Content-Type: multipart/mixed; boundary="-------------0103280150883" This is a multi-part message in MIME format. ---------------0103280150883 Content-Type: text/plain; name="01-113.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-113.keywords" Stochastic PDE, exponential mixing ---------------0103280150883 Content-Type: application/postscript; name="exp.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="exp.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: exp.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Roman Times-Italic %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D 600 -f exp.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.03.28:0946 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A 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/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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y(will)24 b(sho)n(w)f(in)g(this)h(paper)e(that)h(for)g(e)n (v)o(ery)f(probability)f(measure)h Fs(\026)p Fy(,)j(we)e(ha)n(v)o(e)g Fo(P)2926 3791 y Fp(t)2919 3842 y Fj(\003)2957 3821 y Fs(\026)28 b Fo(!)h Fs(\026)3197 3833 y Fj(\003)3259 3821 y Fy(and)515 3921 y(that)d(this)h(con)m(v)o(er)o(gence)c(tak)o(es) k(place)f(with)g(an)h(e)o(xponential)d(rate)i(\(in)h(time\).)43 b(More)26 b(precisely)-5 b(,)515 4021 y(we)24 b(introduce,)g(for)f(a)i (gi)n(v)o(en)e(\(possibly)g(unbounded\))e(Borel)j(function)f Fs(V)49 b Fr(:)31 b Fo(H)g(!)g Fy([)p Fr(1)p Fs(;)14 b Fo(1)p Fy(],)25 b(the)515 4120 y Fl(weighted)19 b(variational)g(norm) h Fy(de\002ned)f(on)h(e)n(v)o(ery)f(signed)g(Borel)i(measure)e Fs(\026)i Fy(by)1168 4343 y Fo(j)-14 b(j)g(j)p Fs(\026)p Fo(j)g(j)g(j)1300 4355 y Fp(V)1381 4343 y Fo(\021)1469 4230 y Fi(Z)1515 4418 y Fj(H)1590 4343 y Fs(V)19 b Fy(\()p Fs(x)p Fy(\))14 b Fs(\026)1824 4355 y Fq(+)1879 4343 y Fy(\()p Fs(dx)p Fy(\))k Fr(+)2126 4230 y Fi(Z)2172 4418 y Fj(H)2247 4343 y Fs(V)h Fy(\()p Fs(x)p Fy(\))14 b Fs(\026)2481 4355 y Fj(\000)2536 4343 y Fy(\()p Fs(dx)p Fy(\))23 b(,)515 4566 y(where)d Fs(\026)789 4578 y Fj(\006)867 4566 y Fy(denotes)h(the)g(positi)n(v)o(e)g(\(resp.)g(ne)o(gati)n(v)o (e\))d(part)j(of)h Fs(\026)p Fy(.)29 b(When)21 b Fs(V)e Fy(\()p Fs(x)p Fy(\))25 b Fr(=)f(1)p Fy(,)e(we)g(reco)o(v)o(er)515 4666 