Content-Type: multipart/mixed; boundary="-------------0006130629344" This is a multi-part message in MIME format. ---------------0006130629344 Content-Type: text/plain; name="00-264.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-264.comments" Revised version of 00-113 ---------------0006130629344 Content-Type: text/plain; name="00-264.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-264.keywords" srochastic Navier-Stokes equation, invariant measure, 2d turbulence ---------------0006130629344 Content-Type: application/postscript; name="NSuni.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="NSuni.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: NSuni.dvi %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips -o NSuni.ps NSuni.dvi %DVIPSParameters: dpi=300, comments removed 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2877 y Fi(\))p Fa(B)r Fi(\()p Fa(!)1475 2861 y Fe(0)1486 2877 y Fa(;)g(E)r Fi(\).)916 3006 y Fs(7)p eop %%Page: 8 8 8 7 bop 617 84 a Fl(P)6 b Fs(\()p Fl(!)r(;)i(U)757 65 y Fq(c)752 95 y(\024)772 85 y Fe(0)785 84 y Fs(\))k Fk(\024)h Fl(C)d Fs(exp)q(\()p Fk(\000)p Fl(c\027)1074 65 y Ft(4)1094 84 y Fl(\024)1120 65 y Ff(0)1137 51 y Fj(2)p 1134 57 21 2 v 1134 78 a Fd(\013)1161 84 y Fs(\))642 b(\(46\))23 284 y(The)12 b(pro)q(of)f(of)h(\(45\))e(is)i(based)g(on)g(a)f(standard) g(argumen)o(t)g(for)g(exp)q(onen)o(tial)i(con)o(v)o(ergence)f(of)g (Mark)o(o)o(v)e(c)o(hains)-47 341 y(\(giv)o(en)i(in)i(Do)q(ob)e([2]\),) g(and)h(the)g(idea)g(is)h(fairly)f(simple.)20 b(If)13 b Fl(Q)g Fs(w)o(as)f(a)h(gen)o(uine)h(transition)f(probabilit)o(y)l(,)h (it)f(w)o(ould)-47 397 y(b)q(e)19 b(enough,)g(in)g(order)g(to)f(pro)o (v)o(e)g(the)g(Prop)q(osition,)i(to)e(sho)o(w)g(that)g Fl(Q)g Fs(has)g(go)q(o)q(d)h(mixing)g(prop)q(erties.)31 b(The)-47 453 y(precise)16 b(prop)q(erties)g(are)f(stated)f(in)i(the)g 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b(that)f(writing,)h(see)g(\(43\),) e Fl(P)23 b Fs(=)17 b Fl(Q)12 b Fs(+)g Fl(R)p Fs(,)-47 733 y(and)j(using)h(\(48\),)d(w)o(e)i(ha)o(v)o(e,)g(for)f(an)o(y)h Fl(!)g Fk(2)e Fl(U)5 b Fs(,)14 b Fl(E)h