Content-Type: multipart/mixed; boundary="-------------0411030830284" This is a multi-part message in MIME format. ---------------0411030830284 Content-Type: text/plain; name="04-354.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-354.comments" S.B.Kuksin@ma.hw.ac.uk, Armen.Shirikyan@math.u-psud.fr ---------------0411030830284 Content-Type: text/plain; name="04-354.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-354.keywords" Ruelle-Perron-Frobenius theorem, uniqueness of stationary measure, ergodicity ---------------0411030830284 Content-Type: application/postscript; name="rpf.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="rpf.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: rpf.dvi %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o rpf.ps rpf.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2004.10.08:1548 %%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 4 get round 4 exch put dup dup 5 get round 5 exch put 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 /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup 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 /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image N /rw ch-width 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 dup 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 dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv} B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 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}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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 dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /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 TeXDict begin 39158280 55380996 1000 300 300 (rpf.dvi) @start /Fa 2 51 df49 DI E /Fb 1 50 df<1218127812981218AC12FF 08107D8F0F>49 D E /Fc 2 50 df<13C0A8B51280A23800C000A811127E8D15>43 D<121812F81218AA12FF080D7D8C0E>49 D E /Fd 1 79 df<387F07E0381081801380EA 1840EA1420A2EA1210EA1108EA1084A213421321A213101308A213041302EA280112FEC7 FC13157F932B>78 D E /Fe 7 117 df34 D<380FC3E038030180EB8200EA01C4EA00C813F0136013F0EA01B8EA 0318EA061CEA080C487E38FC1F80130E7E8D17>88 D<12381218A35A13C0EA3360EA3440 EA7800127E12631320EAC340EAC1800B0E7E8D10>107 D109 D114 D<123E124312421270123C120612C21284127808097D880E> I<120CA3121812FE1218A21230A3123212341238070D7E8C0C>I E /Ff 26 122 df<12381278A212381208A21210A21220A212401280050C7D830C>44 D<1270A212F0126004047C830C>46 D<120C121E121CA21200AA1270A212F0126007127C 910C>58 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y(established)g(in)f(our)h(pap)q(er)g ([3)o(])f(\(see)h(Theorem)f(4.1\).)19 b(The)14 b(result)i(obtained)e (in)f(this)i(pa-)257 2453 y(p)q(er)j(enables)g(one)g(to)f(establish)g (the)h(uniqueness)h(of)d(a)h(stationary)g(measure)g(for)g(some)257 2503 y(in\014nite-dimensional)12 b(RDS.)963 2628 y(2)p eop %%Page: 3 3 3 2 bop 257 262 a Fs(2)67 b(Pro)r(of)22 b(of)f(Theorem)h(1)257 352 y Fj(Step)15 b(1)p Fr(.)j(W)m(e)13 b(b)q(egin)h(with)f(the)h(pro)q (of)f(of)g(assertion)h(\(i\).)k(Let)c Fq(\026)e Fp(2)f(P)s Fr(\()p Fq(X)s Fr(\))j(b)q(e)h(a)e(stationary)257 402 y(measure)h(and)g(let)g Fq(f)i Fp(2)11 b(R)p Fr(.)19 b(Without)13 b(loss)h(of)f(generalit)o(y)m(,)f(w)o(e)j(can)f(assume)f (that)727 507 y(\()p Fq(f)r(;)7 b(\026)p Fr(\))12 b(=)880 450 y Fm(Z)903 545 y Fo(X)942 507 y Fq(f)t Fr(\()p Fq(u)p Fr(\))7 b Fq(\026)p Fr(\()p Fq(du)p Fr(\))12 b(=)g(0)p Fq(:)415 b Fr(\(7\))257 616 y(The)17 b(general)f(case)h(can)f(b)q(e)g (reduced)i(to)d(the)i(former)e(b)o(y)g(the)i(c)o(hange)f