y(the)i(usual)f(v)n(ariational)g(norm)f(which)i(we)g(denote)f(by)g Fo(j)-14 b(j)g(j)14 b(\001)g(j)-14 b(j)g(j)t Fy(.)37 b(W)-7 b(e)25 b(also)g(introduce)d(the)h(f)o(amily)h(of)515 4766 y(norms)19 b Fo(k)f(\001)h(k)887 4778 y Fp(\015)949 4766 y Fy(on)h Fo(H)i Fy(de\002ned)d(by)1668 4865 y Fo(k)p Fs(x)p Fo(k)1799 4877 y Fp(\015)1864 4865 y Fr(=)k Fo(k)p Fs(L)2051 4831 y Fp(\015)2093 4865 y Fs(x)p Fo(k)g Fy(,)p eop %%Page: 3 3 3 2 bop 517 232 a Fy(A)25 b(V)-6 b Fk(A)t(R)t(I)t(A)t(N)t(T)25 b(O)t(F)g(T)t(H)t(E)f Fy(P)t Fk(E)t(R)t(R)q(O)t(N)t Fy(-)t(F)t Fk(R)q(O)t(B)t(E)t(N)t(I)t(U)t(S)j Fy(T)t Fk(H)t(E)t(O)t(R)t(E)t(M)995 b Fu(3)515 523 y Fy(where)19 b Fs(L)i Fy(is)g(the)f(dif)n(ferential)e (operator)h Fr(1)f Fo(\000)g Fs(@)1891 493 y Fq(2)1886 547 y Fp(\030)1948 523 y Fy(and)i Fo(k)e(\001)h(k)h Fy(is)h(the)f (usual)g(norm)f(on)h Fo(H)q Fy(,)g Fl(i.e)o(.)1416 779 y Fo(k)p Fs(u)p Fo(k)1548 745 y Fq(2)1607 779 y Fr(=)1694 666 y Fi(Z)1777 687 y Fq(1)1741 855 y(0)1815 712 y Fi(\000)1853 779 y Fo(j)p Fs(u)p Fo(j)1947 745 y Fq(2)2002 779 y Fr(+)e Fo(j)p Fs(@)2152 791 y Fp(\030)2189 779 y Fs(u)p Fo(j)2260 745 y Fq(2)2296 712 y Fi(\001)2348 779 y Fs(d\030)28 b(:)515 1003 y Fy(The)20 b(e)o(xact)f(formulation)f(of)i(our)f(con)m(v) o(er)o(gence)e(result)j(is)515 1202 y Fu(Theor)o(em)g(1.3)40 b Fl(Ther)m(e)24 b(e)n(xists)g(a)g(constant)e Fs(\025)29 b(>)g Fr(0)23 b Fl(suc)o(h)g(that)g(for)h(e)o(very)f Fs(p)29 b Fo(\025)f Fr(1)p Fl(,)c(e)o(very)g Fs(\015)33 b Fo(\024)c Fs(\013)p Fl(,)515 1302 y(and)19 b(e)o(very)h(pr)l (obability)f(measur)m(e)h Fs(\026)h Fl(on)f Fo(H)q Fl(,)g(one)g(has)957 1484 y Fo(j)-14 b(j)g(jP)1063 1450 y Fp(t)1056 1505 y Fj(\003)1095 1484 y Fs(\026)18 b Fo(\000)g Fs(\026)1296 1496 y Fj(\003)1334 1484 y Fo(j)-14 b(j)g(j)1375 1496 y Fp(V)1414 1504 y Fh(\015)s(;p)1530 1484 y Fo(\024)23 b Fs(C)6 b(e)1722 1450 y Fj(\000)p Fp(\025t)1865 1484 y Fl(,)166 b(with)23 b Fs(V)2266 1496 y Fp(\015)t(;p)2363 1484 y Fy(\()p Fs(u)p Fy(\))f Fr(=)h Fo(k)p Fs(u)p Fo(k)2709 1450 y Fp(p)2709 1505 y(\015)2768 1484 y Fr(+)18 b(1)23 b Fl(,)515 1667 y(for)d(e)o(very)h Fs(t)i Fo(\025)f Fr(1)p Fl(.)j(The)c(constant)e Fs(C)27 b Fl(is)21 b(independent)c(of)k(the)f (pr)l(obability)f(measur)m(e)h Fs(\026)p Fl(.)639 1833 y Fy(In)c(the)h(sequel,)f(we)h(will)g(denote)e(by)h