\000)p FB(c)1091 2308 y Fw(0)1110 2301 y FC(\()1128 2269 y Fs(p)p 1166 2269 44 2 v 32 x FB(a)1190 2308 y Fw(0)1220 2301 y Fs(_)d FB(")1281 2282 y Fw(1)p FA(=)p Fw(3)1336 2301 y FC(\))p Fy(.)348 b FC(\(3.12\))59 2419 y Fd(Pr)o(oof:)23 b FC(Let)17 b FB(c)340 2426 y Fw(0)377 2419 y Fr(>)g FC(1)g(an)o(d)g FB(c)578 2426 y Fw(1)614 2419 y FB(>)g(T)699 2403 y Fw(2)719 2419 y FB(y)741 2426 y Fx(\000)p FA(T)796 2419 y FB(=")h FC(b)q(e)f(const)o(an)o(t)o(s)e(t)o(o)i(b)q(e)g(c)o(h)o(os)q(en)g(la)o (t)o(er,)f(an)o(d)h(d)o(enot)o(e)g(b)o(y)g FB(\034)59 2476 y FC(t)n(h)o(e)c(\014rst)h(exit)f(t)o(im)o(e)g(of)h FB(y)484 2483 y FA(t)513 2476 y FC(f)q(rom)e(t)n(h)o(e)h(str)q(ip)g(0)f FB(<)h(y)902 2483 y FA(t)930 2476 y FB(<)g(c)998 2483 y Fw(1)1018 2476 y FB("=t)1078 2459 y Fw(2)1098 2476 y FC(.)19 b(Set)13 b FB(t)1222 2483 y Fw(0)1255 2476 y FC(=)g Fs(\000)p FB(c)1358 2483 y Fw(0)1377 2476 y FC(\()1395 2447 y Fs(p)p 1433 2447 V 29 x FB(a)1457 2483 y Fw(0)1485 2476 y Fs(_)8 b FB(")1544 2459 y Fw(1)p FA(=)p Fw(3)1599 2476 y FC(\).)19 b(Th)o(en,)13 b(for)59 2532 y Fs(\000)p FB(T)19 b Fr(6)13 b FB(t)g Fr(6)g FB(\034)i Fs(^)10 b FB(t)356 2539 y Fw(0)377 2532 y FC(,)k(w)o(e)g(get)g(f)q(rom) f(\(3.9\))g(an)o(d)h(\(3.5\))f(t)n(h)o(a)o(t)186 2629 y Fs(j)p FB(b)219 2612 y FA(?)238 2629 y FC(\()p FB(y)r(;)8 b(t)p FC(\))p Fs(j)p 186 2649 161 2 v 236 2691 a FB(y)r(t)276 2678 y Fw(2)364 2659 y Fr(6)13 b FB(M)466 2629 y FC(3)p FB(x)515 2612 y FA(?)534 2629 y FC(\()p FB(t)p FC(\))d(+)g FB(y)p 466 2649 199 2 v 544 2691 a Fs(j)p FB(t)p Fs(j)683 2629 y FB(y)p 675 2649 42 2 v 675 2691 a Fs(j)p FB(t)p Fs(j)734 2659 y Fr(6)j FB(M)831 2641 y Fx(0)843 2595 y Fq(\022)876 2659 y FC(1)d(+)g FB(c)974 2666 y Fw(1)1018 2629 y FB(")p 998 2649 62 2 v 998 2691 a Fs(j)p FB(t)p Fs(j)1040 2678 y Fw(3)1064 2595 y Fq(\023)1098 2659 y FB(c)1118 2666 y Fw(1)1162 2629 y FB(")p 1142 2649 V 1142 2691 a Fs(j)p FB(t)p Fs(j)1184 2678 y Fw(3)1221 2659 y Fr(6)j FB(M)1318 2641 y Fx(0)1330 2595 y Fq(\022)1363 2659 y FC(1)d(+)1446 2629 y FB(c)1466 2636 y Fw(1)p 1446 2649 40 2 v 1446 2692 a FB(c)1466 2676 y Fw(3)1466 2705 y(0)1490 2595 y Fq(\023)1529 2629 y FB(c)1549 2636 y Fw(1)p 1529 2649 V 1529 2692 a FB(c)1549 2676 y Fw(3)1549 2705 y(0)1573 2659 y FB(;)130 b FC(\(3.13\))922 2936 y(20)p eop %%Page: 21 21 21 20 bop 59 164 a FC(for)10 b(som)o(e)f(const)o(an)o(t)o(s)f FB(M)r(;)g(M)540 147 y Fx(0)564 164 y FB(>)13 b FC(0.)18 b(Th)o(e)10 b(rela)o(t)o(ions)f(\(3.8\))f(an)o(d)i(\(3.10\))e(yield)j (t)n(h)o(e)f(exi)q(st)o(ence)g(of)g(const)o(an)o(t)o(s)59 220 y FB(c)79 227 y Fx(\006)121 220 y FB(>)k FC(0)g(su)o(c)o(h)h(t)n(h) o(a)o(t)e FB(a)425 204 y FA(?)445 220 y FC(\()p FB(t)p FC(\))g Fr(6)g Fs(\000)p FB(c)613 227 y Fx(\000)642 220 y FB(t)658 204 y Fw(2)694 220 y FC(an)o(d)h Fs(\000)830 202 y Fw(d)p 820 209 33 2 v 820 236 a(d)p FA(t)858 220 y FB(x)884 204 y FA(?)904 220 y FC(\()p FB(t)p FC(\))e Fr(6)i FB(c)1037 227 y Fw(+)1081 220 y FC(for)g FB(t)f Fs(2)h FC([)p Fs(\000)p FB(T)t(;)8 b(t)1339 227 y Fw(0)1357 220 y FC(].)20 b(F)l(rom)14 b(\(3.7\))f(an)o(d)i(\(3.13\))59 277 y(w)o(e)f(obt)o(ain)513 392 y FB(")539 362 y FC(d)p FB(y)p 539 382 49 2 v 543 424 a FC(d)p FB(t)606 392 y Fr(6)f Fs(\000)p FB(c)709 399 y Fx(\000)738 392 y FB(t)754 374 y Fw(2)774 392 y FB(y)798 328 y Fq(\024)822 392 y FC(1)d Fs(\000)905 362 y FB(M)954 345 y Fx(0)p 905 382 61 2 v 911 424 a FB(c)931 431 y Fx(\000)971 328 y Fq(\022)1005 392 y FC(1)f(+)1087 362 y FB(c)1107 369 y Fw(1)p 1087 382 40 2 v 1087 425 a FB(c)1107 409 y Fw(3)1107 438 y(0)1132 328 y Fq(\023)1170 362 y FB(c)1190 369 y Fw(1)p 1170 382 V 1170 425 a FB(c)1190 409 y Fw(3)1190 438 y(0)1214 328 y Fq(\025)1249 392 y FC(+)h FB("c)1335 399 y Fw(+)1364 392 y FB(:)339 b FC(\(3.14\))59 519 y(F)l(or)15 b(an)o(y)f(giv)o(en)h FB(c)363 526 y Fw(1)382 519 y FC(,)g(w)o(e)f(can)h(c)o(h)o(o)q(os)q(e)f FB(c)722 526 y Fw(0)757 519 y FC(large)g(enough)g(for)g(t)n(h)o(e)g(t)o (erm)g(in)h(brac)o(k)o(et)o(s)f(t)o(o)g(b)q(e)h(larger)f(t)n(h)o(an)59 575 y(1)p FB(=)p FC(2.)k(Th)o(en,)c(b)o(y)h(Lemm)o(a)e(3.2,)g(t)n(h)o (ere)h(exi)q(st)o(s)h(a)f(const)o(an)o(t)f FB(c)1070 582 y Fw(2)1102 575 y FC(=)f FB(c)1169 582 y Fw(2)1189 575 y FC(\()p FB(c)1227 582 y Fw(+)1256 575 y FB(;)c(c)1297 582 y Fx(\000)1325 575 y FC(\))k FB(>)h FC(0)i(su)o(c)o(h)f(t)n(h)o(a)o (t)393 706 y FB(y)415 713 y FA(t)442 706 y Fr(6)f FB(y)512 713 y Fx(\000)p FA(T)575 706 y FC(e)595 687 y Fx(\000)p FA(c)637 691 y Fh(\000)664 687 y Fw(\()p FA(t)691 675 y Fz(3)708 687 y Fw(+)p FA(T)761 675 y Fz(3)778 687 y Fw(\))p FA(=)p Fw(6)p FA(")853 706 y FC(+)t FB(c)912 713 y Fw(+)948 644 y Fq(Z)994 657 y FA(t)974 747 y Fx(\000)p FA(T)1036 706 y FC(e)1056 687 y Fx(\000)p FA(c)1098 691 y Fh(\000)1125 687 y Fw(\()p FA(t)1152 675 y Fz(3)1169 687 y Fx(\000)p FA(s)1212 675 y Fz(3)1230 687 y Fw(\))p FA(=)p Fw(6)p FA(")1305 706 y FC(d)p FB(s)g Fr(6)g FB(c)1432 713 y Fw(2)1464 675 y FB(")p 1456 695 37 2 v 1456 737 a(t)1472 724 y Fw(2)1716 706 y FC(\(3.15\))59 834 y(for)h(all)h FB(t)e Fs(2)g FC([)p Fs(\000)p FB(T)t(;)8 b(\034)14 b Fs(^)c FB(t)452 841 y Fw(0)472 834 y FC(].)19 b(Th)o(erefore,)13 b(if)i FB(c)794 841 y Fw(1)826 834 y FB(>)e(c)894 841 y Fw(2)913 834 y FC(,)i(t)n(h)o(en)f FB(\034)k Fr(>)13 b FB(t)1142 841 y Fw(0)1177 834 y FC(follo)o(ws.)130 891 y(Th)o(e)h(lo)o(w)o(er)g(b)q(ou)o(n)o(d)h(can)f(b)q(e)h(obt)o(ain)o (e)q(d)f(in)h(exact)n(ly)f(t)n(h)o(e)h(sam)o(e)e(w)o(ay)l(.)p 1798 891 2 31 v 1800 862 28 2 v 1800 891 V 1827 891 2 31 v 130 985 a(F)l(or)h(t)n(h)o(e)f(rem)o(ain)o(d)o(er)g(of)h(t)n(hi)q (s)f(su)n(bs)q(ect)o(ion,)h(let)g FB(t)937 992 y Fw(0)970 985 y FC(=)e Fs(\000)p FB(c)1072 992 y Fw(0)1092 985 y FC(\()1110 955 y Fs(p)p 1147 955 44 2 v 1147 985 a FB(a)1171 992 y Fw(0)1200 985 y Fs(_)d FB(")1260 968 y Fw(1)p FA(=)p Fw(3)1315 985 y FC(\))14 b(wit)n(h)g FB(c)1467 992 y Fw(0)1500 985 y FC(c)o(h)o(os)q(en)g(accordin)o(g)59 1041 y(t)o(o)j(t)n(h)o(e)g(prece)q(din)o(g)g(pro)o(p)q(os)q(it)o(ion.) 25 b(Not)o(e)16 b(t)n(h)o(a)o(t)g(t)n(hi)q(s)h(pro)o(p)q(os)q(it)o(ion) f(implie)q(s)i(t)n(h)o(a)o(t)d FB(x)1468 1048 y FA(t)1500 1041 y Fs(\020)i FB(x)1578 1025 y FA(?)1598 1041 y FC(\()p FB(t)p FC(\))f Fs(\020)h(j)p FB(t)p Fs(j)g FC(for)59 1097 y Fs(\000)p FB(T)i Fr(6)13 b FB(t)g Fr(6)g FB(t)281 1104 y Fw(0)316 1097 y FC(an)o(d,)h(in)h(part)o(icular,)e(t)n(h)o(a)o (t)g FB(y)801 1104 y FA(t)814 1109 y Fz(0)846 1097 y Fs(\020)g FC(\()p FB("=a)980 1104 y Fw(0)999 1097 y FC(\))d Fs(^)h FB(")1089 1081 y Fw(1)p FA(=)p Fw(3)1144 1097 y FC(.)130 1154 y(W)l(e)k(no)o(w)f(cons)q(id)o(er)h(t)n(h)o(e)g(dyn)o (amics)e(for)h Fs(j)p FB(t)p Fs(j)f Fr(6)g Fs(j)p FB(t)950 1161 y Fw(0)970 1154 y Fs(j)p FC(,)i(st)o(art)o(in)o(g)e(wit)n(h)h(t)n (h)o(e)h(cas)q(e)f(of)h FB(a)1523 1161 y Fw(0)1558 1154 y FC(not)f(t)o(o)q(o)g(sm)o(all,)59 1210 y(i.)7 b(e.,)15 b(t)n(h)o(e)f(cas)q(e)g(of)g FB(y)395 1217 y FA(t)408 1222 y Fz(0)441 1210 y Fs(\020)f FB("=a)557 1217 y Fw(0)576 1210 y FC(.)59 1304 y Fo(Pro)o(p)q(o)q(s)q(it)o(ion)k(3.4.)23 b Fy(Ther)n(e)14 b(exists)h(a)f(c)n(onstant)i FB(\015)930 1311 y Fw(0)961 1304 y FB(>)d FC(0)p Fy(,)i(dep)n(ending)f(only)h(on)g FB(f)21 b Fy(and)15 b FB(y)1579 1311 y FA(t)1592 1316 y Fz(0)1611 1304 y Fy(,)g(such)g(that,)59 1361 y(when)h FB(a)198 1368 y Fw(0)230 1361 y Fr(>)d FB(\015)302 1368 y Fw(0)321 1361 y FB(")342 1344 y Fw(2)p FA(=)p Fw(3)398 1361 y Fy(,)j(then)217 1488 y FB(y)239 1495 y FA(t)267 1488 y FC(=)c FB(C)347 1495 y Fw(1)366 1488 y FC(\()p FB(t)p FC(\)\()p FB(t)452 1469 y FA(?)482 1488 y Fs(\000)e FB(t)p FC(\))g(+)g FB(C)649 1495 y Fw(2)668 1488 y FC(\()p FB(t)p FC(\))92 b Fy(with)f FB(C)1018 1495 y Fw(1)1037 1488 y FC(\()p FB(t)p FC(\))12 b Fs(\020)1183 1457 y FB(")p 1154 1477 80 2 v 1154 1532 a(a)1178 1508 y Fw(3)p FA(=)p Fw(2)1178 1545 y(0)1238 1488 y FB(;)54 b(C)1338 1495 y Fw(2)1357 1488 y FC(\()p FB(t)p FC(\))12 b Fs(\020)1493 1457 y FB(")1514 1441 y Fw(2)p 1474 1477 V 1474 1532 a FB(a)1498 1508 y Fw(5)p FA(=)p Fw(2)1498 1545 y(0)1716 1488 y FC(\(3.16\))59 1629 y Fy(for)k(al)r(l)h Fs(j)p FB(t)p Fs(j)12 b Fr(6)h Fs(j)p FB(t)326 1636 y Fw(0)346 1629 y Fs(j)p Fy(.)59 1723 y Fd(Pr)o(oof:)23 b FC(Again,)15 b(w)o(e)f(will)j(only)e(sh)o(o)o(w)f(h)o(o)o(w)g(t)o(o)h(obt)o(ain)f (an)h(up)o(p)q(er)f(b)q(ou)o(n)o(d,)h(s)q(ince)g(t)n(h)o(e)g(corre)q (sp)q(on)o(din)o(g)59 1779 y(lo)o(w)o(er)f(b)q(ou)o(n)o(d)h(can)f(b)q (e)h(e)q(st)o(a)o(bli)q(sh)o(e)q(d)f(in)h(exact)n(ly)f(t)n(h)o(e)h(sam) o(e)e(w)o(ay)l(.)130 1841 y(First)j(w)o(e)g(\014x)h(a)f(const)o(an)o(t) e FB(c)616 1848 y Fw(1)651 1841 y FB(>)h(a)725 1848 y Fw(0)745 1841 y FB(y)767 1848 y FA(t)780 1853 y Fz(0)800 1841 y FB(=")c FC(+)g(2\()p FB(t)958 1824 y FA(?)988 1841 y Fs(\000)h FB(t)1051 1848 y Fw(0)1071 1841 y FC(\))p FB(=)1112 1812 y Fs(p)p 1149 1812 44 2 v 1149 1841 a FB(a)1173 1848 y Fw(0)1204 1841 y FC(+)f(4)p FB("=)p FC(\()p FB(c)1355 1848 y Fx(\000)1384 1841 y FB(a)1408 1817 y Fw(3)p FA(=)p Fw(2)1408 1854 y(0)1463 1841 y FC(\).)24 b(W)l(e)17 b(d)o(enot)o(e)e(b)o(y)i FB(\034)59 1897 y FC(t)n(h)o(e)d(\014rst)h(exit)f(t)o(im)o(e)g(of)h FB(y)489 1904 y FA(t)519 1897 y FC(f)q(rom)e(t)n(h)o(e)h(str)q(ip)g Fs(j)p FB(y)841 1904 y FA(t)856 1897 y Fs(j)e FB(<)h(c)949 1904 y Fw(1)969 1897 y FB("=a)1037 1904 y Fw(0)1056 1897 y FC(.)20 b(F)l(or)15 b FB(t)1186 1904 y Fw(0)1218 1897 y Fr(6)e FB(t)g Fr(6)g FB(\034)j Fs(^)10 b(j)p FB(t)1448 1904 y Fw(0)1468 1897 y Fs(j)p FC(,)k(w)o(e)g(h)o(a)o(v)o(e)369 1994 y Fs(j)p FB(b)402 1977 y FA(?)421 1994 y FC(\()p FB(y)r(;)8 b(t)p FC(\))p Fs(j)p 369 2014 161 2 v 425 2056 a(j)p FB(y)r Fs(j)547 2024 y Fr(6)13 b FB(M)644 1988 y Fq(\000)665 2024 y FC(3)p FB(x)714 2006 y FA(?)733 2024 y FC(\()p FB(t)p FC(\))d(+)g Fs(j)p FB(y)r Fs(j)890 1988 y Fq(\001)910 2024 y Fs(j)p FB(y)r Fs(j)i Fr(6)h FB(M)1069 2006 y Fx(0)1080 2024 y FB(a)1104 2031 y Fw(0)1124 1960 y Fq(\022)1157 2024 y FC(1)d(+)g FB(c)1255 2031 y Fw(1)1308 1994 y FB(")p 1280 2014 80 2 v 1280 2068 