Fq(f)j Fp(7!)c 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973 y(\014)1138 983 y Fr(\()p Fq(P)1181 989 y Fo(t)1196 983 y Fq(f)t Fr(\))1236 948 y Fm(\014)1236 973 y(\014)1257 983 y Fq(d\026)641 1094 y Fr(=)685 1038 y Fm(Z)708 1132 y Fo(X)739 1059 y Fm(\014)739 1084 y(\014)753 1094 y Fr(\()p Fq(P)796 1100 y Fo(t)810 1094 y Fq(f)t Fr(\))850 1059 y Fm(\014)850 1084 y(\014)872 1094 y Fq(d)p Fr(\()p Fq(P)943 1077 y Fk(\003)937 1105 y Fo(s)961 1094 y Fq(\026)p Fr(\))g(=)1058 1038 y Fm(Z)1081 1132 y Fo(X)1112 1059 y Fm(\014)1112 1084 y(\014)1126 1094 y Fr(\()p Fq(P)1169 1100 y Fo(t)1184 1094 y Fq(f)t Fr(\))1224 1059 y Fm(\014)1224 1084 y(\014)1245 1094 y Fq(d\026)g Fr(=)f Fp(k)p Fq(P)1395 1100 y Fo(t)1409 1094 y Fq(f)t Fp(k)1454 1100 y Fo(\026)1477 1094 y Fq(:)257 1203 y Fr(This)17 b(means)e(that)h Fp(k)p Fq(P)624 1209 y Fo(t)638 1203 y Fq(f)t Fp(k)683 1209 y Fo(\026)722 1203 y Fr(is)g(a)g(non-increasing)h(function)f(of)f Fq(t)p Fr(.)25 b(Hence,)18 b(the)f(con)o(v)o(er-)257 1253 y(gence)g(\(8\))e(will)f(b)q(e)i(established)g(if)e(w)o(e)i(sho)o (w)f(that)h(for)f(an)o(y)f Fq(")h(>)f Fr(0)h(there)i(is)e Fq(t)f Fr(=)g Fq(t)1591 1259 y Fo(")1623 1253 y Fq(>)g Fr(0)257 1303 y(suc)o(h)h(that)857 1389 y Fp(k)p Fq(P)905 1395 y Fo(t)918 1399 y Fe(")935 1389 y Fq(f)t Fp(k)980 1395 y Fo(\026)1015 1389 y Fp(\024)d Fq(":)546 b Fr(\(9\))320 1474 y Fj(Step)15 b(2)p Fr(.)j(W)m(e)c(\014rst)g(assume)g(that)g(there) h(is)f(a)f(sequence)j Fp(f)p Fq(s)1248 1480 y Fo(k)1269 1474 y Fp(g)11 b(\032)h Fn(R)c Fr(for)13 b(whic)o(h)710 1560 y(sup)705 1595 y Fo(u)p Fk(2)p Fo(X)783 1560 y Fq(f)807 1543 y Fl(+)803 1570 y Fo(s)819 1574 y Fe(k)840 1560 y Fr(\()p Fq(u)p Fr(\))e Fp(!)g Fr(0)42 b(as)f Fq(k)13 b Fp(!)e(1)p Fq(:)257 1670 y Fr(In)j(this)g(case,)501 1718 y Fm(Z)524 1812 y Fo(X)555 1741 y Fm(\000)574 1774 y Fq(P)601 1780 y Fo(s)617 1784 y Fe(k)637 1774 y Fq(f)661 1741 y Fm(\001)681 1749 y Fl(+)708 1774 y Fr(\()p Fq(u)p Fr(\))7 b Fq(d\026)k Fr(=)873 1718 y Fm(Z)896 1812 y Fo(X)935 1774 y Fq(f)959 1757 y Fl(+)955 1785 y Fo(s)971 1789 y Fe(k)991 1774 y Fr(\()p Fq(u)p Fr(\))c Fq(d\026)k Fp(!)g Fr(0)41 b(as)h Fq(k)12 b Fp(!)f(1)p Fq(:)169 b Fr(\(10\))257 1883 y(Moreo)o(v)o(er,)14 b(it)g(follo)o(ws)e(from)g (\(7\))i(that)614 1969 y(\()p Fq(f)r(;)7 b(\026)p Fr(\))12 b(=)g(\()p Fq(P)817 1952 y Fk(\003)811 1979 y Fo(t)836 1969 y Fq(\026;)7 b(f)t Fr(\))12 b(=)f(\()p Fq(f)1011 1975 y Fo(t)1026 1969 y Fq(;)c(\026)p Fr(\))12 b(=)f(0)p Fq(;)48 b(t)12 b Fp(2)f(T)f Fq(;)257 2055 y Fr(where)15 b Fq(f)397 2061 y Fo(t)424 2055 y Fr(=)d Fq(P)495 2061 y Fo(t)509 2055 y Fq(f)t Fr(,)i(and)g(therefore)657 2140 y(\()p Fq(f)697 2122 y Fl(+)693 2150 y Fo(t)725 2140 y Fq(;)7 b(\026)p Fr(\))k(=)h(\()p Fq(f)880 2122 y Fk(\000)876 2150 y Fo(t)909 2140 y Fq(;)7 b(\026)p Fr(\))41 b(for)14 b(an)o(y)41 b Fq(t)11 b Fp(2)h(T)e Fq(:)325 b Fr(\(11\))257 2226 y(Com)o(bining)11 b(this)j(with)g(\(10\),)f(w)o(e)h(deriv)o(e)641 2311 y(\()p Fq(f)681 2294 y Fl(+)677 2322 y Fo(s)693 2326 y Fe(k)713 2311 y Fq(;)7 b(\026)p Fr(\))k(=)h(\()p Fq(f)868 2294 y Fk(\000)864 2322 y Fo(s)880 2326 y Fe(k)900 2311 y Fq(;)7 b(\026)p Fr(\))12 b Fp(!)f Fr(0)41 b(as)h Fq(k)12 b Fp(!)f(1)p Fq(;)257 2397 y Fr(whence)16 b(w)o(e)e(conclude)g (that)g(\(9\))g(holds.)k(A)c(similar)d(argumen)o(t)i(sho)o(ws)h(that)g (if)715 2483 y(sup)711 2518 y Fo(u)p Fk(2)p Fo(X)789 2483 y Fq(f)813 2466 y Fk(\000)809 2493 y Fo(s)825 2497 y Fe(k)845 2483 y Fr(\()p Fq(u)p Fr(\))e Fp(!)f Fr(0)41 b(as)h Fq(k)12 b Fp(!)f(1)963 2628 y Fr(3)p eop %%Page: 4 