Fg(\010)h Fy(the)f(Mark)o(o)o(v)f(chain)h(obtained)e(by)i(sampling)g(the)g(solu-) 515 1933 y(tion)g(of)g(\(SGL\))f(at)i(inte)o(ger)e(times)i(and)e(by)h Fo(P)7 b Fy(\()p Fs(x;)28 b Fo(\001)14 b Fy(\))i(the)g(corresponding)d (transition)i(probabilities.)515 2032 y(Theorem)j(1.3)i(is)h(a)f (consequence)e(of)i(the)h(follo)n(wing)d(features)i(of)f(the)i(model)e (\(SGL\).)561 2155 y Fl(A.)47 b Fy(W)-7 b(e)20 b(construct)e(a)h(set)h Fs(K)k Fy(ha)n(ving)18 b(the)h(property)d(that)j(there)g(e)o(xists)g(a) g(probability)e(measure)h Fs(\027)680 2255 y Fy(and)e(a)i(constant)e Fs(\016)26 b(>)d Fr(0)17 b Fy(such)f(that)h Fo(P)7 b Fy(\()p Fs(x;)28 b Fo(\001)14 b Fy(\))22 b Fo(\025)h Fs(\016)17 b(\027)5 b Fy(\()14 b Fo(\001)g Fy(\))i(for)h(e)n(v)o(ery)e Fs(x)24 b Fo(2)f Fs(K)6 b Fy(.)24 b(This)17 b(means)f(that)680 2355 y Fs(K)32 b Fy(beha)n(v)o(es)24 b(\223almost\224)h(lik)o(e)h(an)f (atom)g(for)g(the)h(Mark)o(o)o(v)d(chain)i Fg(\010)p Fy(.)41 b(This)26 b(is)g(sho)n(wn)f(to)h(be)680 2454 y(a)e(consequence)d(of)h(the)i(Strong)e(Feller)h(property)e(and)i(the)g (irreducibility)e(of)i(the)g(Mark)o(o)o(v)680 2554 y(semigroup)18 b(associated)i(to)h(\(SGL\).)561 2677 y Fl(B.)47 b Fy(The)26 b(dynamics)e(has)i(v)o(ery)f(strong)g(contraction)e(properties)i(in)h (the)g(sense)g(that)g(it)g(reaches)680 2777 y(some)33 b(compact)e(set)j(v)o(ery)d(quickly)-5 b(.)61 b(In)32 b(particular)m(,)i(one)e(can)h(bound)d(uniformly)h(from)680 2876 y(belo)n(w)20 b(the)g(transition)f(probabilities)g(to)h(a)h(set)g Fs(K)26 b Fy(satisfying)20 b(property)e Fl(A.)639 3000 y Fy(These)h(conditions)f(yield)h(some)g(strong)g(Doeblin)f(condition)f (and)i(thus)g(lead)g(to)h(e)o(xponential)515 3099 y(con)m(v)o(er)o (gence)h(results.)40 b(The)24 b(intuiti)n(v)o(e)g(reason)h(behind)e (this)j(is)g(that,)g(for)e(an)o(y)g(tw)o(o)i(initial)f(mea-)515 3199 y(sures,)d(their)g(image)g(under)f Fo(P)1400 3211 y Fj(\003)1461 3199 y Fy(has)h(a)h(common)d(part,)i(the)g(amount)f(of)h (which)g(can)g(be)g(bounded)515 3299 y(uniformly)f(from)i(belo)n(w)h (and)f(cancels)h(out.)35 b(This)25 b(will)f(be)g(clari\002ed)g(in)g (the)g(proof)e(of)i(Proposi-)515 3398 y(tion)c(2.1)f(belo)n(w)-5 b(.)639 3498 y(The)22 b(remainder)e(of)h(the)h(paper)f(is)i(or)o (ganized)c(as)j(follo)n(ws.)30 