a(a)1304 2044 y Fw(3)p FA(=)p Fw(2)1304 2081 y(0)1364 1960 y Fq(\023)1397 2024 y FB(c)1417 2031 y Fw(1)1470 1994 y FB(")p 1441 2014 V 1441 2068 a(a)1465 2044 y Fw(3)p FA(=)p Fw(2)1465 2081 y(0)1716 2024 y FC(\(3.17\))59 2166 y(wit)n(h)15 b(const)o(an)o(t)o(s)d FB(M)r(;)c(M)473 2150 y Fx(0)497 2166 y FB(>)13 b FC(0.)19 b(Ch)o(o)q(os)q(in)o(g)14 b FB(\015)820 2173 y Fw(0)855 2166 y FC(large)g(enough,)f(w)o(e)i(get) 614 2287 y FB(")640 2256 y FC(d)p FB(y)p 640 2277 49 2 v 644 2318 a FC(d)p FB(t)706 2287 y Fr(6)e Fs(\000)794 2256 y FB(c)814 2263 y Fx(\000)p 794 2277 50 2 v 808 2318 a FC(2)849 2287 y FB(a)873 2294 y Fw(0)893 2287 y FB(y)f Fs(\000)e FB(c)992 2294 y Fx(\000)1056 2256 y FB(")p 1026 2277 82 2 v 1026 2289 a Fs(p)p 1064 2289 44 2 v 29 x FB(a)1088 2325 y Fw(0)1113 2287 y FC(\()p FB(t)g Fs(\000)g FB(t)1218 2268 y FA(?)1238 2287 y FC(\))p FB(;)447 b FC(\(3.18\))59 2414 y(whic)o(h)15 b(implie)q(s)390 2542 y FB(y)412 2549 y FA(t)440 2542 y Fr(6)e FB(y)510 2549 y FA(t)523 2554 y Fz(0)550 2542 y FC(e)570 2523 y Fx(\000)p FA(c)612 2527 y Fh(\000)638 2523 y FA(a)657 2528 y Fz(0)674 2523 y Fw(\()p FA(t)p Fx(\000)p FA(t)741 2528 y Fz(0)758 2523 y Fw(\))p FA(=)p Fw(2)p FA(")833 2542 y Fs(\000)890 2511 y FB(c)910 2518 y Fx(\000)p 873 2531 82 2 v 873 2544 a Fs(p)p 911 2544 44 2 v 29 x FB(a)935 2580 y Fw(0)968 2480 y Fq(Z)1013 2493 y FA(t)993 2583 y(t)1006 2588 y Fz(0)1036 2542 y FC(e)1056 2523 y Fx(\000)p FA(c)1098 2527 y Fh(\000)1124 2523 y FA(a)1143 2528 y Fz(0)1160 2523 y Fw(\()p FA(t)p Fx(\000)p FA(s)p Fw(\))p FA(=)p Fw(2)p FA(")1298 2542 y FC(\()p FB(s)d Fs(\000)g FB(t)1408 2523 y FA(?)1429 2542 y FC(\))e(d)o FB(s)440 2673 y FC(=)510 2642 y(2)p FB(")p 492 2662 80 2 v 492 2717 a(a)516 2693 y Fw(3)p FA(=)p Fw(2)516 2730 y(0)576 2673 y FC(\()p FB(t)610 2654 y FA(?)640 2673 y Fs(\000)i FB(t)p FC(\))g(+)g FB(\021)r FC(\()p FB(")p FC(\))e(e)883 2654 y Fx(\000)p FA(c)925 2658 y Fh(\000)951 2654 y FA(a)970 2659 y Fz(0)987 2654 y Fw(\()p FA(t)p Fx(\000)p FA(t)1054 2659 y Fz(0)1071 2654 y Fw(\))p FA(=)p Fw(2)p FA(")1146 2673 y FC(+)1222 2642 y(4)p FB(")1266 2625 y Fw(2)p 1190 2662 129 2 v 1190 2717 a FB(c)1210 2724 y Fx(\000)1239 2717 y FB(a)1263 2693 y Fw(5)p FA(=)p Fw(2)1263 2730 y(0)1716 2673 y FC(\(3.19\))922 2936 y(21)p eop %%Page: 22 22 22 21 bop 59 164 a FC(b)o(y)15 b(part)o(ial)e(in)o(t)o(egra)o(t)o(ion.) k(Here)582 271 y FB(\021)r FC(\()p FB(")p FC(\))11 b(=)i FB(y)745 278 y FA(t)758 283 y Fz(0)788 271 y Fs(\000)d FC(2)872 240 y FB(")p 861 260 44 2 v 861 302 a(a)885 309 y Fw(0)909 207 y Fq(\022)948 240 y FB(t)964 223 y FA(?)994 240 y Fs(\000)g FB(t)1055 247 y Fw(0)p 948 260 128 2 v 971 273 a Fs(p)p 1009 273 44 2 v 29 x FB(a)1033 309 y Fw(0)1091 271 y FC(+)1183 240 y(2)p FB(")p 1141 260 129 2 v 1141 315 a(c)1161 322 y Fx(\000)1190 315 y FB(a)1214 291 y Fw(3)p FA(=)p Fw(2)1214 328 y(0)1274 207 y Fq(\023)1716 271 y FC(\(3.20\))59 394 y(sa)o(t)o(i)q(s\014e)q(s)j FB(\021)r FC(\()p FB(")p FC(\))e(=)i Fs(O)q FC(\()p FB("=a)488 401 y Fw(0)508 394 y FC(\).)130 450 y(W)l(e)e(w)o(an)o(t)f(t)o(o)g(e)q (st)o(im)o(a)o(t)o(e)g(t)n(h)o(e)g(con)o(tr)q(ibu)o(t)o(ion)g(of)g(t)n (h)o(e)h(middle)h(t)o(erm)d(on)i(t)n(h)o(e)g(r)q(igh)o(t-h)o(an)o(d)e (s)q(id)o(e)i(of)j(\(3.19\))m(.)59 507 y(Assu)o(m)o(e)g(\014rst)h(t)n (h)o(a)o(t)e FB(\021)r FC(\()p FB(")p FC(\))e FB(>)i FC(0)h(an)o(d)h(cons)q(id)o(er)f FB(t)f Fr(6)g FB(t)944 490 y FA(?)964 507 y FC(.)20 b(By)14 b(con)o(v)o(exit)o(y)l(,)523 613 y(e)543 594 y Fx(\000)p FA(c)585 598 y Fh(\000)612 594 y FA(a)631 599 y Fz(0)648 594 y Fw(\()p FA(t)p Fx(\000)p FA(t)715 599 y Fz(0)732 594 y Fw(\))p FA(=)p Fw(2)p FA(")812 613 y Fr(6)874 582 y FB(t)890 566 y FA(?)921 582 y Fs(\000)c FB(t)p 865 602 128 2 v 865 644 a(t)881 631 y FA(?)911 644 y Fs(\000)g FB(t)972 651 y Fw(0)1007 613 y FC(+)h(e)1073 594 y Fx(\000)p FA(c)1115 598 y Fh(\000)1141 594 y FA(a)1160 599 y Fz(0)1177 594 y Fw(\()p FA(t)1204 582 y Fg(?)1222 594 y Fx(\000)p FA(t)1262 599 y Fz(0)1279 594 y Fw(\))p FA(=)p Fw(2)p FA(")1354 613 y FB(:)349 b FC(\(3.21\))59 714 y(No)o(w,)705 784 y(2)p FB(")p 688 804 80 2 v 688 859 a(a)712 835 y Fw(3)p FA(=)p Fw(2)712 872 y(0)782 815 y FC(+)10 b FB(\021)r FC(\()p FB(")p FC(\))965 784 y(1)p 914 804 128 2 v 914 846 a FB(t)930 833 y FA(?)958 846 y Fs(\000)h FB(t)1020 853 y Fw(0)1058 815 y Fs(\020)1139 784 y FB(")p 1111 804 80 2 v 1111 859 a(a)1135 835 y Fw(3)p FA(=)p Fw(2)1135 872 y(0)1195 815 y FB(:)508 b FC(\(3.22\))59 943 y(Since)16 b FB(X)10 b FC(e)245 927 y Fx(\000)p FA(X)319 943 y Fs(!)j FC(0)i(as)f FB(X)i Fs(!)d(1)p FC(,)i(w)o(e)f(also)h(h)o(a)o(v)o(e)579 1053 y FB(\021)r FC(\()p FB(")p FC(\))8 b(e)687 1034 y Fx(\000)p FA(c)729 1038 y Fh(\000)755 1034 y FA(a)774 1039 y Fz(0)791 1034 y Fw(\()p FA(t)818 1022 y Fg(?)836 1034 y Fx(\000)p FA(t)876 1039 y Fz(0)893 1034 y Fw(\))p FA(=)p Fw(2)p FA(")968 1053 y FC(+)1041 1022 y(4)p FB(")1085 1005 y Fw(2)p 1008 1042 129 2 v 1008 1097 a FB(c)1028 1104 y Fx(\000)1057 1097 y FB(a)1081 1073 y Fw(5)p FA(=)p Fw(2)1081 1110 y(0)1154 1053 y Fs(\020)1226 1022 y FB(")1247 1005 y Fw(2)p 1207 1042 80 2 v 1207 1097 a FB(a)1231 1073 y Fw(5)p FA(=)p Fw(2)1231 1110 y(0)1291 1053 y FB(;)412 b FC(\(3.23\))59 1182 y(pro)o(vid)o(e)q(d)13 b FB(\015)267 1189 y Fw(0)300 1182 y FC(i)q(s)h(large)e(enough.)18 b(Thi)q(s)13 b(sh)o(o)o(ws)f(t)n(h)o(e)h(exi)q(st)o(ence)g(of)g(const)o (an)o(t)o(s)p 1374 1145 22 2 v 11 w FB(C)1396 1189 y Fw(1)1428 1182 y Fr(>)g FC(2)g(an)o(d)p 1608 1145 V 13 w FB(C)1630 1189 y Fw(2)1661 1182 y FB(>)g FC(0)g(su)o(c)o(h)59 1238 y(t)n(h)o(a)o(t)476 1339 y FB(y)498 1346 y FA(t)525 1339 y Fr(6)p 585 1302 V 13 w FB(C)606 1346 y Fw(1)659 1308 y FB(")p 631 1329 80 2 v 631 1383 a(a)655 1359 y Fw(3)p FA(=)p Fw(2)655 1396 y(0)715 1339 y FC(\()p FB(t)749 1320 y FA(?)778 1339 y Fs(\000)e FB(t)p FC(\))f(+)p 925 1302 22 2 v 10 w FB(C)946 1346 y Fw(2)989 1308 y FB(")1010 1292 y Fw(2)p 970 1329 80 2 v 970 1383 a FB(a)994 1359 y Fw(5)p FA(=)p Fw(2)994 1396 y(0)1144 1339 y FC(for)k FB(t)f Fr(6)g FB(t)1306 1320 y FA(?)1336 1339 y Fs(^)e 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FC(\))g(a)o(t)f(a)g(t)o (im)o(e)971 1937 y(~)970 1945 y FB(t)i FC(sa)o(t)o(i)q(sfyin)o(g)1202 1937 y(~)1202 1945 y FB(t)11 b Fs(\000)g FB(t)1291 1929 y FA(?)1325 1945 y Fs(\020)k FB("=a)1443 1952 y Fw(0)1463 1945 y FC(.)22 b(F)l(or)16 b(large)f(enough)59 2002 y FB(\015)83 2009 y Fw(0)102 2002 y FC(,)g(t)n(h)o(e)f(pro)o(p)q(os)q(it) o(ion)f(also)i(sh)o(o)o(ws)e(t)n(h)o(a)o(t)g FB(x)777 2009 y FA(t)805 2002 y Fs(\020)853 1972 y(p)p 891 1972 44 2 v 30 x FB(a)915 2009 y Fw(0)949 2002 y FC(for)h Fs(j)p FB(t)p Fs(j)e Fr(6)h Fs(j)p FB(t)1149 2009 y Fw(0)1169 2002 y Fs(j)i FC(an)o(d)f(t)n(h)o(a)o(t)f FB(y)1398 2011 y Fx(j)p FA(t)1421 2016 y Fz(0)1439 2011 y Fx(j)1463 2002 y Fs(\020)g(\000)p FB("=a)1614 2009 y Fw(0)1634 2002 y FC(.)130 2065 y(W)l(e)18 b(cons)q(id)o(er)f(no)o(w)f(t)n(h)o(e)h (cas)q(e)g FB(a)684 2072 y Fw(0)721 2065 y FB(<)g(\015)797 2072 y Fw(0)816 2065 y FB(")837 2048 y Fw(2)p FA(=)p Fw(3)910 2065 y FC(wit)n(h)g FB(\015)1037 2072 y Fw(0)1074 2065 y 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Fx(\000)p FA(T)1376 2759 y FC(=)13 b(0)p FB(;)256 b FC(\(3.42\))922 2936 y(24)p eop %%Page: 25 25 25 24 bop 59 164 a FC(wh)o(ere)11 b(\026)-24 b FB(a)p FC(\()p FB(t)p FC(\))10 b(i)q(s)h(t)n(h)o(e)g(lin)o(ear)q(iza)o(t)o (ion)f(\(3.35\))e(of)i FB(f)16 b FC(alon)o(g)10 b FB(x)986 147 y Fw(d)o(et)986 175 y FA(t)1037 164 y FC(,)h(an)o(d)f(T)l(aylor's)f (form)n(ula)g(yields)j(t)n(h)o(e)e(rela)o(t)o(ions)456 246 y(\026)457 258 y FB(b)p FC(\()p FB(y)r(;)e(t)p FC(\))j(=)i Fs(\000)668 221 y Fq(\002)687 258 y FC(1)c(+)i Fs(O)q FC(\()p FB(x)846 239 y Fw(d)o(et)846 269 y FA(t)906 258 y FC(+)f Fs(j)p FB(y)r Fs(j)p FC(\))f(+)h Fs(O)q FC(\()p FB(t)p FC(\))1162 221 y Fq(\003)1181 258 y FC(\(3)p FB(x)1248 239 y Fw(d)o(et)1248 269 y FA(t)1307 258 y FC(+)h FB(y)r FC(\))p FB(y)1419 239 y Fw(2)1438 258 y FB(;)432 333 y Fs(j)443 321 y FC(\026)445 333 y FB(b)o FC(\()p FB(y)r(;)d(t)p FC(\))p Fs(j)j Fr(6)i FB(M)5 b FC(\()p FB(x)726 314 y Fw(d)o(et)726 344 y FA(t)786 333 y FC(+)10 b Fs(j)p FB(y)r Fs(j)p FC(\))p 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1214 1221 75 2 v 1214 1259 a FB(\020)s FC(\()p FB(t)p FC(\))o(.)59 1346 y Fo(Pro)o(p)q(o)q(s)q(it)o(ion)k (3.8.)23 b Fy(F)m(or)16 b Fs(\000)p FB(T)i Fr(6)13 b FB(t)g Fr(6)g FB(T)6 b Fy(,)333 1464 y Fl(P)367 1445 y Fx(\000)p FA(T)s(;)p Fw(0)445 1413 y Fq(n)509 1464 y FC(sup)475 1503 y Fx(\000)p FA(T)f Fm(6)p FA(s)p Fm(6)p FA(t)652 1433 y Fs(j)p FB(y)689 1417 y Fw(0)687 1444 y FA(s)709 1433 y Fs(j)p 624 1453 126 2 v 624 1462 a Fq(p)p 670 1462 80 2 v 670 1501 a FB(\020)s FC(\()p FB(s)p FC(\))767 1464 y Fr(>)13 b FB(h)841 1413 y Fq(o)884 1464 y Fr(6)g FB(C)s FC(\()p FB(t;)8 b(")p FC(\))g(exp)1137 1413 y Fq(n)1167 1464 y Fs(\000)1207 1433 y FC(1)p 1207 1453 23 2 v 1207 1495 a(2)1241 1433 y FB(h)1267 1417 y Fw(2)p 1240 1453 48 2 v 1240 1495 a FB(\033)1268 1482 y Fw(2)1292 1427 y Fq(\000)1313 1464 y FC(1)i Fs(\000)g(O)q FC(\()p FB(")p FC(\))1485 1427 y Fq(\001)1506 1413 y(o)1536 1464 y FB(;)167 b FC(\(3.47\))59 1587 y Fy(wher)n(e)703 1693 y FB(C)s FC(\()p FB(t;)8 b(")p FC(\))j(=)897 1662 y Fs(j)p 915 1637 22 2 v FB(\013)p FC(\()p FB(t;)d Fs(\000)p FB(T)e FC(\))p Fs(j)p 897 1682 195 2 v 974 1724 a FB(")995 1711 y Fw(2)1106 1693 y FC(+)k(2)p FB(:)529 b FC(\(3.48\))59 1799 y Fd(Pr)o(oof:)23 b FC(Let)14 b Fs(\000)p FB(T)19 b FC(=)13 b FB(u)472 1806 y Fw(0)504 1799 y FB(<)g(u)578 1806 y Fw(1)610 1799 y FB(<)g Fs(\001)8 b(\001)g(\001)j FB(<)i(u)798 1806 y FA(K)845 1799 y FC(=)f FB(t)p FC(,)j(wit)n(h)f(som) o(e)e FB(K)k(>)d FC(0,)g(b)q(e)h(a)g(part)o(it)o(ion)e(of)i([)p Fs(\000)p FB(T)t(;)8 b(t)p FC(].)17 b(In)59 1855 y([BG,)d(Lemm)o(a)g (3.2],)e(w)o(e)j(sh)o(o)o(w)e(t)n(h)o(a)o(t)g(t)n(h)o(e)i(proba)o (bilit)o(y)f(\(3.47\))e(i)q(s)j(b)q(ou)o(n)o(d)o(e)q(d)h(a)o(b)q(o)o(v) o(e)d(b)o(y)301 1991 y(2)348 