4 4 3 bop 257 262 a Fr(for)14 b(a)f(sequence)k Fp(f)p Fq(s)568 268 y Fo(k)588 262 y Fp(g)11 b(\032)h Fn(R)m Fr(,)f(then)j(\(9\))g(is)g (ful\014lled.)320 311 y Fj(Step)i(3)p Fr(.)21 b(Th)o(us,)16 b(w)o(e)f(can)g(assume)f(that)h(\(2\))g(and)g(\(3\))g(hold)g(with)f(a)h (constan)o(t)g Fq(\013)f(>)f Fr(0.)257 361 y(By)g(condition)f(\(H\),)g (there)i(is)e(an)h(increasing)f(sequence)j Fp(f)p Fq(t)1190 367 y Fo(k)1210 361 y Fp(g)d Fr(suc)o(h)h(that)g(\(4\))f(and)h(\(5\))f (are)257 411 y(satis\014ed.)19 b(W)m(e)14 b(claim)d(that)645 502 y Fp(k)p Fq(P)693 508 y Fo(t)706 512 y Fe(k)725 502 y Fq(f)t Fp(k)770 508 y Fo(\026)804 502 y Fp(\024)h Fq(a)870 485 y Fo(k)q Fk(\000)p Fl(1)870 515 y Fo(f)933 502 y Fp(k)p Fq(f)t Fp(k)999 508 y Fo(\026)1063 502 y Fr(for)i(all)e Fq(k)h Fp(\025)f Fr(1)o Fq(;)321 b Fr(\(12\))257 603 y(where)15 b Fq(a)399 609 y Fo(f)432 603 y Fr(=)d(1)d Fp(\000)h Fq(A)579 585 y Fk(\000)p Fl(1)579 615 y Fo(f)635 603 y Fq(<)i Fr(1.)320 652 y(The)17 b(pro)q(of)f(is)g(b)o(y)g (induction)g(on)g Fq(k)q Fr(.)26 b(Inequalit)o(y)16 b(\(12\))g(is)g(ob) o(vious)g(for)g Fq(k)h Fr(=)f(1.)25 b(As-)257 702 y(suming)13 b(that)i(\(12\))g(is)f(established)i(for)e Fq(k)g Fp(\024)f Fq(r)q Fr(,)h(w)o(e)h(no)o(w)g(pro)o(v)o(e)f(it)h(for)f Fq(k)g Fr(=)f Fq(r)e Fr(+)f(1.)20 b(Note)257 752 y(that,)14 b(b)o(y)g(\(4\))g(and)g(\(5\),)f(w)o(e)h(ha)o(v)o(e)793 737 y Fl(1)381 808 y Fm(Z)404 902 y Fo(X)443 865 y Fq(f)467 847 y Fk(\006)463 875 y Fo(t)476 879 y Fe(r)496 865 y Fq(d\026)d Fr(=)598 808 y Fm(Z)621 902 y Fo(X)659 865 y Fq(P)686 871 y Fo(d)703 875 y Fe(r)722 865 y Fq(f)746 847 y Fk(\006)742 875 y Fo(t)755 879 y Fe(r)775 865 y Fq(d\026)g Fp(\024)16 b Fr(sup)877 899 y Fo(u)p Fk(2)p Fo(X)948 831 y Fm(\000)967 865 y Fq(P)994 871 y Fo(d)1011 875 y Fe(r)1030 865 y Fq(f)1054 847 y Fk(\006)1050 875 y Fo(t)1063 879 y Fe(r)1083 831 y Fm(\001)1102 865 y Fr(\()p Fq(u)p Fr(\))11 b Fp(\024)h Fq(A)1244 871 y Fo(f)1283 865 y Fr(inf)1273 891 y Fo(u)p Fk(2)p Fo(X)1344 831 y Fm(\000)1363 865 y Fq(P)1390 871 y Fo(d)1407 875 y Fe(r)1426 865 y Fq(f)1450 847 y Fk(\006)1446 875 y Fo(t)1459 879 y Fe(r)1479 831 y Fm(\001)1498 865 y Fr(\()p Fq(u)p Fr(\))p Fq(;)257 980 y Fr(whence)562 1030 y Fq(P)589 1036 y Fo(d)606 1040 y Fe(r)625 1030 y Fq(f)649 1012 y Fk(\006)645 1040 y Fo(t)658 1044 y Fe(r)677 1030 y Fr(\()p Fq(u)p Fr(\))e Fp(\000)f Fq(A)815 1012 y Fk(\000)p Fl(1)815 1043 y Fo(f)860 1030 y Fp(k)p Fq(f)905 1012 y Fk(\006)901 1040 y Fo(t)914 1044 y Fe(r)933 1030 y Fp(k)954 1036 y Fo(\026)988 1030 y Fp(\025)j Fr(0)41 b(for)14 b(an)o(y)41 b Fq(u)11 b Fp(2)g Fq(X)q(:)257 1105 y Fr(It)j(follo)o(ws)f(that)333 1150 y Fm(Z)356 1245 y Fo(X)387 1172 y Fm(\014)387 1197 y(\014)401 1207 y Fq(P)428 1213 y Fo(d)445 1217 y Fe(r)464 1207 y Fq(f)488 1189 y Fk(\006)484 1217 y Fo(t)497 1221 y Fe(r)526 1207 y Fp(\000)c Fq(A)598 1189 y Fk(\000)p Fl(1)598 1219 y Fo(f)643 1207 y Fp(k)p Fq(f)688 1189 y Fk(\006)684 1217 y Fo(t)697 1221 y Fe(r)717 1207 y Fp(k)738 1213 y Fo(\026)760 1172 y Fm(\014)760 1197 y(\014)780 1207 y Fq(d\026)j Fr(=)882 1150 y Fm(Z)906 1245 y Fo(X)937 1173 y Fm(\000)956 1207 y Fq(P)983 1213 y Fo(d)1000 1217 y Fe(r)1019 1207 y Fq(f)1043 1189 y Fk(\006)1039 1217 y Fo(t)1052 1221 y Fe(r)1081 1207 y Fp(\000)d Fq(A)1153 1189 y Fk(\000)p Fl(1)1153 1219 y Fo(f)1198 1207 y Fp(k)p Fq(f)1243 1189 y Fk(\006)1239 1217 y Fo(t)1252 1221 y Fe(r)1272 1207 y Fp(k)1293 1213 y Fo(\026)1315 1173 y Fm(\001)1341 1207 y Fq(d\026)i Fr(=)h Fq(a)1465 1213 y Fo(f)1486 1207 y Fp(k)p Fq(f)1531 1189 y Fk(\006)1527 1217 y Fo(t)1540 