Fk(\032)e Fs(\012,)i(that)209 871 y Fk(j)p Fl(P)257 852 y Fq(T)284 871 y Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))g Fk(\000)i Fl(Q)497 852 y Fq(T)525 871 y Fs(\()p Fl(!)r(;)e(E)s Fs(\))p Fk(j)i Fs(=)j Fk(j)741 818 y Fq(T)5 b Ff(\000)p Ft(1)745 830 y Fh(X)746 921 y Fq(t)p Ft(=0)818 871 y Fl(Q)854 852 y Fq(t)869 871 y Fl(RP)939 852 y Fq(T)g Ff(\000)p Fq(t)p Ff(\000)p Ft(1)1052 871 y Fs(\()p Fl(!)r(;)j(E)s Fs(\))p Fk(j)i(\024)j Fl(T)6 b(e)1301 852 y Ff(\000)p Fq(c\027)1362 840 y Fe(\000)p Fd(q)1419 871 y Fk(\021)1472 847 y Ft(1)p 1472 860 20 2 v 1472 889 a(2)1497 871 y Fl(\017)p Fs(\()p Fl(\027)s Fs(\))p Fl(:)232 b Fs(\(53\))-47 1003 y(Then,)329 1102 y Fk(j)p Fl( )372 1110 y Fq(!)q(;!)428 1100 y Fe(0)441 1102 y Fs(\()p Fl(S)490 1083 y Ft(+)519 1102 y Fs(\))9 b(+)i Fl( )622 1110 y Fq(!)q(;!)678 1100 y 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Fl(!)r(;)d(S)672 1350 y Ff(\000)700 1369 y Fs(\))i(+)g Fl(P)808 1350 y Fq(T)836 1369 y Fs(\()p Fl(!)884 1350 y Ff(0)896 1369 y Fl(;)e(S)948 1350 y Ft(+)976 1369 y Fs(\)\))h(+)i Fl(\017)p Fs(\()p Fl(\027)s Fs(\))i Fk(\024)g Fs(1)c Fk(\000)i Fl(\016)r Fs(\()p Fl(\027)s Fs(\))f(+)g Fl(\017)p Fs(\()p Fl(\027)s Fs(\))p Fl(:)306 b Fs(\(55\))-47 1468 y(Th)o(us,)p 193 1545 36 2 v 193 1581 a Fl(Q)o Fs(\()p Fl(t)11 b Fs(+)f Fl(T)t(;)e(E)s Fs(\))g Fk(\000)i Fl(Q)p 478 1596 V Fs(\()p Fl(t)g Fs(+)h Fl(T)t(;)d(E)s Fs(\))18 b(=)j(sup)786 1620 y Fq(!)q(;!)842 1610 y Fe(0)861 1523 y Fh(Z)902 1581 y Fs(\()p Fl(Q)956 1562 y Fq(T)984 1581 y Fs(\()p Fl(!)r(;)8 b(d!)1107 1562 y Ff(00)1126 1581 y Fs(\))i Fk(\000)g Fl(Q)1235 1562 y Fq(T)1263 1581 y Fs(\()p Fl(!)1311 1562 y Ff(0)1322 1581 y Fl(;)e(d!)1397 1562 y Ff(00)1417 1581 y Fs(\)\))p Fl(Q)1489 1562 y Fq(t)1503 1581 y Fs(\()p Fl(!)1551 1562 y Ff(00)1572 1581 y Fl(;)g(E)s Fs(\))729 1705 y(=)21 b(sup)786 1744 y Fq(!)q(;!)842 1734 y Fe(0)861 1647 y Fh(Z)910 1705 y Fl( )940 1712 y Fq(!)q(;!)996 1703 y Fe(0)1009 1705 y Fs(\()p Fl(d!)1081 1686 y Ff(00)1101 1705 y Fs(\))p Fl(Q)1155 1686 y Fq(t)1170 1705 y Fs(\()p Fl(!)1218 1686 y Ff(00)1239 1705 y Fl(;)8 b(E)s Fs(\))729 1810 y Fk(\024)21 