b(In)22 b(Section)f(2,)h(we)g(sho)n(w)g (ho)n(w)f(to)515 3597 y(obtain)f(Theorem)f(1.3)i(from)f(the)h(abo)o(v)o (e)e(properties.)26 b(The)21 b(proof)e(will)j(be)f(strongly)f (reminiscent)515 3697 y(of)h(the)g(standard)f(proof)g(of)h(the)g (Perron-Frobenius)d(theorem.)27 b(In)22 b(Section)e(3)i(we)g(then)e (sho)n(w)i(the)515 3797 y(contraction)h(properties)h(of)h(the)h (dynamics)e(and)h(in)h(Section)f(4)g(we)h(sho)n(w)f(that)h(e)n(v)o(ery) e(compact)515 3896 y(set)d(has)f(the)g(property)e Fl(A.)515 4071 y Fu(Ackno)o(wledgements)515 4200 y Fw(Se)n(v)o(eral)d(proofs)h (in)f(this)g(paper)h(are)g(inspired)g(by)g(the)f(recent)h(w)o(ork)g ([RBT01])f(of)g(Luc)g(Re)o(y-Bellet)g(and)h(Larry)515 4300 y(Thomas.)35 b(The)23 b(author)g(is)g(v)o(ery)g(grateful)g(to)g (them)g(for)g(ha)o(ving)g(communicated)i(their)d(ideas)i(before)f(pub-) 515 4399 y(lication.)37 b(The)23 b(author)i(also)f(bene\002tted)g(from) f(se)n(v)o(eral)h(useful)g(discussions)h(with)e(Jean-Pierre)h(Eckmann) 515 4499 y(and)19 b(Jacques)h(Rougemont.)25 b(This)18 b(w)o(ork)i(w)o(as)f(partially)g(supported)h(by)g(the)f(F)o(onds)g (National)g(Suisse.)515 4727 y Fx(2)99 b(A)25 b(V)-9 b(ariant)25 b(of)g(the)g(P)n(err)n(on-Fr)n(obenius)i(Theor)n(em)515 4902 y Fy(The)20 b(follo)n(wing)f(proposition)g(sho)n(ws,)h (reformulated)e(in)j(a)g(more)f(rigorous)f(w)o(ay)-5 b(,)21 b(why)e(the)i(prop-)515 5001 y(erties)f Fl(A.)h Fy(and)e Fl(B.)h Fy(yield)g(e)o(xponential)e(con)m(v)o(er)o(gence)e (results)21 b(to)n(w)o(ards)f(the)g(in)m(v)n(ariant)e(measure.)p eop %%Page: 4 4 4 3 bop 517 232 a Fy(A)25 b(V)-6 b Fk(A)t(R)t(I)t(A)t(N)t(T)25 b(O)t(F)g(T)t(H)t(E)f Fy(P)t Fk(E)t(R)t(R)q(O)t(N)t Fy(-)t(F)t Fk(R)q(O)t(B)t(E)t(N)t(I)t(U)t(S)j Fy(T)t Fk(H)t(E)t(O)t(R)t(E)t(M)995 b Fu(4)515 523 y(Pr)o(oposition)19 b(2.1)40 b Fl(Let)23 b Fr(\011)f Fl(be)g(a)h(Mark)o(o)o(v)f(c)o(hain)f(on)h(a)g(measur)o (able)f(space)h Fn(X)h Fl(and)f(let)g Fr(\011)h Fl(satisfy)515 623 y(the)d(following)f(pr)l(operties:)571 746 y(a.)46 b(Ther)m(e)23 b(e)n(xist)i(a)e(measur)o(able)f(set)i Fs(K)6 b Fl(,)24 b(a)f(positive)g(constant)f Fs(\016)27 b Fl(and)22 b(a)i(pr)l(obability)e(measur)m(e)680 846 y Fs(\027)721 858 y Fj(\003)786 846 y Fl(suc)o(h)k(that)g(for)h(e)o (very)f(measur)o(able)f(set)i