1221 y Fe(r)1560 1207 y Fp(k)1581 1213 y Fo(\026)1603 1207 y Fq(:)257 1322 y Fr(W)m(e)17 b(no)o(w)f(estimate)h (the)g(expression)h Fp(k)p Fq(P)916 1328 y Fo(t)929 1332 y Fe(r)q Fc(+1)982 1322 y Fq(f)t Fp(k)1027 1328 y Fo(\026)1067 1322 y Fr(=)e Fp(k)p Fq(P)1163 1328 y Fo(d)1180 1332 y Fe(r)1199 1322 y Fq(f)1219 1328 y Fo(t)1232 1332 y Fe(r)1250 1322 y Fp(k)1271 1328 y Fo(\026)1293 1322 y Fr(.)27 b(In)17 b(view)g(of)i(\(11\),)e(w)o(e)257 1371 y(ha)o(v)o(e)665 1463 y Fp(k)p Fq(f)710 1445 y Fl(+)706 1473 y Fo(t)738 1463 y Fp(k)759 1469 y Fo(\026)793 1463 y Fr(=)12 b Fp(k)p Fq(f)882 1445 y Fk(\000)878 1473 y Fo(t)910 1463 y Fp(k)931 1469 y Fo(\026)995 1463 y Fr(for)h(an)o(y)41 b Fq(t)12 b Fp(2)f(T)f Fq(;)341 b Fr(\(13\))257 1554 y(and)14 b(therefore)305 1600 y Fm(Z)328 1694 y Fo(X)359 1621 y Fm(\014)359 1646 y(\014)373 1656 y Fq(P)400 1662 y Fo(d)417 1666 y Fe(r)436 1656 y Fq(f)456 1662 y Fo(t)469 1666 y Fe(r)488 1621 y Fm(\014)488 1646 y(\014)501 1656 y Fq(d\026)e Fr(=)603 1600 y Fm(Z)627 1694 y Fo(X)658 1621 y Fm(\014)658 1646 y(\014)672 1656 y Fq(P)699 1662 y Fo(d)716 1666 y Fe(r)735 1656 y Fr(\()p Fq(f)775 1638 y Fl(+)771 1666 y Fo(t)784 1670 y Fe(r)812 1656 y Fp(\000)e Fq(f)878 1638 y Fk(\000)874 1666 y Fo(t)887 1670 y Fe(r)907 1656 y Fr(\))923 1621 y Fm(\014)923 1646 y(\014)937 1656 y Fq(d\026)560 1767 y Fp(\024)603 1711 y Fm(Z)627 1805 y Fo(X)658 1732 y Fm(\014)658 1757 y(\014)672 1767 y Fq(P)699 1773 y Fo(d)716 1777 y Fe(r)735 1767 y Fq(f)759 1749 y Fl(+)755 1777 y Fo(t)768 1781 y Fe(r)796 1767 y Fp(\000)g Fq(A)869 1749 y Fk(\000)p Fl(1)869 1780 y Fo(f)913 1767 y Fp(k)p Fq(f)958 1749 y Fl(+)954 1777 y Fo(t)967 1781 y Fe(r)987 1767 y Fp(k)1008 1773 y Fo(\026)1030 1732 y Fm(\014)1030 1757 y(\014)1050 1767 y Fq(d\026)f Fr(+)1148 1711 y Fm(Z)1171 1805 y Fo(X)1202 1732 y Fm(\014)1202 1757 y(\014)1216 1767 y Fq(P)1243 1773 y Fo(d)1260 1777 y Fe(r)1279 1767 y Fq(f)1303 1749 y Fk(\000)1299 1777 y Fo(t)1312 1781 y Fe(r)1341 1767 y Fp(\000)g Fq(A)1413 1749 y Fk(\000)p Fl(1)1413 1780 y Fo(f)1458 1767 y Fp(k)p Fq(f)1503 1749 y Fk(\000)1499 1777 y Fo(t)1512 1781 y Fe(r)1532 1767 y Fp(k)1553 1773 y Fo(\026)1575 1732 y Fm(\014)1575 1757 y(\014)1595 1767 y Fq(d\026)560 1857 y Fp(\024)i Fq(a)625 1863 y Fo(f)647 1824 y Fm(\000)666 1857 y Fp(k)p Fq(f)711 1840 y Fl(+)707 1868 y Fo(t)720 1872 y Fe(r)739 1857 y Fp(k)760 1863 y Fo(\026)791 1857 y Fr(+)f Fp(k)p Fq(f)878 1840 y Fk(\000)874 1868 y Fo(t)887 1872 y Fe(r)906 1857 y Fp(k)927 1863 y Fo(\026)949 1824 y Fm(\001)980 1857 y Fr(=)i Fq(a)1046 1863 y Fo(f)1067 1857 y Fp(k)p Fq(f)1108 1863 y Fo(t)1121 1867 y Fe(r)1140 1857 y Fp(k)1161 1863 y Fo(\026)1183 1857 y Fq(:)257 1949 y Fr(Using)i(the)h(induction)e(h)o(yp)q(othesis,)h(w)o(e)g(deriv)o (e)628 2002 y Fm(Z)651 2097 y Fo(X)682 2024 y Fm(\014)682 2048 y(\014)696 2059 y Fq(P)723 2065 y Fo(t)736 2069 y Fe(r)q Fc(+1)790 2059 y Fq(f)814 2024 y Fm(\014)814 2048 y(\014)828 2059 y Fq(d\026)d Fp(\024)h Fq(a)952 2065 y Fo(f)974 2059 y Fp(k)p Fq(P)1022 2065 y Fo(t)1035 2069 y Fe(r)1052 2059 y Fq(f)t Fp(k)1097 2065 y Fo(\026)1132 2059 y Fp(\024)g Fq(a)1198 2042 y Fo(r)1198 2069 y(f)1219 2059 y Fp(k)p Fq(f)t Fp(k)1285 2065 y Fo(\026)1308 2059 y Fq(;)257 2173 y Fr(whic)o(h)18 b(completes)g(the)h(pro)q(of)e(of)k (\(12\).)30 b(Inequalit)o(y)18 b(\(9\))g(is)g(an)g(ob)o(vious)f (consequence)257 2223 y(of)g(\(12\).)320 2273 y Fj(Step)g(4)p Fr(.)26 b(Let)17 b(us)f(turn)h(to)f(assertion)h(\(ii\).)25 b(W)m(e)16 b(\014rst)h(sho)o(w)g(that)f(if)g Fq(\026)1448 2279 y Fl(1)1466 2273 y Fq(;)7 b(\026)1510 