1935 y FA(K)331 1948 y Fq(X)332 2047 y FA(k)q Fw(=1)404 1991 y FB(P)433 1998 y FA(k)455 1991 y FB(;)97 b FC(wh)o(ere)14 b FB(P)723 1998 y FA(k)757 1991 y FC(=)f(exp)873 1941 y Fq(n)904 1991 y Fs(\000)944 1961 y FC(1)p 944 1981 23 2 v 944 2023 a(2)977 1961 y FB(h)1003 1944 y Fw(2)p 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2257 y Fw(2)1447 2175 y Fq(\031)1474 2239 y FB(:)229 b FC(\(3.50\))59 2358 y(Since)16 b(\026)-23 b FB(a)p FC(\()p FB(s)p FC(\))12 b FB(<)h FC(0,)h(w)o(e)g(h)o(a)o(v)o(e)f FB(\020)559 2341 y Fx(0)571 2358 y FC(\()p FB(s)p FC(\))f(=)h([2)q (\026)-24 b FB(a)o FC(\()p FB(s)p FC(\))p FB(\020)s FC(\()p FB(s)p FC(\))9 b(+)h(1])p FB(=")i Fr(6)h FC(1)p FB(=")p FC(,)h(an)o(d)g(t)n(h)n(us)265 2475 y(inf)197 2504 y FA(u)217 2510 y Fg(k)q Fh(\000)p Fz(1)276 2504 y Fm(6)p FA(s)p Fm(6)p FA(u)366 2510 y Fg(k)412 2444 y FB(\020)s FC(\()p FB(s)p FC(\))p 399 2464 106 2 v 399 2506 a FB(\020)s FC(\()p FB(u)466 2513 y FA(k)487 2506 y FC(\))522 2475 y Fr(>)617 2444 y FC(1)p 575 2464 V 575 2506 a FB(\020)s FC(\()p FB(u)642 2513 y FA(k)663 2506 y FC(\))761 2475 y(inf)694 2504 y FA(u)714 2510 y Fg(k)q Fh(\000)p Fz(1)772 2504 y Fm(6)p FA(s)p Fm(6)p FA(u)862 2510 y Fg(k)883 2424 y Fq(h)904 2475 y FB(\020)s FC(\()p FB(u)971 2482 y FA(k)992 2475 y FC(\))c Fs(\000)1071 2444 y FB(u)1097 2451 y FA(k)1128 2444 y 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Fw(\))526 1858 y Fs(j)f FC(=)h FB(\016)r(h)651 1819 y Fq(p)p 696 1819 V 39 x FB(\020)s FC(\()p FB(\034)5 b FC(\()p FB(!)r FC(\)\))15 b(wh)o(en)o(ev)o(er)f FB(\034)5 b FC(\()p FB(!)r FC(\))14 b FB(<)h Fs(1)p FC(,)i(w)o(e)e(conclud)o(e)i(t)n(h)o(a) o(t)d Fl(P)-5 b FC(\()p FB(A)p FC(\))11 b(=)k(0,)59 1922 y(an)o(d)g(t)n(h)n(us)h FB(\034)5 b FC(\()p FB(!)r FC(\))13 b(=)h Fs(1)j FC(for)e(almost)g(all)h FB(!)g Fs(2)f FC(\012)861 1905 y Fw(0)861 1933 y FA(t)880 1922 y FC(\()p FB(h)p FC(\),)g(whic)o(h)h(implie)q(s)g(t)n(h)o(a)o(t)f Fs(j)p FB(y)1384 1929 y FA(s)1402 1922 y FC(\()p FB(!)r FC(\))p Fs(j)e Fr(6)h FC(\(1)c(+)h FB(\016)r FC(\))p FB(h)1706 1883 y Fq(p)p 1751 1883 80 2 v 39 x FB(\020)s FC(\()p FB(s)p FC(\))59 1978 y(for)j Fs(\000)p FB(T)19 b Fr(6)13 b FB(s)g Fr(6)g FB(t)i FC(an)o(d)f(t)n(h)o(e)q(s)q(e)h FB(!)r FC(.)k(Thi)q(s)14 b(complet)o(e)q(s)g(t)n(h)o(e)g(pro)q(of)g(of) j(\(3.58\))m(.)130 2035 y(Th)o(e)g(pro)q(of)f(of)k(\(3.59\))15 b(i)q(s)i(almost)g(t)n(h)o(e)g(sam)o(e.)26 b(In)18 b(t)n(h)o(e)f(cas)q (e)f Fs(\000)p FC(\()1232 2005 y Fs(p)p 1270 2005 44 2 v 30 x FB(a)1294 2042 y Fw(0)1326 2035 y Fs(_)c FB(")1389 2018 y Fw(1)p FA(=)p Fw(3)1444 2035 y FC(\))17 b Fr(6)g FB(t)g Fr(6)h FB(T)6 b FC(,)18 b(w)o(e)e(t)o(ak)o(e)59 2091 y FB(\016)f FC(=)d FB(\045h=)p FC(\()p FB(a)256 2098 y Fw(0)285 2091 y Fs(_)e FB(")346 2075 y Fw(2)p FA(=)p Fw(3)402 2091 y FC(\).)19 b(Th)o(e)14 b(e)q(st)o(im)o(a)o(t)o(e) f(\(3.63\))f(h)o(as)i(t)o(o)g(b)q(e)h(replace)q(d)f(b)o(y)332 2216 y FB(x)358 2197 y Fw(d)o(et)358 2227 y FA(u)421 2216 y Fr(6)f FB(c)489 2223 y Fw(+)518 2216 y FC(\()p Fs(j)p FB(u)p Fs(j)c(_)637 2184 y(p)p 675 2184 V 32 x FB(a)699 2223 y Fw(0)729 2216 y Fs(_)h FB(")790 2197 y Fw(1)p FA(=)p Fw(3)846 2216 y FC(\))p FB(;)96 b Fs(j)p FB(y)1008 2223 y FA(u)1031 2216 y FC(\()p FB(!)r FC(\))p Fs(j)11 b Fr(6)i FC(2)p FB(h)1334 2155 y Fs(p)p 1372 2155 50 2 v 27 x FB(c)1392 2189 y Fw(+)p 1223 2205 311 2 v 1223 2249 a Fs(j)p FB(u)p Fs(j)c(_)1324 2220 y(p)p 1362 2220 44 2 v 29 x FB(a)1386 2256 y Fw(0)1416 2249 y Fs(_)i FB(")1478 2236 y Fw(1)p FA(=)p Fw(3)1538 2216 y FB(;)165 b FC(\(3.69\))59 2346 y(an)o(d)14 b(t)n(h)n(us)h(w)o(e)f (get,)g(inst)o(ead)g(of)k(\(3.65\))m(,)d(t)n(h)o(e)f(b)q(ou)o(n)o(d)341 2476 y Fs(j)p FB(z)375 2483 y FA(s)394 2476 y Fs(j)e Fr(6)h FC(4)p FB(M)766 2443 y(h)792 2427 y Fw(2)812 2443 y FB(c)832 2427 y Fw(3)832 2454 y(+)p 543 2465 541 2 v 543 2509 a FC(\()561 2479 y Fs(p)p 599 2479 44 2 v 30 x FB(a)623 2516 y Fw(0)653 2509 y Fs(_)d FB(")714 2496 y Fw(1)p FA(=)p Fw(3)769 2509 y FC(\)\()p FB(s)826 2496 y Fw(2)855 2509 y Fs(_)h FB(a)920 2516 y Fw(0)950 2509 y Fs(_)f FB(")1011 2496 y Fw(2)p FA(=)p Fw(3)1066 2509 y FC(\))1089 2412 y Fq(\022)1122 2476 y FC(1)g(+)1327 2445 y(2)p FB(h)p 1205 2465 293 2 v 1205 2481 a Fs(p)p 1243 2481 50 2 v 28 x FB(c)1263 2516 y Fw(+)1292 2509 y FC(\()p FB(a)1334 2516 y Fw(0)1363 2509 y Fs(_)h FB(")1425 2496 y Fw(2)p FA(=)p Fw(3)1480 2509 y FC(\))1502 2412 y Fq(\023)1536 2476 y FB(:)167 b FC(\(3.70\))59 2605 y(Th)o(e)14 b(rem)o(ain)o(d)o(er)g(of)g(t)n(h)o(e)h(pro)q(of)e(i)q(s)i (s)q(imilar.)p 1798 2605 2 31 v 1800 2576 28 2 v 1800 2605 V 1827 2605 2 31 v 130 2698 a(No)o(w,)10 b(t)n(h)o(e)h(prece)q (din)o(g)h(t)o(w)o(o)d(pro)o(p)q(os)q(it)o(ions)h(imm)o(e)q(dia)o(t)o (ely)g(imply)h(Th)o(eorem)f(2.2,)g(as)h(Pro)o(p)q(os)q(it)o(ion)e(3.8) 59 2755 y(sh)o(o)o(ws)19 b(t)n(h)o(e)h(d)o(e)q(s)q(ire)q(d)h(b)q(e)o(h) o(a)o(viour)d(for)i(t)n(h)o(e)g(ap)o(pro)o(xim)o(a)o(t)n(ion)e(b)o(y)i (a)g(Ga)n(uss)q(ian)g(pro)q(ce)q(ss)g(an)o(d)g(Pro)o(p)q(os)q(i-)59 2811 y(t)o(ion)14 b(3.10)f(allo)o(ws)h(t)o(o)h(ext)o(en)o(d)f(t)n(hi)q (s)g(re)q(sul)o(t)h(t)o(o)g(t)n(h)o(e)f(or)q(igin)o(al)g(pro)q(ce)q (ss.)922 2936 y(27)p eop %%Page: 28 28 28 27 bop 59 164 a Fn(3.3)56 b(Th)n(e)19 b(trans)q(it)n(ion)f(regim)n (e)59 250 y FC(W)l(e)f(cons)q(id)o(er)e(no)o(w)g(t)n(h)o(e)h(regim)o(e) f(of)g FB(\033)j FC(su\016cien)o(t)n(ly)d(large)g(t)o(o)h(allo)o(w)f (for)g(trans)q(it)o(ions)f(f)q(rom)g(on)o(e)h(st)o(a)o(ble)59 306 y(equilibr)q(iu)o(m)21 b(branc)o(h)e(t)o(o)g(t)n(h)o(e)h(ot)n(h)o (er.)33 b(Here)20 b FB(x)881 290 y Fw(d)o(et)881 317 y FA(t)952 306 y FC(i)q(s)g(t)n(h)o(e)f(solu)o(t)o(ion)g(of)h(t)n(h)o (e)f(d)o(et)o(ermini)q(st)o(ic)g(equa)o(t)o(ion)59 363 y(\(3.6\))13 b(wit)n(h)i(t)n(h)o(e)f(sam)o(e)g(init)o(ial)i(con)o(dit)o (ion)f FB(x)805 346 y Fw(d)o(et)805 376 y Fx(\000)p FA(T)875 363 y FC(as)f(in)i(t)n(h)o(e)f(previous)f(s)q(ect)o(ions,)g(whic)o(h)g (trac)o(ks)f FB(x)1705 346 y FA(?)1725 363 y FC(\()p FB(t)p FC(\))i(a)o(t)59 425 y(di)q(st)o(ance)i(a)o(t)f(most)h Fs(O)q FC(\()p FB(")483 409 y Fw(1)p FA(=)p Fw(3)538 425 y FC(\).)27 b FB(x)622 432 y FA(t)654 425 y FC(d)o(enot)o(e)q(s)17 b(a)g(gen)o(eral)f(solu)o(t)o(ion)h(of)g(t)n(h)o(e)g(SDE)h(\(3.1\))m(.) 27 b(Our)18 b(aim)f(i)q(s)g(t)o(o)59 481 y(e)q(st)o(a)o(bli)q(sh)11 b(an)i(up)o(p)q(er)e(b)q(ou)o(n)o(d)i(for)f(t)n(h)o(e)g(proba)o(bilit)o (y)f(of)h Fy(not)18 b FC(reac)o(hin)o(g)12 b(t)n(h)o(e)g(axi)q(s)g FB(x)h FC(=)f(0,)g(whic)o(h,)h(b)o(y)f(us)q(in)o(g)59 538 y(symm)o(etry)l(,)j(will)j(allo)o(w)e(us)h(t)o(o)g(e)q(st)o(im)o(a) o(t)o(e)e(t)n(h)o(e)i(trans)q(it)o(ion)d(proba)o(bilit)o(y)l(.)26 b(Let)16 b FB(\016)j(>)d FC(0)h(b)q(e)g(t)n(h)o(e)f(const)o(an)o(t)59 594 y(d)o(e\014n)o(e)q(d)f(in)h(\(2.23\))m(,)f(i.e.)f(b)o(y)524 684 y FB(x@)574 691 y FA(xx)616 684 y FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))k Fr(6)g FC(0)90 b(for)14 b Fs(j)p FB(x)p Fs(j)e Fr(6)h FB(\016)k FC(an)o(d)d Fs(j)p FB(t)p Fs(j)e Fr(6)h FB(T)6 b FC(.)350 b(\(3.71\))130 774 y(Th)o(e)17 b(bas)q(ic)g(in)o(gre)q(dien)o(t)h(of)f(our)g(e)q(st)o(im)o(a)o(t)o(e)f (i)q(s)i(t)n(h)o(e)g(follo)o(win)o(g)f(compar)q(i)q(son)e(lemm)o(a)i (whic)o(h)h(allo)o(ws)59 830 y(us)h(t)o(o)f(lin)o(ear)q(ize)h(t)n(h)o (e)g(st)o(o)q(c)o(h)o(ast)o(ic)e(di\013eren)o(t)o(ial)h(equa)o(t)o (ions)f(u)o(n)o(d)o(er)i(cons)q(id)o(era)o(t)o(ion)e(an)o(d,)i(t)n(h)o (erefore,)e(t)o(o)59 887 y(in)o(v)o(e)q(st)o(iga)o(t)o(e)e(Ga)n(uss)q (ian)i(ap)o(pro)o(xim)o(a)o(t)n(ions)d(t)o(o)i(our)g(pro)q(ce)q(ss)q(e) q(s.)25 b(Th)o(e)16 b(lemm)o(a)g(giv)o(e)q(s)h(con)o(dit)o(ions)f(u)o (n)o(d)o(er)59 943 y(whic)o(h)f(rela)o(t)o(ions)e(b)q(et)o(w)o(een)g (init)o(ial)j(con)o(dit)o(ions)e(carry)f(o)o(v)o(er)h(t)o(o)g(t)n(h)o (e)h(sample)e(pa)o(t)n(hs.)59 1027 y Fo(Lemm)o(a)i(3.11.)23 b Fy(Fix)15 b(some)h(initial)g(time)f FB(t)814 1034 y Fw(0)847 1027 y Fs(2)e FC([)p Fs(\000)p FB(T)t(;)8 b(T)e FC(])p Fy(.)18 b(We)d(c)n(onsider)h(the)f(fol)r(lowing)h(pr)n(o)n(c)n (esses)f(on)59 1083 y FC([)p FB(t)88 1090 y Fw(0)108 1083 y FB(;)8 b(T)e FC(])p Fy(:)90 1151 y Fs(\017)24 b Fy(the)18 b(solution)i FB(x)413 1135 y Fw(d)o(et)413 1163 y FA(t)482 1151 y Fy(of)f(the)f(deterministic)f(di\013er)n(ential) h(e)n(quation)j FC(\(3.6\))16 b Fy(with)j(initial)f(c)n(ondition)137 1208 y FB(x)163 1191 y Fw(d)o(et)163 1219 y FA(t)176 1224 y Fz(0)226 1208 y Fs(2)13 b FC([0)p FB(;)8 b(\016)r FC(])p Fy(;)90 1264 y Fs(\017)24 b Fy(the)15 b(solution)i FB(x)407 1271 y FA(t)438 1264 y Fy(of)f(the)g(SDE)22 b FC(\(3.41\))13 b Fy(with)j(initial)g(c)n(ondition)f FB(x)1260 1271 y FA(t)1273 1276 y Fz(0)1305 1264 y Fs(2)e FC([)p FB(x)1387 1248 y Fw(d)o(et)1387 1276 y FA(t)1400 1281 y Fz(0)1437 1264 y FB(;)8 b(\016)r FC(])p Fy(;)90 1321 y Fs(\017)24 b Fy(the)15 b(di\013er)n(enc)n(e)f FB(y)436 1328 y FA(t)464 1321 y FC(=)e FB(x)537 1328 y FA(t)562 1321 y Fs(\000)f FB(x)634 1304 y Fw(d)o(et)634 1332 y FA(t)684 1321 y Fy(,)16 b(which)f(satis\014es)i FB(y)1031 1328 y FA(t)1044 1333 y Fz(0)1076 1321 y FC(=)c FB(x)1150 1328 y FA(t)1163 1333 y Fz(0)1192 1321 y Fs(\000)e FB(x)1264 1304 y Fw(d)o(et)1264 1332 y FA(t)1277 1337 y Fz(0)1327 1321 y Fr(>)i FC(0)p Fy(;)90 1377 y Fs(\017)24 b Fy(the)15 b(solution)i FB(y)405 1361 y Fw(0)403 1389 y FA(t)425 1377 y Fy(of)f(the)f(line)n(ar)h(SDE)414 1486 y FC(d)p FB(y)463 1467 y Fw(0)461 1497 y FA(t)495 1486 y FC(=)548 1455 y(1)p 548 1476 23 2 v 549 1517 a FB(")576 1486 y FC(~)-24 b FB(a)p FC(\()p FB(t)p FC(\))p FB(y)675 1467 y Fw(0)673 1497 y FA(t)702 1486 y FC(d)p FB(t)10 b FC(+)819 1455 y FB(\033)p 803 1476 60 2 v 803 1485 a Fs(p)p 841 1485 22 2 v 32 x FB(")875 1486 y FC(d)p FB(W)943 1493 y FA(t)958 1486 y FB(;)99 b Fy(wher)n(e)15 b FC(~)-23 b FB(a)p FC(\()p FB(t)p FC(\))12 b(=)g FB(@)1356 1493 y FA(x)1378 1486 y FB(f)5 b FC(\()p FB(x)1449 1467 y Fw(d)o(et)1449 1497 y FA(t)1500 1486 y FB(;)j(t)p FC(\))161 b(\(3.72\))137 1605 y Fy(with)15 b(initial)h(c)n(ondition)g FB(y)587 1589 y Fw(0)585 1617 y FA(t)598 