b Fs(sup)786 1849 y Fq(!)q(;!)842 1840 y Fe(0)853 1810 y Fs(\()p Fl( )901 1818 y Fq(!)q(;!)957 1808 y Fe(0)970 1810 y Fs(\()p Fl(S)1019 1791 y Ft(+)1048 1810 y Fs(\))p 1066 1774 V Fl(Q)o Fs(\()p Fl(t;)8 b(E)s Fs(\))h(+)h Fl( )1295 1818 y Fq(!)q(;!)1351 1808 y Fe(0)1364 1810 y Fs(\()p Fl(S)1413 1791 y Ff(\000)1442 1810 y Fs(\))p Fl(Q)p 1460 1826 V -1 w Fs(\()p Fl(t;)e(E)s Fs(\)\))729 1916 y Fk(\024)21 b Fs(\(1)9 b Fk(\000)i Fl(\016)r Fs(\()p Fl(\027)s Fs(\))f(+)g Fl(\017)p Fs(\()p Fl(\027)s Fs(\)\)\()p 1134 1880 V Fl(Q)p Fs(\()p Fl(t;)e(E)s Fs(\))g Fk(\000)i Fl(Q)p 1333 1931 V Fs(\()p Fl(t;)e(E)s Fs(\)\))g(+)j Fl(\017)p Fs(\()p Fl(\027)s Fs(\))-47 2015 y(where,)i(to)g(get)g(the)g (last)g(inequalit)o(y)l(,)i(w)o(e)e(write)h Fl( )811 2022 y Fq(!)q(;!)867 2013 y Fe(0)880 2015 y Fs(\()p Fl(S)929 1998 y Ff(\000)957 2015 y Fs(\))f(=)g Fk(\000)p Fl( )1101 2022 y Fq(!)q(;!)1157 2013 y Fe(0)1170 2015 y Fs(\()p Fl(S)1219 1998 y Ft(+)1248 2015 y Fs(\))6 b(+)g Fl( )1343 2022 y Fq(!)q(;!)1399 2013 y Fe(0)1412 2015 y Fs(\()p Fl(S)1461 1998 y Ft(+)1490 2015 y Fs(\))g(+)g Fl( )1585 2022 y Fq(!)q(;!)1641 2013 y Fe(0)1655 2015 y Fs(\()p Fl(S)1704 1998 y Ff(\000)1732 2015 y Fs(\),)13 b(b)q(ound)-47 2071 y Fl( )-17 2079 y Fq(!)q(;!)39 2069 y Fe(0)51 2071 y Fs(\()p Fl(S)100 2055 y Ft(+)129 2071 y Fs(\))18 b(b)o(y)h(\(55\),)f Fk(j)p Fl( )388 2079 y Fq(!)q(;!)444 2069 y Fe(0)456 2071 y Fs(\()p Fl(S)505 2055 y Ft(+)534 2071 y Fs(\))12 b(+)h Fl( )642 2079 y Fq(!)q(;!)698 2069 y Fe(0)711 2071 y Fs(\()p Fl(S)760 2055 y Ff(\000)789 2071 y Fs(\))p Fk(j)18 b Fs(b)o(y)g(\(54\))f(and)i(use)g Fl(Q)p 1177 2087 V Fs(\()p Fl(t;)8 b(E)s Fs(\))17 b Fk(\024)i Fs(1.)29 b(W)l(e)19 b(conclude)h(that,)f(for)-47 2128 y Fl(\017)p Fs(\()p Fl(\027)s Fs(\))12 b Fl(<)h(\016)r Fs(\()p Fl(\027)s Fs(\),)51 2251 y Fk(j)p Fl(Q)100 2232 y Fq(nT)149 2251 y Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))f Fk(\000)k Fl(Q)362 2232 y Fq(nT)411 2251 y Fs(\(0)p Fl(;)d(E)s Fs(\))p Fk(j)h(\024)p 598 2215 V 13 w Fl(Q)p Fs(\()p Fl(nT)t(;)f(E)s Fs(\))g Fk(\000)i Fl(Q)p 839 2266 V Fs(\()p Fl(nT)t(;)e(E)s Fs(\))i Fk(\024)j Fs(2\(1)c