2279 y Fl(2)1544 2273 y Fp(2)15 b(P)s Fr(\()p Fq(X)s Fr(\))257 2323 y(are)g(t)o(w)o(o)e (stationary)g(measures)h(that)g(do)f(not)h(coincide)g(on)f Fp(R)p Fr(,)h(then)g(they)h(are)f(singular.)p 257 2358 573 2 v 304 2384 a Fb(1)321 2396 y Fh(Here)i(and)g(henceforth)d(a)k (form)o(ula)d(in)o(v)o(olving)g(the)i(sym)o(b)q(ol)f Fw(\006)i Fh(is)f(a)h(brief)e(writing)h(for)g(the)g(t)o(w)o(o)257 2436 y(form)o(ulas)10 b(corresp)q(ondi)o(ng)e(to)j(the)g(upp)q(er)f (and)h(lo)o(w)o(er)g(signs.)963 2628 y Fr(4)p eop %%Page: 5 5 5 4 bop 257 262 a Fr(T)m(o)18 b(this)h(end,)g(w)o(e)g(apply)f(a)g(w)o (ell-kno)o(wn)g(argumen)o(t)f(\(for)h(instance,)i(see)g([1)o(,)g(Prop)q (osi-)257 311 y(tion)14 b(3.2.5]\).)i(Let)e Fq(f)i Fp(2)c(R)i Fr(b)q(e)g(a)g(function)f(suc)o(h)i(that)823 403 y(\()p Fq(f)r(;)7 b(\026)905 409 y Fl(1)924 403 y Fr(\))12 b Fp(6)p Fr(=)f(\()p Fq(f)r(;)c(\026)1077 409 y Fl(2)1096 403 y Fr(\))p Fq(:)492 b Fr(\(14\))257 494 y(By)15 b(assertion)f (\(i\),)482 585 y Fq(P)509 591 y Fo(t)524 585 y Fq(f)i Fp(!)11 b Fr(\()p Fq(f)r(;)c(\026)695 591 y Fo(i)709 585 y Fr(\))41 b(as)h Fq(k)13 b Fp(!)e(1)41 b Fr(in)g Fq(L)1120 568 y Fl(1)1139 585 y Fr(\()p Fq(X)q(;)7 b(\026)1234 591 y Fo(i)1247 585 y Fr(\))p Fq(;)48 b(i)12 b Fr(=)g(1)p Fq(;)7 b Fr(2)p Fq(:)257 677 y Fr(Therefore,)15 b(there)g(is)f(a)g (sequence)i Fq(s)831 683 y Fo(k)863 677 y Fp(!)11 b Fr(+)p Fp(1)j Fr(suc)o(h)g(that)364 768 y Fq(P)391 774 y Fo(s)407 778 y Fe(k)427 768 y Fq(f)i Fp(!)11 b Fr(\()p Fq(f)r(;)c(\026)598 774 y Fo(i)612 768 y Fr(\))42 b(as)f Fq(k)13 b Fp(!)e(1)41 b Fq(\026)944 774 y Fo(i)958 768 y Fr(-almost)12 b(ev)o(erywhere)r Fq(;)48 b(i)11 b Fr(=)h(1)p Fq(;)7 b Fr(2)p Fq(:)106 b Fr(\(15\))257 859 y(Denote)16 b(b)o(y)e Fq(C)489 865 y Fo(i)502 859 y Fr(,)h Fq(i)e Fr(=)g(1)p Fq(;)7 b Fr(2,)13 b(the)j(set)f(of)f(p)q(oin)o(ts)h Fq(u)d Fp(2)h Fq(X)18 b Fr(for)c(whic)o(h)h(\(15\))f(tak)o(es)h(place.)21 b(W)m(e)257 909 y(ha)o(v)o(e)d Fq(\026)382 915 y Fl(1)401 909 y Fr(\()p Fq(C)447 915 y Fl(1)465 909 y Fr(\))g(=)g Fq(\026)574 915 y Fl(2)593 909 y Fr(\()p Fq(C)639 915 y Fl(2)657 909 y Fr(\))h(=)f(1)f(and,)h(in)g(view)f(of)k(\(14\))o(,)e Fq(C)1218 915 y Fl(1)1248 909 y Fp(\\)11 b Fq(C)1317 915 y Fl(2)1353 909 y Fr(=)19 b Fn(?)p Fr(.)30 b(This)17 b(means)257 959 y(that)d Fq(\026)372 965 y Fl(1)405 959 y Fr(and)f Fq(\026)510 965 y Fl(2)543 959 y Fr(are)h(singular.)320 1009 y(W)m(e)d(no)o(w)h(assume)f(that)h Fq(\026)728 1015 y Fl(1)746 1009 y Fq(;)7 b(\026)790 1015 y Fl(2)820 1009 y Fp(2)k(P)s Fr(\()p Fq(X)s Fr(\))i(are)g(t)o(w)o(o)e(stationary)g (measures)i(for)e Fq(P)1583 994 y Fk(\003)1577 1019 y Fo(t)1613 1009 y Fr(that)257 1059 y(do)k(not)h(coincide)f(on)g Fp(R)p Fr(.)22 b(Consider)16 b(the)g(measure)f Fq(\026)f Fr(=)g(\()p Fq(\026)1217 1065 y Fl(1)1246 1059 y Fr(+)c Fq(\026)1313 1065 y Fl(2)1332 1059 y Fr(\))p Fq(=)p Fr(2.)21 b(It)16 b(is)f(clear)g(that)257 1108 y Fq(\026)j Fp(2)f(P)s Fr(\()p Fq(X)s Fr(\))i(is)e(a)g(stationary)g(measure)h(and)f(that)g Fq(\026)h Fr(and)f Fq(\026)1244 1114 y Fl(1)1280 1108 y Fr(do)g(not)h(coincide)g(on)f Fp(R)p Fr(.)257 1158 y(As)g(w)o(as)g(sho)o(wn)f(ab)q(o)o(v)o(e,)g(the)h(measures)g Fq(\026)f Fr(and)h Fq(\026)1069 1164 y Fl(1)1104 1158 y Fr(m)o(ust)e(b)q(e)i(singular.)25 b(On)17 b(the)g(other)257 1208 y(hand,)e(the)g(de\014nition)f(of)g Fq(\026)g Fr(implies)f(that)i (they)g(are)g(not)f(singular.)20 b(The)15 b(con)o(tradiction)257 1258 y(obtained)f(completes)g(the)g(pro)q(of)g(of)f(Theorem)g(1.)257 1395 y Fs(3)67 b(Su\016cien)n(t)28 b(conditions)f(for)f(application)i (of)e(Theo-)358 1470 y(rem)d(1)257 1561 y Fr(Let)17 b Fp(R)g(\032)f Fq(C)464 1567 y Fo(b)480 1561 y Fr(\()p Fq(X)s Fr(\))i(b)q(e)f(a)f(family)e(of)i(functions)h(suc)o(h)g(that)g Fq(f)f Fp(\000)11 b Fq(c)16 b Fp(2)g(R)g Fr(for)h(an)o(y)f Fq(f)21 b Fp(2)15 b(R)257 1611 y Fr(and)f Fq(c)e Fp(2)f Fn(R)m Fr(.)k(W)m(e)f(consider)h(a)f(transition)f(function)h Fq(P)6 b Fr(\()p Fq(t;)h(u;)g Fr(\000\),)12 b Fq(t)f Fp(2)h(T)e Fr(,)j Fq(u)f Fp(2)f Fq(X)s Fr(,)j(\000)e Fp(2)f(B)1646 1617 y Fo(X)1678 1611 y Fr(,)257 1660 y(satisfying)i(the) i(follo)o(wing)c(conditions.)247 1777 y Fg(\(H)303 1783 y Fa(1)324 1777 y Fr(\))21 b Ff(F)m(or)16 b(an)o(y)f Fq(f)20 b Fp(2)14 b(R)i Ff(there)h(is)f(an)g(increasing)g(sequence)i Fq(s)1252 1783 y Fo(m)1298 1777 y Fp(!)d Fr(+)p Fp(1)g Ff(suc)o(h)i(that)f(the)361 1826 y(family)11 b Fp(f)p Fq(P)536 1832 y Fo(s)552 1836 y Fe(m)581 1826 y Fq(f)r(;)c(m)12 b Fp(\025)f Fr(1)p Fp(g)j Ff(is)g(uniformly)d(equicon)o(tin)o(uous.)247 1909 y Fg(\(H)303 1915 y Fa(2)324 1909 y Fr(\))21 b Ff(F)m(or)14 b(ev)o(ery)g Fq(r)f(>)f Fr(0)h Ff(there)i(are)g Fq(")d(>)f Fr(0)j Ff(and)g Fq(l)e Fp(\025)g Fr(1)i Ff(suc)o(h)g(that)651 2001 y Fq(P)6 b Fr(\()p Fq(l)q(;)h(u;)g(B)806 2007 y Fo(X)837 2001 y Fr(\()p Fq(v)q(;)g(r)q Fr(\)\))12 b Fp(\025)f Fq(")42 b Ff(for)14 b(an)o(y)41 b Fq(u;)7 b(v)12 b Fp(2)g Fq(X)q(;)223 b Fr(\(16\))361 2092 y Ff(where)15 b Fq(B)512 2098 y Fo(X)544 2092 y Fr(\()p Fq(v)q(;)7 b(r)q Fr(\))14 b Ff(denotes)h(a)f(ball)f(in)g Fq(X)18 b Ff(of)13 b(radius)h Fq(r)h Ff(cen)o(tred)g(at)f Fq(v)q Ff(.)257 2241 y Fg(Theorem)h(2.)21 b Fj(L)n(et)d(c)n(onditions)h Fr(\(H)842 2247 y Fl(1)860 2241 y Fr(\))g Fj(and)g Fr(\(H)1026 2247 y Fl(2)1045 2241 y Fr(\))f Fj(b)n(e)g(satis\014e)n(d.)29 b(Then)19 b Fr(\(H\))f Fj(is)g(ful\014l)r(le)n(d)257 2291 y(for)f Fp(R)p Fj(,)g(and)h(ther)n(efor)n(e)e(assertions)h Fr(\(i\))g Fj(and)h Fr(\(ii\))e Fj(of)h(The)n(or)n(em)g(1)g(hold.)25 b(Mor)n(e)n(over,)17 b(the)257 2341 y(supp)n(ort)e(of)g Fq(\026)g Fj(c)n(oincides)g(with)f Fq(X)s Fj(,)i(and)f(for)f(any)i Fq(f)g Fp(2)c(R)j Fj(we)f(have)613 2432 y Fq(P)640 2438 y Fo(t)654 2432 y Fq(f)i Fp(!)11 b Fr(\()p Fq(f)r(;)c(\026)p Fr(\))43 b Fj(as)f Fq(t)12 b Fp(!)f(1)42 b Fj(in)g Fq(C)1236 2438 y Fo(b)1253 2432 y Fr(\()p Fq(X)s Fr(\))p Fq(:)282 b Fr(\(17\))963 2628 y(5)p eop %%Page: 6 6 6 5 bop 257 262 a Fj(Pr)n(o)n(of.)20 b Fr(Let)d Fq(f)k Fp(2)15 b(R)i Fr(b)q(e)g(an)f(arbitrary)h(function)f(satisfying)f (inequalities)h(\(2\))h(and)f(\(3\),)257 311 y(where)21 b Fq(f)407 294 y Fk(\006)403 322 y Fo(t)455 311 y Fr(=)g(\()p Fq(P)551 317 y Fo(t)565 311 y Fq(f)t Fr(\))605 296 y Fk(\006)653 311 y Fr(and)e Fq(\013)g Fr(is)f(a)h(p)q(ositiv)o(e)g (constan)o(t.)34 b(W)m(e)18 b(m)o(ust)g(construct)j(a)e(se-)257 361 y(quence)d Fq(t)410 367 y Fo(k)444 361 y Fr(for)d(whic)o(h)h(\(4\)) g(and)g(\(5\))g(hold.)320 411 y(In)e(view)f(of)h(condition)f(\(H)736 417 y Fl(1)755 411 y Fr(\),)h(there)h(is)f Fq(r)g(>)g Fr(0)g(and)f(for)h(an)o(y)g Fq(m)g Fp(\025)f Fr(1)h(there)h(is)f Fq(u)1556 417 y Fo(s)1572 421 y Fe(m)1613 411 y Fp(2)f Fq(X)257 461 y Fr(suc)o(h)k(that)664 552 y(inf)586 581 y Fo(v)q Fk(2)p Fo(B)651 585 y Fe(X)677 581 y Fl(\()p Fo(u)710 585 y Fe(s)724 589 y(m)754 581 y Fo(;r)q Fl(\))800 552 y Fq(f)824 535 y Fl(+)820 562 y Fo(s)836 566 y Fe(m)865 552 y Fr(\()p Fq(v)q Fr(\))e Fp(\025)i Fr(sup)974 587 y Fo(v)q Fk(2)p Fo(X)1051 552 y Fq(f)1075 535 y Fl(+)1071 562 y Fo(s)1087 566 y Fe(m)1117 552 y Fr(\()p Fq(v)q Fr(\))10 b Fp(\000)1226 524 y Fq(\013)p 1226 543 27 2 v 1229 581 a Fr(2)1269 552 y