1622 y Fz(0)630 1605 y Fs(2)d FC([)p FB(y)708 1612 y FA(t)721 1617 y Fz(0)740 1605 y FB(;)8 b(\016)j Fs(\000)f FB(x)863 1589 y Fw(d)o(et)863 1617 y FA(t)876 1622 y Fz(0)914 1605 y FC(])p Fy(.)59 1674 y(If)22 b FC(0)12 b Fr(6)h FB(y)219 1657 y Fw(0)217 1685 y FA(s)251 1674 y Fr(6)g FB(\016)r Fs(\000)q FB(x)383 1657 y Fw(d)o(et)383 1685 y FA(s)446 1674 y Fy(for)e(al)r(l)i FB(s)g Fs(2)g FC([)p FB(t)679 1681 y Fw(0)699 1674 y FB(;)8 b(t)p FC(])p Fy(,)j(then)h FB(y)892 1681 y FA(s)923 1674 y Fr(6)h 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y FA(s)1050 1960 y FB(;)j(s)p FC(\))h(+)i(~)-24 b FB(a)p FC(\()p FB(s)p FC(\))p FB(y)r(:)434 b FC(\(3.73\))59 2054 y(Let)18 b FB(\034)24 b FC(=)19 b(inf)t Fs(f)p FB(s)g Fs(2)g FC([)p FB(t)437 2061 y Fw(0)457 2054 y FB(;)8 b(t)p FC(])d(:)15 b FB(y)562 2061 y FA(s)599 2054 y Fs(62)20 b FC([0)p FB(;)8 b(\016)k Fs(\000)h FB(x)812 2037 y Fw(d)o(et)812 2065 y FA(s)863 2054 y FC(])p Fs(g)18 b(2)h FC([)p FB(t)995 2061 y Fw(0)1015 2054 y FB(;)8 b(t)p FC(])k Fs([)h(f1g)p FC(.)30 b(F)l(or)19 b FB(t)1355 2061 y Fw(0)1394 2054 y Fr(6)g FB(s)g Fr(6)h FB(\034)5 b FC(,)19 b(t)n(h)o(e)f(v)n(ar)q(ia)o (ble)59 2110 y FB(z)80 2117 y FA(s)111 2110 y FC(=)13 b FB(y)181 2117 y FA(s)210 2110 y Fs(\000)d FB(y)279 2094 y Fw(0)277 2121 y FA(s)314 2110 y FC(sa)o(t)o(i)q(s\014e)q(s)385 2222 y FB(z)406 2229 y FA(s)437 2222 y FC(=)j FB(z)506 2229 y FA(t)519 2234 y Fz(0)548 2222 y FC(+)598 2191 y(1)p 598 2211 23 2 v 599 2253 a FB(")634 2160 y Fq(Z)679 2173 y FA(s)659 2263 y(t)672 2268 y Fz(0)697 2185 y Fq(\002)716 2222 y FB(f)5 b FC(\()p FB(x)787 2203 y Fw(d)o(et)787 2233 y FA(u)848 2222 y FC(+)10 b FB(y)915 2229 y FA(u)937 2222 y FB(;)e(u)p FC(\))h Fs(\000)i FB(f)5 b FC(\()p FB(x)1128 2203 y Fw(d)o(et)1128 2233 y FA(u)1178 2222 y FB(;)j(u)p FC(\))h Fs(\000)i FC(~)-24 b FB(a)p FC(\()p FB(u)p FC(\))p FB(y)1407 2203 y Fw(0)1405 2233 y FA(u)1427 2185 y Fq(\003)1454 2222 y FC(d)p FB(u)437 2347 y Fr(6)13 b FB(z)506 2354 y FA(t)519 2359 y Fz(0)549 2347 y FC(+)599 2317 y(1)p 599 2337 V 600 2379 a FB(")634 2285 y Fq(Z)679 2299 y FA(s)659 2389 y(t)672 2394 y Fz(0)706 2347 y FC(~)-24 b FB(a)p FC(\()p FB(u)p FC(\))p FB(z)812 2354 y FA(u)842 2347 y FC(d)p FB(u;)97 b(z)1024 2354 y FA(t)1037 2359 y Fz(0)1069 2347 y Fr(6)13 b FC(0)p FB(:)563 b FC(\(3.74\))59 2462 y(A)n(p)o(plyin)o(g)14 b(Gron)o(w)o(all's)f(in)o(equalit)o(y)l(,)i (w)o(e)f(obt)o(ain)549 2552 y FB(z)570 2559 y FA(s)601 2552 y Fr(6)f FB(z)670 2559 y FA(t)683 2564 y Fz(0)710 2552 y FC(e)733 2533 y Fw(~)-21 b FA(\013)p Fw(\()p FA(s;t)806 2538 y Fz(0)822 2533 y Fw(\))p FA(=")885 2552 y Fr(6)13 b FC(0)89 b Fs(8)p FB(s)13 b Fs(2)g FC([)p FB(t)1176 2559 y Fw(0)1196 2552 y FB(;)8 b(\034)14 b Fs(^)d FB(t)p FC(])p FB(;)382 b FC(\(3.75\))59 2642 y(wh)o(ere)17 b(~)-28 b FB(\013)p FC(\()p FB(s;)8 b(t)291 2649 y Fw(0)310 2642 y FC(\))13 b(=)388 2605 y Fq(R)418 2618 y FA(s)410 2658 y(t)423 2663 y Fz(0)450 2642 y FC(~)-24 b FB(a)p FC(\()p FB(u)p FC(\))8 b(d)o FB(u)p FC(.)19 b(Thi)q(s)12 b(pro)o(v)o(e)q(s)g(t) n(h)o(e)g(re)q(sul)o(t)h(for)f FB(t)1144 2649 y Fw(0)1176 2642 y Fr(6)h FB(s)g Fr(6)g FB(\034)e Fs(^)6 b FB(t)p FC(.)20 b(No)o(w)11 b(if)j FB(y)1584 2649 y FA(s)1615 2642 y FC(i)q(s)f(n)o(ega)o(t)o(iv)o(e,)59 2698 y(t)n(h)o(e)k(re)q(sul) o(t)h(i)q(s)g(tr)q(ivially)g(sa)o(t)o(i)q(s\014e)q(d,)e(an)o(d)h(if)h FB(y)831 2705 y FA(s)868 2698 y FC(b)q(ecom)o(e)q(s)e(p)q(os)q(it)o(iv) o(e)i(again,)f(t)n(h)o(e)g(a)o(b)q(o)o(v)o(e)f(argu)o(m)o(en)o(t)f(can) 59 2755 y(b)q(e)g(rep)q(ea)o(t)o(e)q(d.)j(Not)o(e)13 b(t)n(h)o(a)o(t)g FB(y)542 2762 y FA(\034)577 2755 y Fr(6)g FB(\016)e Fs(\000)f FB(x)727 2738 y Fw(d)o(et)727 2766 y FA(\034)793 2755 y FC(i)q(s)15 b(imm)o(e)q(dia)o(t)o(e.)j(Thi)q (s)c(pro)o(v)o(e)q(s)f(t)n(h)o(e)h(\014rst)g(ass)q(ert)o(ion,)f(an)o(d) h(t)n(h)o(e)59 2811 y(s)q(econ)o(d)g(ass)q(ert)o(ion)f(can)i(b)q(e)g (pro)o(v)o(e)q(d)e(direct)n(ly)l(,)i(wit)n(h)o(ou)o(t)f(us)q(e)h(of)f FB(\034)5 b FC(.)p 1798 2811 2 31 v 1800 2782 28 2 v 1800 2811 V 1827 2811 2 31 v 922 2936 a(28)p eop %%Page: 29 29 29 28 bop 130 164 a FC(W)l(e)14 b(will)i(no)o(w)d(pro)q(cee)q(d)h(as)g (follo)o(ws.)j(Let)d FB(\020)s FC(\()p FB(t)p FC(\))g(b)q(e)g(t)n(h)o (e)g(fu)o(nct)o(ion)g(d)o(e\014n)o(e)q(d)h(in)f(\(3.46\))m(,)g(an)o(d)g (let)h FB(h)f FC(b)q(e)59 220 y(su)o(c)o(h)g(t)n(h)o(a)o(t)g FB(x)279 204 y Fw(d)o(et)279 232 y FA(s)339 220 y FC(+)c FB(h)410 181 y Fq(p)p 456 181 80 2 v 456 220 a FB(\020)s FC(\()p FB(s)p FC(\))i FB(<)h(\016)k FC(for)d(all)h FB(s)e Fs(2)g FC([)p FB(t)870 227 y Fw(0)890 220 y FB(;)8 b(t)p FC(].)19 b(Giv)o(en)c FB(x)1128 227 y Fw(0)1160 220 y Fs(2)e FC(\(0)p FB(;)8 b(\016)r FC(\),)k(w)o(e)j(can)f(wr)q(it)o(e)136 332 y Fl(P)170 314 y FA(t)182 319 y Fz(0)197 314 y FA(;x)227 319 y Fz(0)246 296 y Fq(\010)272 332 y FB(x)298 339 y FA(s)329 332 y FB(>)f FC(0)25 b Fs(8)p FB(s)13 b Fs(2)g FC([)p FB(t)556 339 y Fw(0)576 332 y FB(;)8 b(t)p FC(])626 296 y Fq(\011)664 332 y Fr(6)13 b Fl(P)746 314 y FA(t)758 319 y Fz(0)773 314 y FA(;x)803 319 y Fz(0)822 282 y Fq(n)875 332 y FC(sup)852 371 y FA(t)865 376 y Fz(0)882 371 y Fm(6)p FA(s)p Fm(6)p FA(t)979 302 y FB(x)1005 309 y FA(s)1033 302 y Fs(\000)e FB(x)1105 285 y Fw(d)o(et)1105 313 y FA(s)p 979 322 177 2 v 1004 331 a Fq(p)p 1050 331 80 2 v 1050 370 a FB(\020)s FC(\()p FB(s)p FC(\))1173 332 y FB(>)i(h)1247 282 y Fq(o)722 445 y FC(+)d Fl(P)801 426 y FA(t)814 431 y Fz(0)828 426 y FA(;x)858 431 y Fz(0)877 408 y Fq(\010)904 445 y FC(0)i FB(<)h(x)1013 452 y FA(s)1044 445 y Fr(6)g FB(x)1118 426 y Fw(d)o(et)1118 456 y FA(s)1178 445 y FC(+)d FB(h)1249 404 y Fq(p)p 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FB(Q)669 2758 y FA(k)690 2751 y FB(:)1013 b FC(\(3.82\))922 2936 y(29)p eop %%Page: 30 30 30 29 bop 78 167 a FC(2.)22 b(In)15 b(ord)o(er)e(t)o(o)g(e)q(st)o(im)o (a)o(t)o(e)g FB(Q)574 174 y FA(k)595 167 y FC(,)h(w)o(e)f(in)o(tro)q(d) o(u)o(ce)g(t)n(h)o(e)g(st)o(o)q(c)o(h)o(ast)o(ic)f(pro)q(ce)q(ss)i(\()p FB(y)1361 143 y Fw(\()p FA(k)q Fw(\))1359 173 y FA(s)1409 167 y FC(\))1427 176 y FA(s)p Fx(2)p Fw([)p FA(u)1497 182 y Fg(k)1516 176 y FA(;u)1546 182 y Fg(k)q Fz(+1)1604 176 y Fw(])1630 167 y FC(d)o(e\014n)o(e)q(d)g(b)o(y)557 280 y FB(y)581 261 y Fw(\()p FA(k)q Fw(\))579 291 y FA(s)642 280 y FC(=)e FB(\032)713 287 y FA(k)742 280 y FC(e)p 765 244 18 2 v -19 x FA(\013)p Fw(\()p FA(s;u)845 267 y Fg(k)864 261 y Fw(\))p FA(=")921 280 y FC(+)977 249 y FB(\033)p 961 269 60 2 v 961 279 a Fs(p)p 999 279 22 2 v 32 x FB(")1033 218 y Fq(Z)1078 231 y FA(s)1058 321 y(u)1078 327 y Fg(k)1107 280 y FC(e)p 1131 244 18 2 v -19 x FA(\013)p Fw(\()p FA(s;u)p Fw(\))p FA(=")1267 280 y FC(d)p FB(W)1341 261 y Fw(\()p FA(k)q Fw(\))1335 291 y FA(u)1390 280 y FB(;)313 b FC(\(3.83\))136 407 y(wh)o(ere)16 b(\()p FB(W)334 383 y Fw(\()p FA(k)q Fw(\))328 413 y FA(u)383 407 y FC(\))401 416 y FA(u)p Fx(2)p Fw([)p FA(u)475 422 y Fg(k)494 416 y FA(;u)524 422 y Fg(k)q Fz(+1)582 416 y Fw(])610 407 y FC(i)q(s)h(t)n(h)o(e)g(Bro)o(wnian)e(mot)o(ion)g FB(W)1142 383 y Fw(\()p FA(k)q Fw(\))1136 413 y FA(u)1207 407 y FC(=)h FB(W)1301 414 y FA(u)1335 407 y Fs(\000)c FB(W)1425 414 y FA(u)1445 420 y Fg(k)1467 407 y FC(.)25 b(Not)o(e)16 b(t)n(h)o(a)o(t)f FB(y)1734 391 y Fw(\()p FA(k)q Fw(\))1800 407 y FC(i)q(s)136 464 y(t)n(h)o(e)e(solu)o(t)o(ion)f (of)h(t)n(h)o(e)g(SDE)g(\(3.72\))d(wit)n(h)j(init)o(ial)h(con)o(dit)o (ion)e FB(y)1172 440 y Fw(\()p FA(k)q Fw(\))1170 469 y FA(u)1190 475 y Fg(k)1234 464 y FC(=)g FB(\032)1305 471 y FA(k)1339 464 y FC(a)o(t)h(t)o(im)o(e)f FB(u)1516 471 y FA(k)1537 464 y FC(.)19 b(W)l(e)14 b(d)o(e\014n)o(e)f(t)n(h)o(e) 136 520 y(st)o(o)o(p)o(pin)o(g)i(t)o(im)o(e)q(s)589 604 y FB(\034)614 585 y Fw(0)646 604 y FC(=)e(inf)749 567 y Fq(\010)775 604 y FB(s)g Fs(2)g FC([)p FB(u)891 611 y FA(k)912 604 y FB(;)8 b(u)959 611 y FA(k)q Fw(+1)1024 604 y FC(])d(:)15 b FB(y)1094 585 y Fw(\()p FA(k)q Fw(\))1092 615 y FA(s)1155 604 y FC(=)e(0)1226 567 y Fq(\011)579 682 y FB(\034)604 663 y Fw(+)646 682 y FC(=)g(inf)749 645 y Fq(\010)775 682 y FB(s)g Fs(2)g FC([)p FB(u)891 689 y FA(k)912 682 y FB(;)8 b(u)959 689 y FA(k)q Fw(+1)1024 682 y FC(])d(:)15 b FB(y)1094 663 y Fw(\()p FA(k)q Fw(\))1092 693 y FA(s)1155 682 y FC(=)e FB(h)1229 641 y Fq(p)p 1274 641 80 2 v 41 x FB(\020)s FC(\()p FB(s)p FC(\))1361 645 y Fq(\011)1716 641 y FC(\(3.84\))136 779 y(d)o(e)q(scr)q(ibin)o(g)j(t)n (h)o(e)e(t)o(im)o(e)g(wh)o(en)h FB(y)665 755 y Fw(\()p FA(k)q Fw(\))663 785 y FA(s)729 779 y FC(e)q(it)n(h)o(er)g(reac)o(h)o (e)q(s)f(t)n(h)o(e)g FB(t)p FC(-axi)q(s)h(or)f(t)n(h)o(e)h(up)o(p)q(er) f(b)q(ou)o(n)o(d)o(ary)g FB(h)1693 740 y Fq(p)p 1739 740 V 1739 779 a FB(\020)s FC(\()p FB(s)p FC(\))o(.)136 846 y(No)o(w,)g(Lemm)o(a)h(3.11)f(implie)q(s)i(t)n(h)o(a)o(t)e(if)i FB(y)821 853 y FA(u)841 859 y Fg(k)877 846 y Fr(6)e FB(\032)950 853 y FA(k)971 846 y FC(,)h(t)n(h)o(en)h FB(y)1122 853 y FA(s)1154 846 y Fr(6)f FB(y)1228 822 y Fw(\()p FA(k)q Fw(\))1226 852 y FA(s)1292 846 y FC(for)g FB(u)1388 853 y FA(k)1423 846 y Fr(6)f FB(s)g Fr(6)h FB(\034)1582 830 y Fw(0)1612 846 y Fs(^)c FB(\034)1678 830 y Fw(+)1707 846 y FC(.)22 b(Thi)q(s)136 903 y(sh)o(o)o(ws)14 b(t)n(h)o(a)o(t)224 986 y FB(Q)260 993 y FA(k)294 986 y Fr(6)f Fl(P)376 967 y FA(u)396 973 y Fg(k)413 967 y FA(;\032)441 973 y Fg(k)462 949 y Fq(\010)488 986 y FB(\034)513 967 y Fw(0)545 986 y FB(<)g(u)619 993 y FA(k)q Fw(+1)686 949 y Fq(\011)722 986 y FC(+)e Fl(P)801 967 y FA(u)821 973 y Fg(k)838 967 y FA(;\032)866 973 y Fg(k)887 949 y Fq(\010)913 986 y FB(\034)938 967 y Fw(+)980 986 y FB(<)i(u)1054 993 y FA(k)q Fw(+1)1121 949 y Fq(\011)1157 986 y FC(+)e Fl(P)1236 967 y FA(u)1256 973 y Fg(k)1273 967 y FA(;\032)1301 973 y Fg(k)1322 949 y Fq(\010)1348 986 y FB(y)1372 967 y Fw(\()p FA(k)q Fw(\))1370 997 y FA(u)1390 1003 y Fg(k)q Fz(+1)1463 986 y FB(>)i(\032)1535 993 y FA(k)q Fw(+1)1601 949 y Fq(\011)1716 986 y FC(\(3.85\))136 1070 y(for)h(0)e Fr(6)h FB(k)h(<)f(K)g Fs(\000)d