Fk(\000)i Fl(\016)r Fs(\()p Fl(\027)s Fs(\))e(+)i Fl(\017)p Fs(\()p Fl(\027)s Fs(\)\))1439 2232 y Fq(n)p Ff(\000)p Ft(1)1517 2251 y Fs(+)1637 2220 y Fl(\017)p Fs(\()p Fl(\027)s Fs(\))p 1568 2240 218 2 v 1568 2282 a Fl(\016)r Fs(\()p Fl(\027)s Fs(\))f Fk(\000)g Fl(\017)p Fs(\()p Fl(\027)s Fs(\))1790 2251 y Fl(:)-47 2378 y Fs(Recall)21 b(that)f Fl(\016)r Fs(\()p Fl(\027)s Fs(\))h(=)g(exp\()p Fk(\000)p Fl(C)s(\027)541 2361 y Ff(\000)p Ft(3)589 2378 y Fs(\(log)8 b Fl(\027)698 2361 y Ff(\000)p Ft(1)745 2378 y Fs(\))763 2361 y Fq(c)780 2378 y Fs(\))20 b(and)g(that)g Fl(T)27 b Fs(=)21 b Fl(T)6 b Fs(\()p Fl(\027)s Fs(\))20 b(=)i Fl(C)s(\027)1357 2361 y Ff(\000)p Ft(1)1411 2378 y Fs(log)9 b Fl(\027)1503 2361 y Ff(\000)p Ft(1)1550 2378 y Fs(,)21 b(hence,)h(see)f(\(53\),)-47 2434 y Fl(\017)p Fs(\()p Fl(\027)s Fs(\))12 b Fk(\024)h Fl(e)113 2418 y Ff(\000)p Fq(c)155 2406 y Fe(0)167 2418 y Fq(\027)186 2406 y Fe(\000)p Fd(q)230 2434 y Fs(;)i(so,)f(since)j(w)o (e)e(assume)g Fl(\027)j Fs(to)d(b)q(e)h(small)g(and)f Fl(q)f(>)f Fs(3,)i(part)g(\(a\))f(of)h(the)g(Prop)q(osition)h(follo)o (ws.)23 2514 y(Let)h(us)f(no)o(w)g(pro)o(v)o(e)f(part)h(\(b\).)23 b(It)16 b(su\016ces)g(to)g(assume)g Fl(\024)f Fk(\025)f Fl(C)s(\027)1152 2498 y Ff(\000)p Ft(2)p Fq(\013)1239 2514 y Fs(since)j(the)f(LHS)h(of)f(\(46\))f(is)i(b)q(ounded)-47 2571 y(b)o(y)e(one.)20 b(Using)15 b(\(10\))f(for)h Fl(n)e Fs(=)g(0,)h(w)o(e)h(ha)o(v)o(e)558 2670 y Fk(k)p Fl(!)r Fs(\(1\))p Fk(k)693 2677 y Fq(\024)713 2668 y Fe(0)737 2670 y Fk(\024)e(k)p Fl(F)6 b Fs(\()p Fl(!)r Fs(\(0\)\))p Fk(k)991 2677 y Fq(\024)1011 2668 y Fe(0)1033 2670 y Fs(+)11 b Fk(k)p Fl(g)r Fs(\(1\))p Fk(k)1208 2677 y Fq(\024)1228 2668 y Fe(0)1821 2670 y Fs(\(56\))-47 2768 y(and)453 2867 y(\010\(1\))g(=)609 2844 y Ft(1)p 609 2857 20 2 v 609 2885 a(2)634 2867 y Fk(k)p Fl(!)r Fs(\(1\))p Fk(k)769 2849 y Ft(2)769 2881 y Fq(L)793 2871 y Fj(2)823 2867 y Fk(\024)i(k)p Fl(F)6 b Fs(\()p Fl(!)r Fs(\(0\)\))p Fk(k)1077 2849 y Ft(2)1077 2881 y Fq(L)1101 2871 y Fj(2)1129 2867 y Fs(+)k Fk(k)p Fl(g)r Fs(\(1\))p Fk(k)1303 2849 y Ft(2)1303 2881 y Fq(L)1327 2871 y Fj(2)1821 