Fp(\025)1318 524 y Fq(\013)p 1318 543 V 1321 581 a Fr(2)1350 552 y Fq(:)254 b Fr(\(18\))257 670 y(Let)15 b Fq(")e(>)f Fr(0)i(and)g Fq(l)f Fp(\025)g Fr(1)h(b)q(e)h(the)f(constan)o(ts)i(en)o(tering)e(condition)g(\(H)1332 676 y Fl(2)1351 670 y Fr(\).)19 b(Since)c(\(16\))f(holds)257 720 y(uniformly)h(with)i(resp)q(ect)i(to)e Fq(u)g Fp(2)g Fq(X)s Fr(,)h(the)f(Chapman{Kolmo)o(goro)o(v)d(relation)i(implies)257 770 y(that)550 861 y Fq(P)6 b Fr(\()p Fq(l)612 844 y Fk(0)624 861 y Fq(;)h(u;)g(B)717 867 y Fo(X)747 861 y Fr(\()p Fq(v)q(;)g(r)q Fr(\)\))12 b Fp(\025)g Fq(")42 b Fr(for)14 b(an)o(y)f Fq(u;)7 b(v)12 b Fp(2)g Fq(X)s Fr(,)i Fq(l)1305 846 y Fk(0)1328 861 y Fp(\025)e Fq(l)q(:)219 b Fr(\(19\))320 952 y(W)m(e)12 b(no)o(w)h(construct)i(a)d(subsequence)k Fp(f)p Fq(t)961 958 y Fo(k)981 952 y Fp(g)11 b(\032)h(f)p Fq(s)1097 958 y Fo(m)1129 952 y Fp(g)h Fr(p)q(ossessing)h(the)g (required)g(prop-)257 1002 y(ert)o(y)m(.)j(W)m(e)10 b(set)h Fq(t)498 1008 y Fl(1)528 1002 y Fr(=)h Fq(s)591 1008 y Fl(1)620 1002 y Fr(and)e(assume)g(that)g Fq(t)938 1008 y Fl(2)956 1002 y Fq(;)d(:)g(:)g(:)e(;)i(t)1064 1008 y Fo(k)1094 1002 y Fr(are)k(already)f(constructed.)18 b(Let)11 b Fq(t)1627 1008 y Fo(k)q Fl(+1)257 1052 y Fr(b)q(e)16 b(the)f(smallest)f(elemen)o(t)h(of)f Fp(f)p Fq(s)791 1058 y Fo(m)823 1052 y Fp(g)g Fr(greater)i(than)f Fq(t)1113 1058 y Fo(k)1143 1052 y Fr(+)c Fq(l)q Fr(.)21 b(By)15 b(de\014nition,)g(there)h(is)f(an)257 1102 y(in)o(teger)g Fq(m)d Fp(\025)f Fr(1)j(suc)o(h)h(that)f Fq(s)724 1108 y Fo(m)767 1102 y Fr(=)e Fq(t)826 1108 y Fo(k)846 1102 y Fr(.)18 b(In)c(view)g(of)i(\(18\))e(and)g(\(19\),)f(w)o(e)h(ha)o(v)o (e)370 1181 y Fm(\000)389 1214 y Fq(P)416 1220 y Fo(d)433 1224 y Fe(k)453 1214 y Fq(f)477 1197 y Fl(+)473 1225 y Fo(t)486 1229 y Fe(k)507 1181 y Fm(\001)526 1214 y Fr(\()p Fq(u)p Fr(\))d(=)637 1158 y Fm(Z)660 1252 y Fo(X)698 1214 y Fq(P)6 b Fr(\()p Fq(d)769 1220 y Fo(k)789 1214 y Fq(;)h(u;)g(dv)q Fr(\))p Fq(f)934 1197 y Fl(+)930 1225 y Fo(t)943 1229 y Fe(k)962 1214 y Fr(\()p Fq(v)q Fr(\))13 b Fp(\025)1071 1158 y Fm(Z)1095 1252 y Fo(B)1120 1256 y Fe(X)1146 1252 y Fl(\()p Fo(u)1179 1256 y Fe(t)1191 1263 y(k)1211 1252 y Fo(;r)q Fl(\))1259 1214 y Fq(P)6 b Fr(\()p Fq(d)1330 1220 y Fo(k)1350 1214 y Fq(;)h(u;)g(dv)q Fr(\))p Fq(f)1495 1197 y Fl(+)1491 1225 y Fo(t)1504 1229 y Fe(k)1523 1214 y Fr(\()p Fq(v)q Fr(\))593 1326 y Fp(\025)12 b Fq(P)670 1293 y Fm(\000)688 1326 y Fq(l)q(;)7 b(u;)g(B)794 1332 y Fo(X)825 1326 y Fr(\()p Fq(u)865 1332 y Fo(t)878 1336 y Fe(k)898 1326 y Fq(;)g(r)q Fr(\))953 1293 y Fm(\001)1051 1326 y Fr(inf)978 1355 y Fo(v)q Fk(2)p Fo(B)1043 1359 y Fe(X)1070 1355 y Fl(\()p Fo(u)1103 1359 y Fe(t)1115 1366 y(k)1135 1355 y Fo(;r)q Fl(\))1181 1326 y Fq(f)1205 1308 y Fl(+)1201 1336 y Fo(t)1214 1340 y Fe(k)1234 1326 y Fr(\()p Fq(v)q Fr(\))12 b Fp(\025)1348 1298 y Fq(\013")p 1348 1317 47 2 v 1361 1355 a Fr(2)1399 1326 y Fq(;)205 b Fr(\(20\))257 1445 y(where)15 b Fq(d)399 1451 y Fo(k)431 1445 y Fr(=)d Fq(t)490 1451 y Fo(k)q Fl(+1)561 1445 y Fp(\000)e Fq(t)618 1451 y Fo(k)638 1445 y Fr(.)18 b(On)d(the)f(other)h (hand,)491 1503 y Fm(\000)510 1536 y Fq(P)537 1542 y Fo(d)554 1546 y Fe(k)574 1536 y Fq(f)598 1518 y Fl(+)594 1546 y Fo(t)607 1550 y Fe(k)627 1503 y Fm(\001)646 1536 y Fr(\()p Fq(u)p Fr(\))d Fp(\024)k Fr(sup)758 1571 y Fo(u)p Fk(2)p Fo(X)836 1536 y Fq(f)860 1518 y Fl(+)856 1546 y Fo(t)869 1550 y Fe(k)889 1536 y Fr(\()p Fq(u)p Fr(\))c Fp(\024)k Fr(sup)1001 1571 y Fo(u)p Fk(2)p Fo(X)1072 1501 y Fm(\014)1072 1526 y(\014)1086 1536 y Fq(f)1106 1542 y Fo(t)1119 