FC(1,)k(an)o(d)420 1153 y FB(Q)456 1160 y FA(K)r Fx(\000)p Fw(1)548 1153 y Fr(6)f Fl(P)630 1135 y FA(u)650 1141 y Fg(K)q Fh(\000)p Fz(1)716 1135 y FA(;\032)744 1141 y Fg(K)r Fh(\000)p Fz(1)814 1116 y Fq(\010)841 1153 y FB(\034)866 1135 y Fw(0)898 1153 y FB(<)g(u)972 1160 y FA(K)1006 1116 y Fq(\011)1043 1153 y FC(+)d Fl(P)1122 1135 y FA(u)1142 1141 y Fg(K)q Fh(\000)p Fz(1)1208 1135 y FA(;\032)1236 1141 y Fg(K)r Fh(\000)p Fz(1)1306 1116 y Fq(\010)1333 1153 y FB(\034)1358 1135 y Fw(+)1400 1153 y FB(<)j(u)1474 1160 y FA(K)1508 1116 y Fq(\011)1534 1153 y FB(:)169 b FC(\(3.86\))136 1243 y(Eac)o(h)14 b(of)h(t)n(h)o(e)q(s)q(e)f(t)o(erms)f(d)o(ep)q(en)o (ds)i(only)f(on)h FB(y)895 1226 y Fw(\()p FA(k)q Fw(\))944 1243 y FC(,)f(an)o(d)g(can)h(b)q(e)g(eas)q(ily)g(e)q(st)o(im)o(a)o(t)o (e)q(d.)i(Let)658 1352 y FB(v)682 1333 y Fw(\()p FA(k)q Fw(\))680 1363 y FA(u)700 1369 y Fg(k)q Fz(+1)773 1352 y FC(=)825 1321 y FB(\033)853 1304 y Fw(2)p 825 1341 48 2 v 839 1383 a FB(")885 1290 y Fq(Z)931 1303 y FA(u)951 1309 y Fg(k)q Fz(+1)911 1393 y FA(u)931 1399 y Fg(k)1018 1352 y FC(e)1038 1333 y Fw(2)p 1059 1315 18 2 v FA(\013)p Fw(\()p FA(u)1113 1339 y Fg(k)q Fz(+1)1170 1333 y FA(;u)p Fw(\))p FA(=")1258 1352 y FC(d)p FB(u)407 b FC(\(3.87\))136 1479 y(d)o(enot)o(e)14 b(t)n(h)o(e)h(v)n(ar)q(iance)e(of)h FB(y)606 1455 y Fw(\()p FA(k)q Fw(\))604 1485 y FA(u)624 1491 y Fg(k)q Fz(+1)685 1479 y FC(.)19 b(Th)o(en)14 b(b)o(y)h(symm)o (etry)d(\(as)i(in)h(\(2.22\))m(\),)f(w)o(e)h(h)o(a)o(v)o(e)353 1569 y Fl(P)387 1550 y FA(u)407 1556 y Fg(k)423 1550 y FA(;\032)451 1556 y Fg(k)472 1532 y Fq(\010)499 1569 y FB(\034)524 1550 y Fw(0)556 1569 y FB(<)e(u)630 1576 y FA(k)q Fw(+1)696 1532 y Fq(\011)736 1569 y FC(=)f(2)t Fl(P)843 1550 y FA(u)863 1556 y Fg(k)880 1550 y FA(;\032)908 1556 y Fg(k)929 1532 y Fq(\010)955 1569 y FB(y)979 1550 y Fw(\()p FA(k)q Fw(\))977 1580 y FA(u)997 1586 y Fg(k)q Fz(+1)1070 1569 y FB(<)h FC(0)1141 1532 y Fq(\011)736 1702 y FC(=)821 1671 y(2)p 788 1692 88 2 v 788 1701 a Fs(p)p 826 1701 51 2 v 37 x FC(2)p FB(\031)889 1640 y Fq(Z)934 1653 y Fx(\000)p FA(\032)979 1659 y Fg(k)1005 1653 y Fw(e)p 1022 1625 17 1 v 1020 1638 a Fg(\013)p Fz(\()p Fg(u)1070 1646 y(k)q Fz(+1)1128 1638 y Fg(;u)1155 1646 y(k)1175 1638 y Fz(\))p Fg(=")1220 1617 y Fq(\000)1240 1653 y FA(v)1258 1636 y Fz(\()p Fg(k)q Fz(\))1257 1659 y Fg(u)1275 1667 y(k)q Fz(+1)1336 1617 y Fq(\001)1357 1625 y Fh(\000)p Fz(1)p Fg(=)p Fz(2)914 1743 y Fx(\0001)1439 1702 y FC(e)1459 1683 y Fx(\000)p FA(z)1503 1672 y Fz(2)1521 1683 y FA(=)p Fw(2)1566 1702 y FC(d)p FB(z)736 1836 y Fr(6)g FC(exp)852 1772 y Fq(\032)886 1836 y Fs(\000)926 1806 y FC(1)p 926 1826 23 2 v 926 1867 a(2)959 1805 y FB(\032)983 1788 y Fw(2)983 1819 y FA(k)1012 1805 y FC(e)1032 1788 y Fw(2)p 1053 1771 18 2 v FA(\013)p Fw(\()p FA(u)1107 1794 y Fg(k)q Fz(+1)1164 1788 y FA(;u)1194 1794 y Fg(k)1213 1788 y Fw(\))p FA(=")p 959 1826 304 2 v 1060 1880 a FB(v)1084 1856 y Fw(\()p FA(k)q Fw(\))1082 1886 y FA(u)1102 1892 y Fg(k)q Fz(+1)1268 1772 y Fq(\033)736 1981 y Fr(6)g FC(exp)852 1917 y Fq(\032)886 1981 y Fs(\000)926 1950 y FC(1)p 926 1970 23 2 v 926 2012 a(8)962 1981 y(e)982 1962 y Fx(\000)p Fw(2)1042 1950 y FB(h)1068 1933 y Fw(2)p 1042 1970 48 2 v 1042 2012 a FB(\033)1070 1999 y Fw(2)1132 1950 y FB(\020)s FC(\()p FB(u)1199 1957 y FA(k)1220 1950 y FC(\))p 1099 1970 173 2 v 1099 2025 a FB(v)1123 2001 y Fw(\()p FA(k)q Fw(\))1121 2030 y FA(u)1141 2036 y Fg(k)q Fz(+1)1201 2025 y FB(=\033)1252 2011 y Fw(2)1276 1917 y Fq(\033)1310 1981 y FB(:)393 b FC(\(3.88\))136 2105 y(Th)o(e)10 b(s)q(econ)o(d)f(t)o(erm)g(on)h(t)n(h)o(e)g(r)q(igh)o(t-h)o (an)o(d)e(s)q(id)o(e)j(of)i(\(3.85\))7 b(or)j(\(3.86\))m(,)h(re)q(sp)q 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59 164 a FC(wh)o(ere)15 b FB(\024)f Fs(2)g FC(\(0)p FB(;)8 b FC(1\))13 b(i)q(s)j(a)f(const)o(an)o(t.)k(Not)o(e)14 b(t)n(h)o(a)o(t)g(t)n(h)o(e)h(up)o(p)q(er)g(b)q(ou)o(n)o(d)o(ary)j(~) -26 b FB(x)p FC(\()p FB(t)p FC(\))15 b(of)g Fs(D)q FC(\()p FB(\024)p FC(\))g(sa)o(t)o(i)q(s\014e)q(s)i(~)-26 b FB(x)p FC(\()p FB(t)p FC(\))14 b(=)59 185 y Fs(p)p 97 185 105 2 v 35 x FC(1)c Fs(\000)g FB(\024)t FC(\(1)f Fs(\000)h(O)q FC(\()p FB(t)p FC(\)\))p FB(x)433 204 y FA(?)452 220 y FC(\()p FB(t)p FC(\).)19 b(Our)c(\014rst)f(st)o(ep)g(t)o(o)o(w)o (ards)e(e)q(st)o(im)o(a)o(t)o(in)o(g)h FB(\034)1219 230 y Fx(D)q Fw(\()p FA(\024)p Fw(\))1312 220 y FC(i)q(s)i(t)o(o)g(e)q(st)o (im)o(a)o(t)o(e)e(t)n(h)o(e)h(\014rst)g(exit)59 277 y(t)o(im)o(e)g FB(\034)179 284 y Fx(S)220 277 y FC(f)q(rom)f(a)h(sm)o(aller)h(str)q (ip)f Fs(S)s FC(,)h(d)o(e\014n)o(e)q(d)g(b)o(y)458 401 y Fs(S)h FC(=)549 337 y Fq(\032)583 401 y FC(\()p FB(x;)8 b(t)p FC(\))k Fs(2)h FC([)p Fs(\000)p FB(\016)o(;)8 b(\016)r FC(])h Fs(\002)h FC([)p 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FC([0)p FB(;)8 b(T)e FC(].)17 b(Thi)q(s)c(sh)o(o)o(ws)f(t)n(h)o(a)o(t)g(t)n(h)o(ere)g (exi)q(st)o(s)59 476 y(a)j(const)o(an)o(t)d FB(L)g(>)h FC(0)i(su)o(c)o(h)f(t)n(h)o(a)o(t)g(t)n(h)o(e)g(con)o(dit)o(ion)g (\(3.122\))e(i)q(s)j(sa)o(t)o(i)q(s\014e)q(d)e(wh)o(en)o(ev)o(er)599 555 y Fq(\020)632 575 y FB(h)p 631 595 28 2 v 631 637 a(\033)664 555 y Fq(\021)691 566 y Fw(3+)p FA(\026)759 555 y Fq(\020)786 606 y FC(1)d(+)869 575 y FB(\013)p FC(\()p FB(u)942 582 y FA(k)q Fw(+1)1008 575 y FB(;)e(u)1055 582 y FA(k)1076 575 y FC(\))p 869 595 225 2 v 963 638 a FB(t)979 622 y Fw(3)979 651 y(2)1098 555 y Fq(\021)1138 606 y Fr(6)13 b FB(L)1231 575 y(t)1247 558 y Fw(4)1247 587 y(2)p 1226 595 48 2 v 1226 637 a FB(\033)1254 624 y Fw(2)1278 606 y FB(:)402 b FC(\(3.124\))59 735 y(Thi)q(s)14 b(con)o(dit)o(ion)h(i)q(s)f(equiv)n(alen)o(t)g(t)o(o)h(\(3.118\))l(.) 130 791 y(Assu)o(m)o(e)f Fs(j)p FB(x)334 798 y FA(u)354 804 y Fg(k)375 791 y Fs(j)388 752 y Fq(p)p 433 752 107 2 v 39 x FB(a)p FC(\()p FB(u)501 798 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y(lik)o(ely)e(t)o(o)f(lea)o(v)o(e)g Fs(D)q FC(\()p FB(\024)p FC(\))g(via)g(t)n(h)o(e)f(up)o(p)q(er)h(b)q (ou)o(n)o(d)o(ary)f(an)o(d)h(so)f(i)q(s)h FB(x)1170 2584 y FA(t)1185 2577 y FC(.)27 b(So)16 b(w)o(e)h(are)f(left)h(wit)n(h)g(t)n (h)o(e)g(cas)q(e)f(of)59 2634 y FB(x)85 2617 y Fw(0)85 2645 y FA(t)120 2634 y FC(ret)o(ur)q(nin)o(g)d(t)o(o)i(zero.)k(Thi)q(s) 14 b(ev)o(en)o(t)g(h)o(as)g(a)h(sm)o(all)f(bu)o(t)h(not)f(n)o (egligible)i(proba)o(bilit)o(y)l(.)j(Not)o(e)13 b(t)n(h)o(a)o(t)h(if)h FB(x)1811 2617 y Fw(0)1811 2645 y FA(t)59 2690 y FC(ret)o(ur)q(ns)e(t)o (o)i(zero,)g(t)n(h)o(en)g FB(x)504 2697 y FA(t)534 2690 y FC(i)q(s)h(st)o(ill)g(non-n)o(ega)o(t)o(iv)o(e.)j(If)c FB(x)1020 2697 y FA(t)1051 2690 y FC(h)o(as)f(n)o(ev)o(ert)n(h)o(ele)q (ss)h(left)h Fs(D)q FC(\()p FB(\024)p FC(\),)e(w)o(e)h(are)g(don)o(e.) 59 2747 y(If)g(not,)f FB(x)223 2754 y FA(t)252 2747 y FC(i)q(s)h(e)q(it)n(h)o(er)g(in)g Fs(S)j FC(or)c(in)i Fs(D)q FC(\()p FB(\024)p FC(\))9 b Fs(n)h(S)s FC(.)19 b(Since)d(w)o(e)e(m)o(ay)g(assu)o(m)o(e)g(t)n(h)o(a)o(t,)f(aft)o(er)g (a)h(sh)o(ort)g(t)o(im)o(e,)f FB(x)1770 2754 y FA(t)1800 2747 y FC(i)q(s)59 2803 y(in)j Fs(D)q FC(\()p FB(\024)p FC(\))9 b Fs(n)h(S)18 b FC(again,)c(w)o(e)g(can)g(rep)q(ea)o(t)g(t)n(h) o(e)g(a)o(b)q(o)o(v)o(e)g(argu)o(m)o(en)o(t.)922 2936 y(36)p eop %%Page: 37 37 37 36 bop 130 164 a FC(Makin)o(g)17 b(t)n(h)o(e)f(a)o(b)q(o)o(v)o (e-said)g(preci)q(s)q(e,)g(w)o(e)h(obt)o(ain)f(an)h(in)o(t)o(egral)e (equa)o(t)o(ion)h(for)g(an)h(up)o(p)q(er)f(b)q(ou)o(n)o(d)h(on)59 220 y(t)n(h)o(e)c(proba)o(bilit)o(y)f(t)n(h)o(a)o(t)g FB(x)477 227 y FA(u)513 220 y FC(do)q(e)q(s)h(not)f(lea)o(v)o(e)h Fs(D)q FC(\()p FB(\024)p FC(\))g(up)h(t)o(o)f(t)o(im)o(e)f FB(t)p FC(,)i(whic)o(h)f(will)i(b)q(e)e(solv)o(e)q(d)f(b)o(y)h(it)o (era)o(t)o(ions.)59 277 y(W)l(e)i(will)h(cit)o(e)f(t)n(h)o(e)f(in)o(t)o (egral)g(equa)o(t)o(ion)f(f)q(rom)g([BG],)g(as)i(t)n(h)o(e)f(gen)o (eral)g(argu)o(m)o(en)o(t)o(s)e(leadin)o(g)k(t)o(o)e(it)h(do)g(not)59 333 y(require)g(ad)o(apt)o(a)o(t)o(ion.)20 b(Let)c(us)g(\014rst)f(in)o (tro)q(d)o(u)o(ce)f(t)n(h)o(e)i(n)o(ece)q(ssary)e(not)o(a)o(t)o(ions.) 20 b(W)l(e)d(c)o(h)o(o)q(os)q(e)e FB(h)f FC(=)g FB(K)s(\033)k FC(for)59 390 y(som)o(e)c(\(p)q(oss)q(ibly)g(large\))g(const)o(an)o(t)e FB(K)k(>)d FC(0.)19 b(F)l(or)14 b FB(\024)f Fs(2)g FC(\(0)p FB(;)8 b FC(1\),)k(w)o(e)i(c)o(h)o(o)q(os)q(e)g FB(\026)f(>)g FC(0)i(in)g(su)o(c)o(h)g(a)f(w)o(ay)g(t)n(h)o(a)o(t)380 469 y(1)p 380 489 23 2 v 380 531 a(2)451 469 y FB(\026)p 412 489 106 2 v 412 531 a FC(1)c(+)g FB(\026)522 449 y Fq(h)544 500 y FC(1)f Fs(\000)i(O)659 449 y Fq(\020)751 469 y FC(1)p 691 489 143 2 v 691 531 a FB(\026)d FC(log)g FB(K)839 449 y Fq(\021i)900 500 y Fr(6)13 b FB(\024)g Fr(6)1079 469 y FB(\026)p 1040 489 106 2 v 1040 531 a FC(1)c(+)i FB(\026)1150 449 y Fq(h)1171 500 y FC(1)f Fs(\000)g(O)1286 449 y Fq(\020)1379 469 y FC(1)p 1319 489 143 2 v 1319 531 a FB(\026)e FC(log)g FB(K)1467 449 y Fq(\021i)1693 500 y FC(\(3.129\))59 610 y(for)17 b(all)h(large)g (enough)f FB(K)s FC(.)28 b(Not)o(e)17 b(t)n(h)o(a)o(t)f(c)o(h)o(o)q(os) q(in)o(g)h FB(\024)i FC(t)o(o)q(o)e(clos)q(e)h(t)o(o)f(1)h(require)q(s) f(large)g FB(\026)i FC(an)o(d)e(i)q(s)h(t)n(h)n(us)59 666 y(not)d(d)o(e)q(s)q(ira)o(ble.)23 b(Since)17 b(w)o(e)e(w)o(an)o(t)g (t)o(o)g(ap)o(ply)h(Pro)o(p)q(os)q(it)o(ion)e(3.14)g(on)i(t)n(h)o(e)g (\014rst)f(exit)h(t)o(im)o(e)f(f)q(rom)g Fs(S)k FC(wit)n(h)59 723 y FB(t)75 730 y Fw(2)115 723 y Fr(>)i FB(c)191 730 y Fw(2)210 690 y Fs(p)p 248 690 28 2 v 33 x FB(\033)g FC(for)e(a)g(suit)o(a)o(bly)f(c)o(h)o(os)q(en)h FB(c)753 730 y Fw(2)772 723 y FC(,)h(Con)o(dit)o(ion)e(\(3.118\))e(m)n(ust)j(b)q (e)g(sa)o(t)o(i)q(s\014e)q(d.)32 b(Th)o(erefore,)17 b(w)o(e)59 779 y(c)o(h)o(o)q(os)q(e)d FB(c)222 786 y Fw(2)254 779 y FC(=)f FB(c)322 786 y Fw(2)341 779 y FC(\()p FB(K)s FC(\))h(large)g(enough)g(for)641 889 y FB(K)683 871 y Fw(3+)p FA(\026)752 839 y Fq(\020)779 889 y FC(1)9 b(+)861 859 y(1)h(+)g FB(\026)p 861 879 106 2 v 894 921 a(c)914 906 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a