Fs(A)g Fl(and)f(e)o(very)g Fs(x)35 b Fo(2)g Fs(K)6 b Fl(,)28 b(one)d(has)i Fo(P)7 b Fy(\()p Fs(x;)14 b(A)p Fy(\))33 b Fo(\025)680 945 y Fs(\016)17 b(\027)775 957 y Fj(\003)813 945 y Fy(\()p Fs(A)p Fy(\))p Fl(.)574 1068 y(b)m(.)46 b(Ther)m(e)20 b(e)n(xists)i(a)e(constant)f Fs(\016)1499 1038 y Fj(0)1546 1068 y Fs(>)j Fr(0)f Fl(suc)o(h)e(that)h Fo(P)7 b Fy(\()p Fs(x;)14 b(K)6 b Fy(\))22 b Fo(\025)h Fs(\016)2449 1038 y Fj(0)2493 1068 y Fl(for)d(e)o(very)h Fs(x)i Fo(2)h Fn(X)p Fl(.)515 1192 y(Then)d Fr(\011)g Fl(has)g(a)g(unique)f(in)m (variant)g(measur)m(e)h Fs(\026)1910 1204 y Fj(\003)1970 1192 y Fl(and)f(one)h(has)g(for)g(e)o(very)h(pr)l(obability)e(measur)m (e)515 1291 y Fs(\026)h Fl(the)f(estimate)g Fo(j)-14 b(j)g(jP)1111 1261 y Fp(n)1104 1312 y Fj(\003)1156 1291 y Fs(\026)19 b Fo(\000)f Fs(\026)1358 1303 y Fj(\003)1396 1291 y Fo(j)-14 b(j)g(j)28 b(\024)23 b Fr(2)p Fy(\()p Fr(1)17 b Fo(\000)h Fs(\016)s(\016)1845 1261 y Fj(0)1868 1291 y Fy(\))1896 1261 y Fj(\000)p Fp(n=)p Fq(2)2060 1291 y Fl(.)515 1468 y(Pr)l(oof)o(.)40 b Fy(The)24 b(\002rst)h(observ)n (ation)c(we)k(mak)o(e)e(is)i(that)f(for)g(e)n(v)o(ery)e(probability)g (measure)h Fs(\026)i Fy(one)e(has)515 1568 y(by)d(property)e Fl(a.)p Fy(,)1307 1640 y Fi(\000)1345 1707 y Fo(P)1403 1719 y Fj(\003)1441 1707 y Fs(\026)1491 1640 y Fi(\001)1529 1707 y Fy(\()p Fs(K)6 b Fy(\))22 b Fr(=)1771 1594 y Fi(Z)1817 1783 y Ff(X)1876 1707 y Fo(P)7 b Fy(\()p Fs(x;)14 b(K)6 b Fy(\))14 b Fs(\026)p Fy(\()p Fs(dx)p Fy(\))22 b Fo(\025)h Fs(\016)2518 1673 y Fj(0)2564 1707 y Fs(:)515 1894 y Fy(As)k(a)f(consequence)e(of)i(this)h(and)e(of)h(property)e Fl(b)m(.)p Fy(,)j(one)f(has)g(for)g(e)n(v)o(ery)f(measurable)g(set)i Fs(A)g Fy(the)515 1994 y(bound)1104 2049 y Fi(\000)1142 2116 y Fo(P)1207 2082 y Fq(2)1200 2137 y Fj(\003)1244 2116 y Fs(\026)1294 2049 y Fi(\001)1332 2116 y Fy(\()p Fs(A)p Fy(\))c Fo(\025)1560 2003 y Fi(Z)1606 2192 y Fp(K)1684 2116 y Fo(P)7 b Fy(\()p Fs(x;)14 b(A)p Fy(\))1965 2049 y Fi(\000)2003 2116 y Fo(P)2061 2128 y Fj(\003)2099 2116 y Fs(\026)2149 2049 y Fi(\001)2187 2116 y Fy(\()p Fs(dx)p Fy(\))23 b Fo(\025)f Fs(\016)s(\016)2523 2082 y Fj(0)2547 2116 y Fs(\027)2588 2128 y Fj(\003)2626 2116 y Fy(\()p Fs(A)p Fy(\))h Fs(:)430 b Fy(\(2.1\))515 2309 y(De\002ne)20 b(the)g(constant)g