1546 y Fe(k)1139 1536 y Fr(\()p Fq(u)p Fr(\))1195 1501 y Fm(\014)1195 1526 y(\014)1220 1536 y Fp(\024)h Fr(sup)1264 1571 y Fo(u)p Fk(2)p Fo(X)1336 1501 y Fm(\014)1336 1526 y(\014)1350 1536 y Fq(f)t Fr(\()p Fq(u)p Fr(\))1430 1501 y Fm(\014)1430 1526 y(\014)1444 1536 y Fq(:)160 b Fr(\(21\))257 1654 y(Com)o(bining)12 b(\(20\))i(and)f(\(21\),)g(w)o(e)i(arriv)o(e)e(at)h(\(4\))g(with)748 1746 y Fq(A)779 1752 y Fo(f)813 1746 y Fr(=)d(2\()p Fq(\013")p Fr(\))955 1728 y Fk(\000)p Fl(1)1011 1746 y Fr(sup)1007 1780 y Fo(u)p Fk(2)p Fo(X)1078 1710 y Fm(\014)1078 1735 y(\014)1092 1746 y Fq(f)t Fr(\()p Fq(u)p Fr(\))1172 1710 y Fm(\014)1172 1735 y(\014)1187 1746 y Fq(:)257 1864 y Fr(Inequalit)o(y)j(\(5\))g(can)g(b)q(e)g(pro)o(v)o(ed)g(in)g(a)g (similar)d(w)o(a)o(y)m(.)320 1913 y(W)m(e)17 b(no)o(w)h(assume)f(that)h Fq(\026)g Fr(is)g(a)g(stationary)g(measure)f(for)h Fq(P)1323 1898 y Fk(\003)1317 1924 y Fo(t)1360 1913 y Fr(and)f(sho)o(w)h(that)g (its)257 1963 y(supp)q(ort)d(coincides)g(with)e Fq(X)s Fr(.)19 b(T)m(o)13 b(this)h(end,)g(it)f(su\016ces)i(to)f(c)o(hec)o(k)h (that)578 2054 y Fq(\026)603 2021 y Fm(\000)622 2054 y Fq(B)653 2060 y Fo(X)686 2054 y Fr(\()p Fq(v)q(;)7 b(r)q Fr(\))778 2021 y Fm(\001)808 2054 y Fq(>)12 b Fr(0)41 b(for)14 b(an)o(y)f Fq(v)g Fp(2)f Fq(X)17 b Fr(and)d Fq(r)e(>)g Fr(0)p Fq(:)247 b Fr(\(22\))257 2146 y(In)14 b(view)g(of)f(the)i(in)o(v)n(ariance)e(of)g Fq(\026)h Fr(and)g(inequalit)o(y)f(\(16\),)g(w)o(e)h(ha)o(v)o(e)558 2258 y Fq(\026)583 2225 y Fm(\000)602 2258 y Fq(B)633 2264 y Fo(X)665 2258 y Fr(\()p Fq(v)q(;)7 b(r)q Fr(\))757 2225 y Fm(\001)788 2258 y Fr(=)832 2202 y Fm(Z)855 2296 y Fo(X)893 2258 y Fq(P)926 2225 y Fm(\000)945 2258 y Fq(l)q(;)g(u;)g(B)1051 2264 y Fo(X)1082 2258 y Fr(\()p Fq(v)q(;)g(r)q Fr(\))1174 2225 y Fm(\001)1200 2258 y Fq(\026)p Fr(\()p Fq(du)p Fr(\))k Fp(\025)h Fq(";)257 2373 y Fr(whic)o(h)i(implies)e(\(22\).)320 2423 y(It)18 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Fr(m)o(ust)d(v)n(anish)g(on)h (the)h(supp)q(ort)g(of)f Fq(\026)p Fr(.)18 b(Since)13 b(supp)8 b Fq(\026)k Fr(=)f Fq(X)s Fr(,)257 843 y(w)o(e)k(ha)o(v)o(e)f Fq(g)g Fp(\021)f Fr(0.)19 b(In)c(remains)e(to)i(note)g(that)f(the)h (function)f Fq(t)f Fp(7!)f Fr(sup)1368 853 y Fo(X)1407 843 y Fp(j)p Fq(P)1446 849 y Fo(t)1460 843 y Fq(f)t Fr(\()p Fq(u)p Fr(\))p Fp(j)i Fr(is)h(non-)257 892 y(increasing,)g(and)g (therefore)h(w)o(e)f(obtain)f(\(17\))o(.)21 b(The)15 b(pro)q(of)f(of)g(Theorem)h(2)f(is)h(complete.)p 1659 942 2 29 v 1661 916 25 2 v 1661 942 V 1686 942 2 29 v 257 1080 a Fs(References)257 1170 y Fr([1])20 b(G.)12 b(Da)g(Prato,)g(J.)h(Zab)q(czyk,)g(Ergo)q(dicit)o(y)f(for)h (In\014nite-Dimensional)d(Systems,)i(Lon-)322 1220 y(don)g (Mathematical)e(So)q(ciet)o(y)j(Lecture)h(Note)f(Series,)g(v)o(ol.)e (229,)g(Cam)o(bridge)f(Univ)o(er-)322 1270 y(sit)o(y)k(Press,)h(Cam)o (bridge,)c(1996.)257 1353 y([2])20 b(W.)h(F)m(eller,)i(An)e(In)o(tro)q (duction)h(to)g(Probabilit)o(y)e(Theory)i(and)f(Its)i(Applications,)322 1403 y(V)m(ol.)12 b(I)q(I,)i(John)g(Wiley)f(&)h(Sons,)f(New)i(Y)m (ork{London{Sydney)m(,)d(1971.)257 1486 y([3])20 b(S.)14 b(Kuksin,)g(A.)f(Shiriky)o(an,)g(Sto)q(c)o(hastic)i(dissipativ)o(e)f (PDE's)g(and)g(Gibbs)g(measures,)322 1536 y(Comm.)c(Math.)k(Ph)o(ys.)f (213)h(\(2000\),)e(291{330.)257 1619 y([4])20 b(D.)e(Ruelle,)i (Statistical)e(mec)o(hanics)h(of)f(a)h(one-dimensional)e(lattice)i (gas,)h(Comm.)322 1669 y(Math.)13 b(Ph)o(ys.)28 b(9)14 b(\(1968\),)e(p.)i(267{278.)963 2628 y(7)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0411030830284--