FB(")1151 1608 y FC(e)1172 1589 y Fx(\000)p FA(\024\013)p Fw(\()p FA(u;s)p Fw(\))p FA(=")1352 1571 y Fq(\002)1371 1608 y FC(1)h(+)i FB(g)r FC(\()p FB(u;)d(s)p FC(\))1577 1571 y Fq(\003)1601 1608 y FC(d)p FB(u:)28 b FC(\(3.133\))59 1724 y(Th)o(en,)14 b([BG,)g(\(4.95\))e(an)o(d)i(\(4.107\)])e(sh)o(o)o (w)h(t)n(h)o(a)o(t)g(for)h(an)o(y)g FB(n)h FC(an)o(d)g FB(s)e Fr(>)g FB(t)1261 1731 y Fw(2)1293 1724 y Fr(>)g FB(c)1361 1731 y Fw(2)1381 1692 y Fs(p)p 1419 1692 28 2 v 32 x FB(\033)q FC(,)170 1843 y(sup)121 1885 y FA(x)t Fw(:)e(\()p FA(x;s)p Fw(\))p Fx(2S)295 1843 y Fl(P)328 1825 y FA(s;x)373 1807 y Fq(\010)400 1843 y FB(\034)420 1853 y Fx(D)q Fw(\()p FA(\024)p Fw(\))510 1843 y Fr(>)i FB(t)574 1807 y Fq(\011)614 1843 y Fr(6)g FC(2)685 1793 y Fq(\020)717 1813 y FB(h)p 717 1833 V 717 1875 a(\033)749 1793 y Fq(\021)776 1804 y FA(\026)807 1843 y FC(e)827 1825 y Fx(\000)p FA(\024\013)p Fw(\()p FA(t;s)p Fw(\))p FA(=")1008 1843 y FC(+)p FB(\024)1069 1793 y Fq(\020)1102 1813 y FB(h)p 1101 1833 V 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n(h)o(a)o(t)h(if)i FB(x)1657 2001 y FA(t)1689 1994 y FC(reac)o(h)o(e)q(s)59 2050 y FB(\016)79 2057 y Fw(0)99 2050 y FC(,)e(an)o(d)g FB(\016)234 2057 y Fw(0)269 2050 y FC(i)q(s)h(clos)q(e)f(enough)g(t)o(o)f FB(x)661 2034 y FA(?)661 2062 y Fx(\000)691 2050 y FC(\()p FB(t)p FC(\))h(\(bu)o(t)g (it)g(m)o(ay)f(st)o(ill)j(b)q(e)e(a)o(t)f(a)h(di)q(st)o(ance)f(of)h (ord)o(er)f(1\),)g(t)n(h)o(en)h(it)h(i)q(s)59 2107 y(lik)o(ely)g(t)o(o) e(reac)o(h)h(a)f(n)o(e)q(igh)o(b)q(ourh)o(o)q(o)q(d)g(of)g FB(x)773 2090 y FA(?)773 2118 y Fx(\000)802 2107 y FC(\()p FB(t)p FC(\))h(as)f(w)o(ell.)130 2163 y(Let)g FB(\016)230 2170 y Fw(0)263 2163 y FB(<)f(\016)331 2170 y Fw(1)363 2163 y FB(<)g(x)437 2170 y Fw(c)468 2163 y FB(<)g(\016)536 2170 y Fw(2)571 2163 y FC(b)q(e)h(t)n(h)o(e)h(const)o(an)o(t)o(s)d(sa)o (t)o(i)q(sfyin)o(g)i(\(2.46\))m(,)h(t)n(h)o(a)o(t)e(i)q(s,)476 2261 y FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))k Fs(\020)h(\000)p FC(1)179 b(for)14 b FB(\016)988 2268 y Fw(0)1021 2261 y Fr(6)f FB(x)f Fr(6)h FB(\016)1175 2268 y Fw(1)1210 2261 y FC(an)o(d)i Fs(j)p FB(t)p Fs(j)d Fr(6)h FB(T)446 2330 y(@)470 2337 y FA(xx)511 2330 y FB(f)5 b FC(\()p FB(x;)j(t)p FC(\))k Fr(6)h FC(0)179 b(for)14 b FB(\016)988 2337 y Fw(1)1021 2330 y Fr(6)f FB(x)f Fr(6)h FB(\016)1175 2337 y Fw(2)1210 2330 y FC(an)o(d)i Fs(j)p FB(t)p Fs(j)d Fr(6)h FB(T)6 b FC(.)1716 2296 y(\(4.20\))59 2426 y(Th)o(e)14 b(bas)q(ic)h(in)o(gre)q(dien)o(t)f(of)h(our)f(e)q(st)o(im)o(a)o(t)o(e)f (i)q(s)i(t)n(h)o(e)f(follo)o(win)o(g)g(an)o(alogue)f(of)i(Lemm)o(a)e (3.11:)59 2517 y Fo(Lemm)o(a)i(4.4.)23 b Fy(Fix)17 b(some)h(initial)f (time)g FB(t)795 2524 y Fw(0)831 2517 y Fs(2)e FC([)p Fs(\000)p FB(T)t(;)8 b(T)e FC(])p Fy(.)23 b(We)17 b(c)n(onsider)g(the)h (fol)r(lowing)g(pr)n(o)n(c)n(esses)e(on)59 2573 y FC([)p FB(t)88 2580 y Fw(0)108 2573 y FB(;)8 b(T)e FC(])p Fy(:)90 2642 y Fs(\017)24 b Fy(the)16 b(solution)h FB(x)408 2625 y Fw(d)o(et)408 2653 y FA(t)475 2642 y Fy(of)f(the)g(deterministic)f (di\013er)n(ential)g(e)n(quation)k 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Fy(,)16 b(which)f(satis\014es)i FB(y)1031 2818 y FA(t)1044 2823 y Fz(0)1076 2811 y FC(=)c FB(x)1150 2818 y FA(t)1163 2823 y Fz(0)1192 2811 y Fs(\000)e FB(x)1264 2795 y Fw(d)o(et)1264 2822 y FA(t)1277 2827 y Fz(0)1327 2811 y Fr(>)i FC(0)p Fy(;)922 2936 y FC(40)p eop %%Page: 41 41 41 40 bop 90 164 a Fs(\017)24 b Fy(the)15 b(solution)i FB(y)405 147 y Fw(0)403 175 y FA(t)425 164 y Fy(of)f(the)f(line)n(ar)h (SDE)414 285 y FC(d)p FB(y)463 266 y Fw(0)461 296 y FA(t)495 285 y FC(=)548 254 y(1)p 548 274 23 2 v 549 316 a FB(")576 285 y FC(~)-24 b FB(a)p FC(\()p FB(t)p FC(\))p FB(y)675 266 y Fw(0)673 296 y FA(t)702 285 y FC(d)p FB(t)10 b FC(+)819 254 y FB(\033)p 803 274 60 2 v 803 284 a Fs(p)p 841 284 22 2 v 32 x FB(")875 285 y FC(d)p FB(W)943 292 y FA(t)958 285 y FB(;)99 b Fy(wher)n(e)15 b FC(~)-23 b FB(a)p FC(\()p FB(t)p FC(\))12 b(=)g FB(@)1356 292 y FA(x)1378 285 y FB(f)5 b FC(\()p FB(x)1449 266 y Fw(d)o(et)1449 296 y FA(t)1500 285 y FB(;)j(t)p FC(\))161 b(\(4.21\))137 417 y Fy(with)15 b(initial)h(c)n(ondition)g FB(y)587 400 y Fw(0)585 428 y FA(t)598 433 y Fz(0)630 417 y Fs(2)d FC([)p FB(y)708 424 y FA(t)721 429 y Fz(0)740 417 y FB(;)8 b(\016)781 424 y Fw(2)810 417 y Fs(\000)i FB(x)881 400 y Fw(d)o(et)881 428 y FA(t)894 433 y Fz(0)932 417 y FC(])p Fy(.)59 485 y(If)24 b FB(\016)134 492 y Fw(1)167 485 y Fr(6)13 b FB(y)239 468 y Fw(0)237 496 y FA(s)264 485 y FC(+)6 b FB(x)331 468 y Fw(d)o(et)331 496 y FA(s)395 485 y Fr(6)13 b FB(\016)463 492 y Fw(2)497 485 y Fy(for)h(al)r(l)h FB(s)e Fs(2)g FC([)p FB(t)735 492 y Fw(0)755 485 y FB(;)8 b(t)p FC(])p Fy(,)13 b(then)i FB(y)953 492 y FA(s)984 485 y Fr(6)e FB(y)1056 468 y Fw(0)1054 496 y FA(s)1090 485 y Fy(for)h(those)g FB(s)p Fy(.)20 b(Similarly,)15 b(if)23 b FB(\016)1605 492 y Fw(1)1638 485 y Fr(6)13 b FB(x)1712 492 y FA(s)1743 485 y Fr(6)g FB(\016)1811 492 y Fw(2)59 541 y Fy(for)j(al)r(l)h FB(s)c Fs(2)g FC([)p FB(t)301 548 y Fw(0)321 541 y FB(;)8 b(t)p FC(])p Fy(,)15 b(then)g FB(y)523 525 y Fw(0)521 553 y FA(s)556 541 y Fr(>)e FB(y)626 548 y FA(s)660 541 y Fy(for)j(those)g FB(s)p Fy(.)21 b(The)16 b(r)n(esult)g(r)n(emains)f(true)h(when)g FB(t)g Fy(is)h(r)n(eplac)n(e)n(d)d(by)i(a)59 598 y(stopping)g(time.)130 692 y FC(W)l(e)e(will)h(pro)q(cee)q(d)f(as)f(follo)o(ws.)k(Let)d FB(\020)s FC(\()p FB(t)p FC(\))g(b)q(e)f(t)n(h)o(e)h(fu)o(nct)o(ion)f (d)o(e\014n)o(e)q(d)h(in)h(\(4.18\))m(,)e(an)o(d)h(let)g FB(h)g FC(b)q(e)f(su)o(c)o(h)59 748 y(t)n(h)o(a)o(t)j FB(x)181 732 y Fw(d)o(et)181 759 y FA(s)244 748 y FC(+)c FB(h)317 709 y Fq(p)p 363 709 80 2 v 363 748 a FB(\020)s FC(\()p FB(s)p FC(\))17 b FB(<)h(\016)533 755 y Fw(2)571 748 y FC(for)f(all)i FB(s)f Fs(2)g FC([)p FB(t)825 755 y Fw(0)844 748 y FB(;)8 b(t)p FC(].)28 b(Giv)o(en)19 b FB(x)1095 755 y Fw(0)1132 748 y Fs(2)f FC(\()p FB(\016)1218 755 y Fw(1)1238 748 y FB(;)8 b(\016)1279 755 y Fw(2)1298 748 y FC(\))17 b(an)o(d)h(t)o(im)o(e)q(s)f FB(t)1561 755 y Fw(0)1599 748 y FB(<)h(t)1668 755 y Fw(1)1706 748 y FB(<)g(t)g FC(in)59 804 y([)p Fs(\000)p FB(T)t(;)8 b(T)e FC(],)15 b(w)o(e)h(cons)q(id)o(er)g(t)n(h)o(e)h(solu)o(t)o(ion)f FB(x)751 811 y FA(t)783 804 y FC(of)g(t)n(h)o(e)g(SDE)h(\(4.1\))e(wit)n (h)h(init)o(ial)i(con)o(dit)o(ion)e FB(x)1585 811 y FA(t)1598 816 y Fz(0)1633 804 y FC(=)g FB(x)1710 811 y Fw(0)1730 804 y FC(.)25 b(W)l(e)59 861 y(in)o(tro)q(d)o(u)o(ce)14 b(t)n(h)o(e)g(st)o(o)o(p)o(pin)o(g)g(t)o(im)o(e)499 963 y FB(\034)k FC(=)12 b(inf)639 926 y Fq(\010)665 963 y FB(s)h Fs(2)g FC([)p FB(t)771 970 y Fw(0)791 963 y FB(;)8 b(t)828 970 y Fw(1)847 963 y FC(])d(:)14 b FB(x)918 970 y FA(s)949 963 y Fr(6)f FB(\016)1017 970 y Fw(1)1037 926 y Fq(\011)1076 963 y Fs(2)g FC([)p FB(t)1148 970 y Fw(0)1168 963 y FB(;)8 b(t)1205 970 y Fw(1)1224 963 y FC(])i Fs([)g(f1g)p FB(:)325 b FC(\(4.22\))59 1065 y(W)l(e)16 b(can)e(t)n(h)n(us)h(wr)q(it)o(e)151 1167 y Fl(P)184 1148 y FA(t)197 1153 y Fz(0)211 1148 y FA(;x)241 1153 y Fz(0)260 1130 y Fq(\010)287 1167 y FB(x)313 1174 y FA(s)344 1167 y FB(>)e(\016)412 1174 y Fw(0)457 1167 y Fs(8)p FB(s)g Fs(2)g FC([)p FB(t)588 1174 y Fw(0)608 1167 y FB(;)8 b(t)p FC(])658 1130 y Fq(\011)696 1167 y Fr(6)13 b Fl(P)778 1148 y FA(t)791 1153 y Fz(0)805 1148 y FA(;x)835 1153 y Fz(0)854 1130 y Fq(\010)880 1167 y FB(x)906 1174 y FA(s)937 1167 y FB(>)g(\016)1005 1174 y Fw(1)1051 1167 y Fs(8)p FB(s)g Fs(2)g FC([)p FB(t)1182 1174 y Fw(0)1201 1167 y FB(;)8 b(t)1238 1174 y Fw(1)1258 1167 y FC(])1271 1130 y Fq(\011)754 1253 y FC(+)i Fl(E)830 1234 y FA(t)843 1239 y Fz(0)863 1234 y FA(;x)893 1239 y Fz(0)912 1203 y Fq(n)942 1253 y FC(1)965 1262 y Fx(f)p FA(\034)t Fm(6)p FA(t)1043 1267 y Fz(1)1059 1262 y Fx(g)1079 1253 y Fl(P)1112 1234 y FA(\034)r(;\016)1155 1239 y Fz(1)1172 1216 y Fq(\010)1198 1253 y FB(x)1224 1260 y FA(s)1255 1253 y FB(>)j(\016)1323 1260 y Fw(0)1368 1253 y Fs(8)p FB(s)h Fs(2)f FC([)p FB(\034)s(;)8 b(t)p FC(])1557 1216 y Fq(\011)1581 1203 y(o)1611 1253 y FB(:)92 b FC(\(4.23\))59 1364 y(Th)o(e)14 b(\014rst)g(t)o(erm)g(can)g(b)q(e)h(furt)n(h)o(er)e(e) q(st)o(im)o(a)o(t)o(e)q(d)h(b)o(y)99 1493 y Fl(P)133 1474 y FA(t)145 1479 y Fz(0)160 1474 y FA(;x)190 1479 y Fz(0)209 1456 y Fq(\010)235 1493 y FB(x)261 1500 y FA(s)292 1493 y FB(>)f(\016)360 1500 y Fw(1)405 1493 y Fs(8)p FB(s)g Fs(2)g FC([)p FB(t)536 1500 y Fw(0)556 1493 y FB(;)8 b(t)593 1500 y Fw(1)612 1493 y FC(])625 1456 y Fq(\011)664 1493 y Fr(6)13 b Fl(P)746 1474 y FA(t)758 1479 y Fz(0)773 1474 y FA(;x)803 1479 y Fz(0)822 1443 y Fq(n)884 1493 y FC(sup)852 1531 y FA(t)865 1536 y Fz(0)882 1531 y Fm(6)p FA(s)p Fm(6)p FA(t)965 1536 y Fz(1)996 1462 y FB(x)1022 1469 y FA(s)1050 1462 y Fs(\000)e FB(x)1122 1446 y Fw(d)o(et)1122 1473 y FA(s)p 996 1483 177 2 v 1022 1492 a Fq(p)p 1067 1492 80 2 v 38 x FB(\020)s FC(\()p FB(s)p FC(\))1190 1493 y FB(>)i(h)1264 1443 y Fq(o)722 1606 y FC(+)d Fl(P)801 1587 y FA(t)814 1592 y Fz(0)828 1587 y FA(;x)858 1592 y Fz(0)877 1569 y Fq(\010)904 1606 y FB(\016)924 1613 y Fw(1)956 1606 y FB(<)j(x)1030 1613 y FA(s)1061 1606 y Fr(6)g FB(x)1135 1587 y Fw(d)o(et)1135 1617 y FA(s)1196 1606 y FC(+)d FB(h)1267 1565 y Fq(p)p 1312 1565 V 41 x FB(\020)s FC(\()p FB(s)p FC(\))25 b Fs(8)p FB(s)13 b Fs(2)g FC([)p FB(t)1548 1613 y Fw(0)1568 1606 y FB(;)8 b(t)1605 1613 y Fw(1)1624 1606 y FC(])1637 1569 y Fq(\011)1663 1606 y FB(:)40 b FC(\(4.24\))59 1708 y(Th)o(e)16 b(t)o(w)o(o)e(su)o(mm)o(an)o(ds)g(in)j(\(4.24\))d(can)h(b)q (e)h(e)q(st)o(im)o(a)o(t)o(e)q(d)f(in)i(a)f(s)q(imilar)f(w)o(ay)g(as)h (in)h(t)n(h)o(e)e(symm)o(etr)q(ic)f(cas)q(e.)59 1764 y(Th)o(e)g(\014rst)g(on)o(e)h(i)q(s)g(d)o(eal)o(t)g(wit)n(h)g(in)g(t)n (h)o(e)g(follo)o(win)o(g)f(re)q(sul)o(t.)59 1858 y Fo(Pro)o(p)q(o)q(s)q (it)o(ion)j(4.5.)23 b Fy(Assume)16 b FB(\016)627 1865 y Fw(1)660 1858 y Fr(6)d FB(x)734 1865 y Fw(0)766 1858 y Fr(6)g FB(x)840 1842 y Fw(d)o(et)840 