Fs(")i Fr(=)h Fs(\016)s(\016)1406 2279 y Fj(0)1429 2309 y Fy(.)j(An)20 b(immediate)f(consequence)f(of)i (\(2.1\))f(is)i(that)f(for)g(an)o(y)f(proba-)515 2408 y(bility)h(measure)f Fs(\026)p Fy(,)i(one)e(has)1548 2508 y Fo(j)-14 b(j)g(jP)1654 2474 y Fq(2)1647 2529 y Fj(\003)1692 2508 y Fs(\026)18 b Fo(\000)g Fs("\027)1923 2520 y Fj(\003)1961 2508 y Fo(j)-14 b(j)g(j)28 b Fr(=)22 b(1)c Fo(\000)g Fs(")23 b(:)515 2654 y Fy(No)n(w)29 b(tak)o(e)h(an)o(y) f(tw)o(o)h(probability)d(measures)j Fs(\026)g Fy(and)f Fs(\027)5 b Fy(.)54 b(Denote)29 b(by)g Fs(\021)2684 2666 y Fj(\006)2771 2654 y Fy(the)g(positi)n(v)o(e)g(\(resp.)515 2754 y(ne)o(gati)n(v)o(e\))e(part)i(of)g Fs(\026)c Fo(\000)g Fs(\027)5 b Fy(.)53 b(Since)30 b Fs(\026)g Fy(and)f Fs(\027)35 b Fy(are)29 b(probability)e(measures,)k(one)e(has)h Fo(j)-14 b(j)g(j)p Fs(\021)3173 2766 y Fq(+)3229 2754 y Fo(j)g(j)g(j)44 b Fr(=)515 2853 y Fo(j)-14 b(j)g(j)p Fs(\021)597 2865 y Fj(\000)654 2853 y Fo(j)g(j)g(j)27 b Fr(=)c(\001)p Fy(,)e(say)-5 b(.)24 b(Then,)19 b(since)i Fo(P)1534 2865 y Fj(\003)1592 2853 y Fy(preserv)o(es)e(probability)-5 b(,)18 b(one)i(has)674 3030 y Fo(j)-14 b(j)g(jP)780 2996 y Fq(2)773 3051 y Fj(\003)817 3030 y Fs(\026)19 b Fo(\000)f(P)1034 2996 y Fq(2)1027 3051 y Fj(\003)1070 3030 y Fs(\027)6 b Fo(j)-14 b(j)g(j)28 b Fr(=)22 b Fo(j)-14 b(j)g(jP)1379 2996 y Fq(2)1372 3051 y Fj(\003)1416 3030 y Fs(\021)1457 3042 y Fq(+)1531 3030 y Fo(\000)18 b(P)1679 2996 y Fq(2)1672 3051 y Fj(\003)1716 3030 y Fs(\021)1757 3042 y Fj(\000)1813 3030 y Fo(j)-14 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a Fd(P)2466 1400 y Fp(n)2511 1388 y Fy(\()p Fs(v)s Fy(\))2610 1330 y currentpoint grestore moveto 2610 1330 a 1353 1564 a Fy(Figure)20 b(1:)25 b(Construction)19 b(of)h Fs(D)j Fy(and)c Fs(E)5 b Fy(.)515 1828 y(Since)27 b Fo(P)7 b Fy(\()p Fs(x;)14 b Fn(X)p Fy(\))34 b Fr(=)h(1)27 b Fy(for)f(e)n(v)o(ery)f Fs(x)36 b Fo(2)g Fn(X)p Fy(,)29 b(one)d(has)h Fs(\026)2136 1840 y Fq(0)2173 1828 y Fy(\()p Fs(S)2252 1840 y Fp(x)2293 1828 y Fy(\))35 b Fs(>)g Fr(0)27 b Fy(for)f(e)n(v)o(ery)f Fs(x)j Fy(and)e(therefore)515 1928 y Fs(\026)565 1897 y Fq(2)565 1948 y(0)602 1928 y Fy(\()p Fs(S)686 1897 y Fq(2)723 1928 y Fy(\))c Fr(=)861 1861 y Fi(R)900 1957 y Ff(X)959 1928 y Fs(\026)1009 1940 y Fq(0)1047 1928 y Fy(\()p Fs(S)1126 1940 y Fp(x)1167 1928 y Fy(\))14 b Fs(d\026)1302 1940 y Fq(0)1339 1928 y Fy(\()p Fs(x)p Fy(\))23 b Fs(>)f Fr(0)p Fy(,)e(where)g Fs(\026)1909 1897 y Fq(2)1909 1948 y(0)1969 1928 y Fr(=)j Fs(\026)2107 1940 y Fq(0)2163 1928 y Fo(\002)18 b Fs(\026)2296 1940 y Fq(0)2333 1928 y Fy(.)25 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(Equation)f(Under)902 2636 y(Random)i(P)-6 b(erturbations)p Fw(,)19 b(Commun.)h(Math.)f(Phys.)f Fa(172)i Fw(\(1995\),)g(no.)f(1,)f (119\226141.)515 2794 y([H)6 b(\250)-31 b(or67])147 b(L.)21 b(H)6 b(\250)-31 b(ormander)m(,)23 b Ft(Hypoelliptic)f(Second)i(Or)m (der)e(Dif)o(fer)m(ential)f(Equations)p Fw(,)i(Acta)f(Math.)g Fa(119)902 2885 y Fw(\(1967\),)e(147\226171.)515 3043 y([H)6 b(\250)-31 b(or85])147 b(L.)14 b(H)6 b(\250)-31 b(ormander)m(,)17 b Ft(The)e(Analysis)g(of)h(Linear)f(P)-6 b(artial)15 b(Dif)o(fer)m(ential)f(Oper)o(ator)o(s)j(I\226IV)p Fw(,)d(Springer,)902 3134 y(Ne)n(w)19 b(Y)-8 b(ork,)19 b(1985.)515 3292 y([KS00])167 b(S.)18 b(B.)h(K)o(uksin)h(and)g(A.)e (Shirik)o(yan,)h Ft(Stoc)o(hastic)h(Dissipative)g(PDE')m(s)e(and)i (Gibbs)g(Measur)m(es)p Fw(,)902 3383 y(Commun.)g(Math.)f(Phys.)f Fa(213)i Fw(\(2000\),)f(291\226230.)515 3541 y([KS01])167 b(S.)19 b(B.)f(K)o(uksin)i(and)g(A.)f(Shirik)o(yan,)h Ft(A)f(Coupling)h(Appr)m(oac)o(h)h(r)m(o)e(Randomly)h(F)-8 b(or)m(ced)21 b(Nonlin-)902 3632 y(ear)e(PDE')m(s)p Fw(,)f(Preprint,)g (2001.)515 3790 y([Mat01])143 b(J.)30 b(C.)g(Mattingly)-5 b(,)33 b Ft(Exponential)f(Con)m(ver)m(g)o(ence)h(for)d(the)h(Stoc)o (hastically)g(F)-8 b(or)m(ced)31 b(Navier)o(-)902 3881 y(Stok)o(es)20 b(Equations)g(and)f(Other)g(P)-6 b(artially)19 b(Dissipative)g(Dynamics)p Fw(,)g(Preprint,)f(2001.)515 4039 y([MT94])151 b(S.)22 b(P)-8 b(.)22 b(Me)o(yn)i(and)g(R.)e(L.)g(T) -6 b(weedie,)24 b Ft(Mark)o(o)o(v)h(Chains)e(and)h(Stoc)o(hastic)g (Stability)p Fw(,)g(Springer,)902 4130 y(Ne)n(w)19 b(Y)-8 b(ork,)19 b(1994.)515 4288 y([RBT01])117 b(L.)24 b(Re)o(y-Bellet)g(and) h(L.)f(Thomas,)i Ft(Exponential)f(Con)m(ver)m(g)o(ence)i(to)e (Non-Equilibrium)g(Sta-)902 4379 y(tionary)20 b(States)f(in)g (Classical)g(Statistical)f(Mec)o(hanics)p Fw(,)i(Preprint,)e(2001.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0103280150883--