1869 y FA(t)853 1874 y Fz(0)901 1858 y FC(+)951 1840 y Fw(1)p 951 1847 18 2 v 951 1874 a(2)974 1858 y FB(h)1000 1819 y Fq(p)p 1045 1819 95 2 v 39 x FB(\020)s FC(\()p FB(t)1102 1865 y Fw(0)1122 1858 y FC(\))p Fy(.)20 b(Then)349 1994 y Fl(P)383 1975 y FA(t)396 1980 y Fz(0)410 1975 y FA(;x)440 1980 y Fz(0)459 1944 y Fq(n)521 1994 y FC(sup)489 2032 y FA(t)502 2037 y Fz(0)520 2032 y Fm(6)p FA(s)p Fm(6)p FA(t)603 2037 y Fz(1)633 1963 y FB(x)659 1970 y FA(s)688 1963 y Fs(\000)10 b FB(x)759 1947 y Fw(d)o(et)759 1974 y FA(s)p 633 1984 177 2 v 659 1993 a Fq(p)p 704 1993 80 2 v 39 x FB(\020)s FC(\()p FB(s)p FC(\))827 1994 y FB(>)j(h)901 1944 y Fq(o)944 1994 y Fr(6)997 1963 y FC(5)p 997 1984 23 2 v 997 2025 a(2)1025 1944 y Fq(\020)1057 1963 y Fs(j)p 1075 1938 22 2 v FB(\013)p FC(\()p FB(t)1133 1970 y Fw(1)1153 1963 y FB(;)8 b(t)1190 1970 y Fw(0)1209 1963 y FC(\))p Fs(j)p 1057 1984 182 2 v 1137 2025 a FB(")1254 1994 y FC(+)i(1)1322 1944 y Fq(\021)1356 1994 y FC(e)1376 1975 y Fx(\000)p FA(\024h)1443 1964 y Fz(2)1462 1975 y FA(=\033)1501 1964 y Fz(2)1520 1994 y FB(;)183 b FC(\(4.25\))59 2139 y Fy(wher)n(e)15 b FB(\024)h Fy(is)g(a)g(p)n(ositive)f(c)n (onstant)i(and)p 749 2114 22 2 v 16 w FB(\013)p FC(\()p FB(t)807 2146 y Fw(1)827 2139 y FB(;)8 b(t)864 2146 y Fw(0)883 2139 y FC(\))k(=)961 2102 y Fq(R)991 2116 y FA(t)1004 2121 y Fz(1)982 2155 y FA(t)995 2160 y Fz(0)1032 2139 y FC(\026)-24 b FB(a)p FC(\()p FB(s)p FC(\))8 b(d)o FB(s)p Fy(.)130 2233 y FC(Th)o(e)j(pro)q(of)g(i)q(s)h(almost)f(t)n(h)o (e)h(sam)o(e)f(as)g(t)n(h)o(e)h(pro)q(of)e(of)i(Pro)o(p)q(os)q(it)o (ion)e(3.12.)17 b(Inst)o(ead)11 b(of)k(\(3.84\))m(,)e(w)o(e)e(m)o(ay)59 2289 y(d)o(e\014n)o(e)j FB(\034)211 2273 y Fw(0)244 2289 y FC(an)o(d)f FB(\034)354 2273 y Fw(+)397 2289 y FC(as)g(t)n(h)o(e)g (\014rst)f(t)o(im)o(e)q(s)h(wh)o(en)g FB(y)872 2265 y Fw(\()p FA(k)q Fw(\))870 2295 y FA(s)934 2289 y FC(e)q(it)n(h)o(er)g (reac)o(h)o(e)q(s)g FB(\016)1234 2296 y Fw(1)1261 2289 y Fs(\000)7 b FB(x)1329 2273 y Fw(d)o(et)1329 2301 y FA(s)1393 2289 y FC(or)13 b(t)n(h)o(e)g(up)o(p)q(er)g(b)q(ou)o(n)o(d)o (ary)59 2346 y FB(h)85 2307 y Fq(p)p 131 2307 80 2 v 131 2346 a FB(\020)s FC(\()p FB(s)p FC(\))o(.)25 b(Th)o(en)16 b(Lemm)o(a)f(4.4)g(implie)q(s)j FB(y)779 2353 y FA(s)813 2346 y Fr(6)e 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(of)k(\(4.18\))12 b(an)o(d)i(Pro)o(p)q(os)q(it)o(ion)f(4.3.)922 2936 y(41)p eop %%Page: 42 42 42 41 bop 130 164 a FC(Let)14 b(us)h(no)o(w)f(examin)o(e)g(t)n(h)o(e)g (s)q(econ)o(d)g(t)o(erm)g(in)h(\(4.24\))m(.)59 266 y Fo(Pro)o(p)q(o)q(s)q(it)o(ion)i(4.6.)23 b Fy(Ther)n(e)14 b(exist)h(c)n(onstants)g FB(c)888 273 y Fw(1)920 266 y FB(>)e FC(0)i Fy(and)i FC(\026)-25 b FB(\024)13 b(>)g FC(0)i Fy(such)g(that,)g(if)g FB(c)1486 242 y Fw(3)p FA(=)p Fw(2)1486 279 y(1)1541 266 y FB(\033)f Fr(>)f FB(a)1653 242 y Fw(3)p FA(=)p Fw(4)1653 279 y(0)1716 266 y Fs(_)c FB(")1776 249 y Fw(3)p FA(=)p Fw(4)59 322 y Fy(and)16 b FB(h)d(>)g FC(2)p FB(\033)r Fy(,)i(then)284 442 y Fl(P)318 424 y Fx(\000)p FA(c)360 429 y Fz(1)375 424 y FA(\033)396 412 y Fz(2)p Fg(=)p Fz(3)445 424 y FA(;x)475 429 y Fz(0)494 405 y Fq(\010)520 442 y FB(\016)540 449 y Fw(1)573 442 y FB(<)e(x)647 449 y FA(s)678 442 y Fr(6)g FB(x)752 424 y Fw(d)o(et)752 454 y FA(s)812 442 y FC(+)e FB(h)884 401 y Fq(p)p 929 401 80 2 v 41 x FB(\020)s FC(\()p FB(s)p FC(\))25 b Fs(8)p FB(s)13 b Fs(2)g FC([)p Fs(\000)p FB(c)1204 449 y Fw(1)1223 442 y FB(\033)1251 424 y Fw(2)p FA(=)p Fw(3)1306 442 y FB(;)8 b(t)1343 449 y Fw(1)1362 442 y FC(])1375 405 y Fq(\011)468 548 y Fr(6)521 517 y FC(3)p 521 538 23 2 v 521 579 a(2)556 548 y(exp)625 484 y Fq(\032)659 548 y Fs(\000)r FC(\026)-25 b FB(\024)891 517 y FC(1)p 733 538 339 2 v 733 579 a(log\()p FB(h=\033)r FC(\))9 b Fs(_)h(j)p FC(log)d FB(\033)r Fs(j)1085 517 y Fq(b)-29 b FB(\013)q FC(\()p FB(t)1145 524 y Fw(1)1164 517 y FB(;)8 b Fs(\000)p FB(c)1240 524 y Fw(1)1259 517 y FB(\033)1287 501 y Fw(2)p FA(=)p Fw(3)1342 517 y FC(\))p 1081 538 278 2 v 1210 579 a FB(")1364 484 y Fq(\033)1716 548 y FC(\(4.26\))59 685 y Fy(holds)21 b(with)i Fq(b)-28 b FB(\013)p FC(\()p FB(t;)8 b(s)p FC(\))19 b(=)477 648 y Fq(R)507 661 y FA(t)498 701 y(s)529 685 y Fq(b)-24 b FB(a)p FC(\()p FB(u)p FC(\))8 b(d)n FB(u)p Fy(,)21 b(for)f Fs(\000)p FB(c)839 692 y Fw(1)859 685 y FB(\033)887 668 y Fw(2)p FA(=)p Fw(3)961 685 y Fr(6)h FB(t)1033 692 y Fw(1)1073 685 y Fr(6)f FB(c)1148 692 y Fw(1)1168 685 y FB(\033)1196 668 y Fw(2)p FA(=)p Fw(3)1270 685 y Fy(and)h(al)r(l)g (initial)f(c)n(onditions)g FB(x)1811 692 y Fw(0)59 749 y Fy(satisfying)c FB(\016)279 756 y Fw(1)311 749 y Fr(6)d FB(x)385 756 y Fw(0)418 749 y Fr(6)g FB(x)492 733 y Fw(d)o(et)492 769 y Fx(\000)p FA(c)534 774 y Fz(1)552 769 y FA(\033)573 759 y Fz(2)p Fg(=)p Fz(3)634 749 y FC(+)d FB(h)705 709 y Fq(p)p 750 709 216 2 v 40 x FB(\020)s FC(\()p Fs(\000)p FB(c)846 756 y Fw(1)866 749 y FB(\033)894 736 y Fw(2)p FA(=)p Fw(3)949 749 y FC(\))o Fy(.)59 881 y Fd(Pr)o(oof:)78 949 y FC(1.)22 b(Let)14 b Fq(b)-27 b FB(x)244 932 y Fw(d)o(et)240 960 y FA(t)307 949 y FC(b)q(e)12 b(t)n(h)o(e)f(d)o(et)o(ermini)q(st)o (ic)g(solu)o(t)o(ion)g(trac)o(kin)o(g)g(t)n(h)o(e)h(saddle)g(a)o(t)f FB(x)1322 932 y FA(?)1322 961 y Fw(0)1341 949 y FC(\()p FB(t)p FC(\))h(an)o(d)g(s)q(et)f FB(x)1583 956 y FA(t)1610 949 y FC(=)k Fq(b)-27 b FB(x)1688 932 y Fw(d)o(et)1684 960 y FA(t)1743 949 y FC(+)t FB(z)1803 956 y FA(t)1818 949 y FC(.)136 1005 y(Th)o(en)599 1115 y(d)p FB(z)645 1122 y FA(t)673 1115 y FC(=)725 1084 y(1)p 725 1105 23 2 v 726 1146 a FB(")753 1078 y Fq(\002)771 1115 y(b)j FB(a)p FC(\()p FB(t)p FC(\))p FB(z)869 1122 y FA(t)893 1115 y FC(+)935 1103 y Fq(b)938 1115 y FB(b)p FC(\()p FB(z)997 1122 y FA(t)1011 1115 y FB(;)8 b(t)p FC(\))1066 1078 y Fq(\003)1092 1115 y FC(d)p FB(t)i FC(+)1209 1084 y FB(\033)p 1194 1105 60 2 v 1194 1114 a Fs(p)p 1231 1114 22 2 v 1231 1147 a FB(")1265 1115 y FC(d)p FB(W)1333 1122 y FA(t)1348 1115 y FB(;)355 b FC(\(4.27\))136 1241 y(wh)o(ere)14 b(\(4.17\))m(,)h(\(4.18\))d(an)o(d)i(\(4.20\))e(imply)701 1363 y Fq(b)-25 b FB(a)p FC(\()p FB(t)p FC(\))12 b Fs(\020)h(j)p FB(t)p Fs(j)d(_)929 1331 y(p)p 967 1331 44 2 v 32 x FB(a)991 1370 y Fw(0)1021 1363 y Fs(_)1062 1328 y(p)p 1099 1328 22 2 v 1099 1363 a FB(")j Fs(\020)1212 1332 y FC(1)p 1186 1352 75 2 v 1186 1394 a FB(\020)s FC(\()p FB(t)p FC(\))1716 1363 y(\(4.28\))136 1487 y(an)o(d)582 1577 y Fq(b)585 1589 y FB(b)o FC(\()p FB(z)r(;)8 b(t)p FC(\))j Fr(6)i FC(0)180 b(for)15 b Fq(b)-26 b FB(x)1061 1570 y Fw(d)o(et)1057 1600 y FA(t)1121 1589 y FC(+)10 b FB(z)15 b Fs(2)e FC([)p FB(\016)1278 1596 y Fw(1)1297 1589 y FB(;)8 b(\016)1338 1596 y Fw(2)1358 1589 y FC(].)332 b(\(4.29\))136 1697 y(Let)13 b FB(\045)f FC(=)g FB(\045)p FC(\()p FB(h=\033)r FC(\))f Fr(>)i FC(1)f(an)o(d)g(d)o(e\014n)o(e)g(a)h (part)o(it)o(ion)d Fs(\000)p FB(c)1011 1704 y Fw(1)1031 1697 y FB(\033)1059 1680 y Fw(2)p FA(=)p Fw(3)1126 1697 y FC(=)j FB(u)1200 1704 y Fw(0)1232 1697 y FB(<)g Fs(\001)8 b(\001)g(\001)j FB(<)i(u)1420 1704 y FA(K)1467 1697 y FC(=)f FB(t)1530 1704 y Fw(1)1563 1697 y FC(of)g([)p Fs(\000)p FB(c)1680 1704 y Fw(1)1699 1697 y FB(\033)1727 1680 y Fw(2)p FA(=)p Fw(3)1782 1697 y FB(;)c(t)p FC(])136 1753 y(b)o(y)431 1884 y Fq(b)-29 b FB(\013)q FC(\()p FB(u)501 1891 y FA(k)521 1884 y FB(;)8 b(u)568 1891 y FA(k)q Fx(\000)p Fw(1)634 1884 y FC(\))k(=)h FB(\045")90 b FC(for)13 b(1)f Fr(6)h FB(k)h(<)f(K)j FC(=)1186 1820 y Fq(\030)1221 1853 y(b)-28 b FB(\013)p FC(\()p FB(t)1281 1860 y Fw(1)1301 1853 y FB(;)8 b Fs(\000)p FB(c)1377 1860 y Fw(1)1396 1853 y FB(\033)1424 1837 y Fw(2)p FA(=)p Fw(3)1478 1853 y FC(\))p 1218 1874 278 2 v 1334 1915 a FB(\045")1501 1820 y Fq(\031)1527 1884 y FB(:)176 b FC(\(4.30\))136 2008 y(Let)137 2110 y FB(Q)173 2117 y FA(k)207 2110 y FC(=)205 b(sup)255 2161 y FA(\016)271 2166 y Fz(1)288 2161 y FA(<)q Fc(b)-21 b FA(x)339 2152 y Fz(det)335 2170 y Fg(u)353 2178 y(k)383 2161 y Fw(+)p FA(z)426 2167 y Fg(k)446 2161 y Fm(6)p FA(x)493 2152 y Fz(det)493 2170 y Fg(u)511 2178 y(k)537 2161 y Fw(+)p FA(h)584 2130 y Fs(p)p 623 2130 85 2 v 623 2161 a FA(\020)r Fw(\()p FA(u)675 2167 y Fg(k)694 2161 y Fw(\))715 2110 y Fl(P)749 2092 y FA(u)769 2098 y Fg(k)785 2092 y FA(;z)811 2098 y Fg(k)832 2073 y Fq(\010)859 2110 y FB(\016)879 2117 y Fw(1)912 2110 y FB(<)14 b Fq(b)-26 b FB(x)990 2092 y Fw(d)o(et)986 2122 y FA(s)1050 2110 y FC(+)10 b FB(z)1116 2117 y FA(s)1147 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Fq(\011)1390 2473 y Fr(6)1438 2416 y FA(K)r Fx(\000)p Fw(1)1447 2430 y Fq(Y)1444 2528 y FA(k)q Fw(=0)1522 2473 y FB(Q)1558 2480 y FA(k)1580 2473 y FB(:)123 b FC(\(4.32\))136 2614 y(Th)o(e)20 b(re)q(sul)o(t)h(will)h (t)n(h)n(us)e(b)q(e)g(pro)o(v)o(e)q(d)f(if)i(w)o(e)f(m)o(an)o(age)f(t)o (o)h(c)o(h)o(o)q(os)q(e)f FB(\045)i FC(in)g(su)o(c)o(h)f(a)g(w)o(ay)f (t)n(h)o(a)o(t)g FB(Q)1758 2621 y FA(k)1800 2614 y FC(i)q(s)136 2671 y(b)q(ou)o(n)o(d)o(e)q(d)d(aw)o(ay)d(f)q(rom)g(1)h(for)g FB(k)g FC(=)f(0)p FB(;)8 b(:)g(:)g(:)t(;)g(K)k Fs(\000)f FC(2.)922 2936 y(42)p eop %%Page: 43 43 43 42 bop 78 164 a FC(2.)22 b(W)l(e)13 b(will)g(e)q(st)o(im)o(a)o(t)o (e)e(t)n(h)o(e)g FB(Q)575 171 y FA(k)609 164 y FC(in)h(a)g(s)q(imilar)g (w)o(ay)f(as)g(in)i(Pro)o(p)q(os)q(it)o(ion)d(3.13,)g(bu)o(t)i(w)o(e)g (sh)o(all)g(di)q(st)o(in)o(gui)q(sh)136 220 y(t)n(hree)j(cas)q(e)q(s)g (inst)o(ead)g(of)g(t)o(w)o(o.)20 b(Th)o(e)q(s)q(e)15 b(cas)q(e)q(s)f(corre)q(sp)q(on)o(d)g(t)o(o)h FB(x)1236 227 y FA(s)1270 220 y 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y FC(t)o(ur)q(ns)e(ou)o(t)i(t)o(o)g (play)g(a)g(cru)o(cial)g(role.)29 b(Th)o(e)136 1015 y(a)o(b)q(o)o(v)o (e)14 b(proba)o(bilit)o(y)g(i)q(s)i(in)o(d)o(ee)q(d)g(clos)q(e)g(t)o(o) f(1)p FB(=)p FC(2)f(wh)o(en)g FB(\045)h FC(i)q(s)h(larger)e(t)n(h)o(an) g(a)h(const)o(an)o(t)e(t)o(im)o(e)q(s)i Fs(j)p FC(log)7 b FB(\033)r Fs(j)p FC(,)136 1072 y(whic)o(h)19 b(sh)o(o)o(ws)d(t)n(h)o (a)o(t)h(once)h(a)g(pa)o(t)n(h)g(h)o(as)g(reac)o(h)o(e)q(d)f(t)n(h)o(e) h(saddle,)h(it)g(also)f(h)o(as)f(a)o(b)q(ou)o(t)h(\014ft)o(y)f(p)q (ercen)o(t)136 1128 y(c)o(h)o(ance)d(t)o(o)h(reac)o(h)f(t)n(h)o(e)h (lev)o(el)g FB(\016)652 1135 y Fw(1)687 1128 y FC(in)h(a)e(t)o(im)o(e)g (of)g(ord)o(er)g FB(")p Fs(j)p FC(log)8 b FB(\033)r Fs(j)p FB(=)o Fq(b)-24 b FB(a)o FC(\()p FB(u)p FC(\).)78 1184 y(4.)22 b(F)l(rom)c(\(4.36\))m(,)h(\(4.38\))c(an)o(d)j(\(4.41\))e(an)o (d)h(t)n(h)o(e)h(f)q(act)f(t)n(h)o(a)o(t)g FB(\045)h Fr(>)h FC(1,)f(w)o(e)g(obt)o(ain)f(t)n(h)o(e)h(exi)q(st)o(ence)f(of)h (a)136 1241 y(const)o(an)o(t)13 b FB(C)i(>)e FC(0)i(su)o(c)o(h)f(t)n(h) o(a)o(t)212 1340 y FB(Q)248 1347 y FA(k)282 1340 y Fr(6)335 1309 y FC(1)p 335 1330 23 2 v 335 1371 a(2)373 1340 y(+)423 1309 y FB(C)p 423 1330 36 2 v 427 1371 a(\033)533 1340 y FC(sup)471 1379 y FA(u)491 1385 y Fg(k)510 1379 y Fm(6)p FA(u)p Fm(6)p FA(u)604 1385 y Fg(k)q Fz(+1)671 1299 y Fq(p)p 717 1299 86 2 v 41 x(b)-24 b FB(a)p FC(\()p FB(u)p FC(\))802 1290 y Fq(h)823 1299 y(p)p 869 1299 106 2 v 869 1340 a FB(\020)s FC(\()p FB(u)936 1347 y FA(k)957 1340 y FC(\))7 b FB(h)h FC(e)1036 1321 y Fx(\000)p FA(\045=)p Fw(3)p FA(L)1716 1340 y FC(\(4.42\))833 1461 y(+)73 b(sup)878 1500 y FA(u)898 1506 y Fg(k)918 1500 y Fm(6)p FA(u)p Fm(6)p FA(u)1012 1506 y Fg(k)q Fz(+1)1071 1461 y FC(\()p FB(x)1115 1442 y Fw(d)o(et)1115 1472 y FA(u)1175 1461 y Fs(\000)12 b Fq(b)-26 b FB(x)1251 1442 y Fw(d)o(et)1247 1472 y FA(u)1301 1461 y FC(\))10 b(+)g(\()r Fq(b)-27 b FB(x)1422 1442 y Fw(d)o(et)1418 1472 y FA(u)1438 1478 y Fg(k)q Fz(+1)1508 1461 y Fs(\000)10 b FB(\016)1573 1468 y Fw(1)1593 1461 y FC(\))e(e)1638 1442 y Fx(\000)p FA(\045=)p Fw(3)1721 1411 y Fq(i)1742 1461 y FB(:)136 1590 y FC(Since)17 b Fs(j)p FB(u)p Fs(j)c Fr(6)h FB(c)389 1597 y Fw(1)409 1590 y FB(\033)437 1574 y Fw(2)p FA(=)p Fw(3)491 1590 y FC(,)i(t)n(h)o(e)f(pro)o(p)q(ert)o(ie)q(s)f(of)h FB(x)884 1574 y Fw(d)o(et)934 1590 y FC(,)j Fq(b)-27 b FB(x)993 1574 y Fw(d)o(et)1043 1590 y FC(,)16 b FB(\020)j FC(an)o(d)14 b Fq(b)-24 b FB(a)16 b FC(imply)f(t)n(h)o(e)h(exi)q(st)o (ence)e(of)i(anot)n(h)o(er)136 1647 y(const)o(an)o(t)d FB(C)350 1654 y Fw(1)384 1647 y FC(su)o(c)o(h)h(t)n(h)o(a)o(t)556 1761 y FB(Q)592 1768 y FA(k)626 1761 y Fr(6)679 1731 y FC(1)p 679 1751 23 2 v 679 1793 a(2)716 1761 y(+)c FB(C)794 1768 y Fw(1)814 1711 y Fq(h)841 1731 y FB(h)p 840 1751 28 2 v 840 1793 a(\033)880 1761 y FC(e)900 1743 y Fx(\000)p FA(\045=)p Fw(3)p FA(L)1015 1761 y FC(+)p FB(c)1070 1738 y Fw(3)p FA(=)p Fw(2)1070 1774 y(1)1134 1761 y FC(+)1180 1730 y Fs(p)p 1217 1730 40 2 v 1217 1761 a FB(c)1237 1768 y 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635 2098 a(2)725 2036 y(1)p 668 2056 137 2 v 668 2098 a(\(3)p FB(=)p FC(2\))773 2084 y FA(K)822 2066 y Fr(6)875 2036 y FC(3)p 875 2056 23 2 v 875 2098 a(2)910 2066 y(exp)979 2016 y Fq(n)1009 2066 y Fs(\000)1044 2016 y Fq(\020)1072 2066 y FC(log)1143 2036 y(3)p 1143 2056 V 1143 2098 a(2)1170 2016 y Fq(\021)1209 2036 y(b)-28 b FB(\013)p FC(\()p FB(t)1269 2043 y Fw(1)1289 2036 y FB(;)8 b Fs(\000)p FB(c)1365 2043 y Fw(1)1384 2036 y FB(\033)1412 2019 y Fw(2)p FA(=)p Fw(3)1466 2036 y FC(\))p 1206 2056 278 2 v 1323 2098 a FB(\045")1489 2016 y Fq(o)1519 2066 y FB(;)184 b FC(\(4.44\))136 2194 y(an)o(d)15 b(t)n(h)o(e)f(re)q(sul)o(t)h(follo)o(ws)e(f)q(rom)g(our)h (c)o(h)o(oice)h(of)f FB(\045)p FC(.)p 1798 2194 2 31 v 1800 2165 28 2 v 1800 2194 V 1827 2194 2 31 v 130 2285 a(It)f(rem)o(ains)f(t)o(o)h(e)q(st)o(im)o(a)o(t)o(e)e(t)n(h)o(e)i(s)q (econ)o(d)f(t)o(erm)g(in)i(\(4.23\))m(,)f(d)o(e)q(scr)q(ibin)o(g)h(t)n (h)o(e)f(proba)o(bilit)o(y)f(not)h(t)o(o)f(reac)o(h)59 2342 y FB(\016)79 2349 y Fw(0)115 2342 y FC(wh)o(en)k(st)o(art)o(in)o (g)e(in)j FB(\016)473 2349 y Fw(1)493 2342 y FC(.)23 b(Thi)q(s)16 b(i)q(s)g(don)o(e)g(b)o(y)g(us)q(in)o(g)g(t)n(h)o(e)f(f)q (act)g(t)n(h)o(a)o(t,)g(b)o(y)g(assu)o(mpt)o(ion,)g(t)n(h)o(e)h(dr)q (ift)f(t)o(erm)59 2398 y(i)q(s)h(b)q(ou)o(n)o(d)o(e)q(d)h(aw)o(ay)e(f)q (rom)f(zero)i(on)g(t)n(h)o(e)g(in)o(t)o(erv)n(al)f([)p FB(\016)948 2405 y Fw(0)967 2398 y FB(;)8 b(\016)1008 2405 y Fw(1)1027 2398 y FC(].)23 b(W)l(e)17 b(will)h(n)o(ee)q(d)e(t)o (o)g(assu)o(m)o(e)g(t)n(h)o(a)o(t)f(it)h(can)g(b)q(e)59 2455 y(b)q(ou)o(n)o(d)o(e)q(d)f(aw)o(ay)f(f)q(rom)f(zero)h(on)g(a)h (sligh)o(t)n(ly)g(larger)f(in)o(t)o(erv)n(al,)e(whic)o(h)j(i)q(s)g(p)q (oss)q(ible)g(b)o(y)g(con)o(t)o(in)n(uit)o(y)f(of)g FB(f)5 b FC(.)59 2536 y Fo(Pro)o(p)q(o)q(s)q(it)o(ion)17 b(4.7.)23 b Fy(L)n(et)d FC(0)g FB(<)i(\032)e Fr(6)h FB(\016)742 2543 y Fw(1)776 2536 y Fs(\000)14 b FB(\016)845 2543 y Fw(0)885 2536 y Fy(b)n(e)20 b(a)g(c)n(onstant)h(such)g(that)f FB(f)5 b FC(\()p FB(x;)j(t)p 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1246 a FC(2)p FB(c\033)1294 1233 y Fw(2)1317 1165 y Fq(o)1716 1215 y FC(\(4.49\))59 1345 y(b)o(y)i(Do)q(ob's)f(su)n(bm)o(art)o(in)o(gale)f(in)o(equalit)o(y)l(.) p 1798 1345 2 31 v 1800 1316 28 2 v 1800 1345 V 1827 1345 2 31 v 59 1488 a FD(Reference)r(s)59 1590 y FC([BPSV])65 b(R.)19 b(Benzi,)i(A.)e(Su)o(t)o(era,)h(A.)f(V)l(ulpiani,)j Fy(The)e(me)n(chanism)g(of)g(sto)n(chastic)g(r)n(esonanc)n(e)p FC(,)g(J.)272 1646 y(Ph)o(ys.)13 b(A)i Fo(14)p FC(:L453{L45)o(7)d (\(1981\).)272 1721 y(R.)i(Benzi,)h(G.)g(P)o(ar)q(i)q(s)q(i,)e(A.)h(Su) o(t)o(era,)f(A.)h(V)l(ulpiani,)i Fy(A)g(the)n(ory)f(of)h(sto)n(chastic) f(r)n(esonanc)n(e)h(in)272 1778 y(climatic)f(change)p FC(,)e(SIAM)j(J.)e(A)n(p)o(pl.)g(Ma)o(t)n(h.)g Fo(43)p FC(:565{5)o(78)d(\(1983\).)59 1853 y([BG])120 b(N.)16 b(Berglu)o(n)o(d,)h(B.)g(Gen)o(tz,)g Fy(Pathwise)h(description)g(of)g (dynamic)f(pitchfork)g(bifur)n(c)n(ations)272 1909 y(with)e(additive)g (noise)p FC(,)g(WIAS)g(prepr)q(in)o(t)e(no.)h(604)g(\(2000\).)272 1966 y Fb(http://w)o(ww.)o(wia)o(s-)o(ber)o(li)o(n.d)o(e/p)o(ub)o(lic)o (at)o(ion)o(s/p)o(re)o(pri)o(nt)o(s/6)o(04)272 2022 y(arXiv:ma)o(th.)o (PR/)o(00)o(082)o(08)59 2097 y FC([ET])123 b(J.-P)l(.)21 b(Ec)o(km)o(ann,)g(L.)7 b(E.)22 b(Th)o(om)o(as,)f Fy(R)n(emarks)g(on)i (sto)n(chastic)f(r)n(esonanc)n(e)p FC(,)h(J.)e(Ph)o(ys.)g(A)272 2154 y Fo(15)p FC(:L261{L26)o(6)12 b(\(1982\).)59 2229 y([FH])124 b(S.)24 b(F)l(a)n(uv)o(e,)i(F.)e(He)q(slot,)i Fy(Sto)n(chastic)e(r)n(esonanc)n(e)g(in)h(a)f(bistable)h(system)p FC(,)h(Ph)o(ys.)d(Let)n(t.)272 2285 y Fo(97A)p FC(:5{7)12 b(\(1983\).)59 2361 y([F)l(o)o(x])116 b(R.)7 b(F.)19 b(F)l(o)o(x,)h Fy(Sto)n(chastic)g(r)n(esonanc)n(e)g(in)g(a)h(double)f (wel)r(l)p FC(,)h(Ph)o(ys.)d(Rev.)h(A)h Fo(39)p FC(:4148{4)o(15)o(3)272 2417 y(\(1989\).)59 2492 y([F)l(r])144 b(M.)7 b(I.)18 b(F)l(re)q(idlin,)k Fy(Quasi-deterministic)d(appr)n(oximation,)g (metastability)g(and)g(sto)n(chastic)272 2549 y(r)n(esonanc)n(e)p FC(,)13 b(Ph)o(ys)q(ica)h(D)h Fo(137)p FC(:333{35)o(2)d(\(2000\).)59 2624 y([Gr])134 b(I.)7 b(S.)12 b(Grad)n(\024)-20 b(st)o(e)-5 b(\025)-19 b(\020n,)11 b Fy(Applic)n(ations)j(of)f(A.)7 b(M.)14 b(Lyapunov's)g(the)n(ory)e(of)i(stability)f(to)h(the)f(the)n (ory)272 2680 y(of)j(di\013er)n(ential)g(e)n(quations)g(with)h(smal)r (l)h(c)n(o)n(e\016cients)d(in)i(the)g(derivatives)p FC(,)d(Ma)o(t.)h (Sb)q(or)q(nik)272 2737 y(N.S.)e Fo(32)p FC(:263{286)e(\(1953\).)922 2936 y(45)p eop %%Page: 46 46 46 45 bop 59 164 a FC([JH])130 b(P)l(.)9 b(Ju)o(n)o(g,)j(P)l(.)e(H\177) -23 b(an)o(ggi,)10 b Fy(Sto)n(chastic)h(nonline)n(ar)h(dynamics)f(mo)n (dulate)n(d)h(by)f(external)h(p)n(erio)n(dic)272 220 y(for)n(c)n(es)p FC(,)h(Euro)o(ph)o(ys.)f(Let)n(t)o(ers)i Fo(8)p FC(:505{51)o(0)e(\(1989\).)272 295 y(P)l(.)e(Ju)o(n)o(g,)i(P)l (.)e(H\177)-23 b(an)o(ggi,)11 b Fy(A)o(mpli\014c)n(ation)h(of)h(smal)r (l)g(signals)h(via)e(sto)n(chastic)g(r)n(esonanc)n(e)p FC(,)e(Ph)o(ys.)272 352 y(Rev.)k(A)h Fo(44)p FC(:8032{)o(80)o(42)c (\(1991\).)59 427 y([MW])100 b(F.)17 b(Moss,)h(K.)f(Wie)q(s)q(enfeld,)i Fy(The)f(b)n(ene\014ts)g(of)g(b)n(ackgr)n(ound)f(noise)p FC(,)h(Scien)o(t)o(i\014c)f(Am)o(er)q(ican)272 483 y Fo(273)p FC(:50{53)11 b(\(1995\).)272 559 y(K.)h(Wie)q(s)q(enfeld,)i (F.)g(Moss,)f Fy(Sto)n(chastic)h(r)n(esonanc)n(e)f(and)i(the)f(b)n (ene\014ts)g(of)h(noise:)k(fr)n(om)14 b(ic)n(e)272 615 y(ages)h(to)h(cr)n(ay\014sh)g(and)g(SQUIDs)p FC(,)h(Na)o(t)o(ure)c Fo(373)p FC(:33{36)e(\(1995\).)59 690 y([McNW])46 b(B.)20 b(McNam)o(ara,)f(K.)h(Wie)q(s)q(enfeld,)j Fy(The)n(ory)d(of)h(sto)n (chastic)g(r)n(esonanc)n(e)p FC(,)g(Ph)o(ys.)e(Rev.)h(A)272 747 y Fo(39)p FC(:4854{)o(48)o(69)11 b(\(1989\).)59 822 y([SNAS])69 b(B.)18 b(Sh)n(ulgin,)j(A.)d(Ne)q(im)o(an,)i(V.)e(Ani)q (shc)o(h)o(enk)o(o,)g Fy(Me)n(an)i(switching)f(fr)n(e)n(quency)f(lo)n (cking)i(in)272 878 y(sto)n(chastic)10 b(bistable)i(systems)g(driven)e (by)i(a)f(p)n(erio)n(dic)f(for)n(c)n(e)p FC(,)f(Ph)o(ys.)g(Rev.)g(Let)n (t)o(ers)g Fo(75)p FC(:4157{)272 935 y(4160)k(\(1995\).)272 1010 y(A.)g(Ne)q(im)o(an,)g(A.)h(Silc)o(h)o(enk)o(o,)f(V.)g(Ani)q(shc)o (h)o(enk)o(o,)f(L.)i(Sc)o(him)o(ansky-Ge)q(ier,)f Fy(Sto)n(chastic)i(r) n(es-)272 1066 y(onanc)n(e:)20 b(Noise-enhanc)n(e)n(d)15 b(phase)h(c)n(oher)n(enc)n(e)p FC(,)c(Ph)o(ys.)h(Rev.)i(E)f Fo(58)p FC(:7118{71)o(25)d(\(1998\).)59 1141 y([Ti])142 b(A.)7 b(N.)13 b(Tih)o(ono)o(v,)g Fy(Systems)j(of)g(di\013er)n(ential)e (e)n(quations)i(c)n(ontaining)f(smal)r(l)i(p)n(ar)n(ameters)e(in)272 1198 y(the)g(derivatives)p FC(,)e(Ma)o(t.)h(Sb)q(or)q(nik)g(N.S.)g Fo(31)p FC(:575{58)o(6)e(\(1952\).)59 1397 y FG(Nils)h(Berglu)o(n)o(d) 59 1447 y Fv(Dep)m(ar)m(tment)k(of)f(Ma)m(thema)m(tics,)h(ETH)d(Z)835 1444 y(\177)834 1447 y(urich)59 1497 y FG(ETH)g(Zen)o(tru)o(m,)g(8092)f (Z)q(\177)-22 b(ur)q(ic)o(h,)13 b(Switzerlan)o(d)59 1547 y FE(E-mail)h(addr)n(ess:)19 b Fa(berglund@math.ethz.)o(ch)59 1653 y FG(Barbara)14 b(Gen)o(tz)59 1709 y Fv(Weierstra\031)h(Institute) g(f)o(or)i(Applied)e(Anal)m(ysis)i(and)g(Stochastics)59 1766 y FG(Mohrenstra\031e)e(39,)e(10117)g(Berlin,)h(Germ)o(an)o(y)59 1822 y FE(E-mail)g(addr)n(ess:)19 b Fa(gentz@wias-berlin.d)o(e)922 2936 y FC(46)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0012291202823--