2867 y Fs(\(57\))916 3006 y(9)p eop %%Page: 10 10 10 9 bop -47 -2 a Fs(and,)15 b(b)o(y)g(Prop)q(osition)h(1,)f(w)o(e)g (kno)o(w)g(that,)g(if)g Fl(!)r Fs(\(0\))e Fk(2)g Fl(U)928 5 y Fq(\024)950 -2 y Fs(,)i Fl(F)6 b Fs(\()p Fl(!)r Fs(\(0\)\))12 b Fk(2)i Fl(U)1225 8 y Fq(\024)p Ft(\(1\))1292 -2 y Fs(,)h(with)h Fl(\024)p Fs(\(1\))c(=)1606 -19 y Fq(\024)p 1575 -12 84 2 v 1575 14 a Ft(1+)p Fq(\021)q(\027)1679 -2 y Fs(\(recall)k(that) -47 74 y Fl(\024)c Fk(\025)h Fl(C)s(\027)100 57 y Ff(\000)p Ft(2)p Fq(\013)183 74 y Fk(\025)g Fs(1)i(here\).)20 b(Then,)15 b(letting)h Fl(\020)g Fs(=)d(\(1)c(+)i Fl(\021)r(\027)s Fs(\))926 52 y Ff(\000)958 40 y Fj(1)p 957 44 18 2 v 957 62 a(2)994 74 y Fl(<)i Fs(1,)i(w)o(e)g(get,)f(for)h Fl(\024)1344 57 y Ff(0)1368 74 y Fk(\025)e Fl(\020)s(\024)p Fs(,)i(that)624 184 y Fk(k)p Fl(F)6 b Fs(\()p Fl(!)r Fs(\(0\)\))p Fk(k)830 192 y Fq(\024)850 182 y Fe(0)875 184 y Fk(\024)13 b Fs(\(1)c(+)i Fl(\021)r(\027)s Fs(\))1087 162 y Ff(\000)1119 151 y Fj(1)p 1118 154 V 1118 172 a(2)1143 184 y Fl(\027)s(a\024)1218 165 y Ff(0)-47 286 y Fs(and)584 343 y Fk(k)p Fl(F)6 b Fs(\()p Fl(!)r Fs(\(0\)\))p Fk(k)790 324 y Ft(2)790 356 y Fq(L)814 347 y Fj(2)844 343 y Fk(\024)13 b Fs(\(1)d(+)g Fl(\021)r(\027)s Fs(\))1056 324 y Ff(\000)1090 310 y Fj(2)p 1088 316 21 2 v 1088 337 a Fd(\013)1115 343 y Fl(\027)1140 324 y Ft(2)1160 343 y Fl('\024)1216 324 y Ff(0)1233 310 y Fj(2)p 1230 316 V 1230 337 a Fd(\013)1257 343 y Fl(:)-47 426 y Fs(No)o(w,)k(assume)h(that)f Fl(g)r Fs(\(1\))g(satis\014es,)h Fk(8)p Fm(k)p Fs(,)598 585 y Fk(j)p Fl(g)633 592 y Fg(k)656 585 y Fs(\(1\))p Fk(j)c(\024)i Fl(\017)805 592 y Ft(1)825 585 y Fl(\027)850 566 y Ft(2)870 585 y Fl(\024)896 566 y Ff(0)913 552 y Fj(1)p 911 558 V 911 579 a Fd(\013)938 585 y Fl(b)958 562 y Ff(\000)990 551 y Fj(1)p 990 555 18 2 v 990 573 a(2)1014 585 y Fk(j)p Fm(k)p Fk(j)p Fs(\()p Fl(\015)1110 592 y Fg(k)1132 585 y Fs(\))1155 551 y Fj(1)p 1155 555 V 1155 573 a(2)1179 585 y Fl(;)629 b Fs(\(58\))-47 687 y(with)16 b Fl(\017)75 694 y Ft(1)111 687 y Fs(small)h(\(dep)q(ending)i(on)d Fl(\021)h Fs(but)g(indep)q(enden)o(t)h(of)e Fl(\027)k Fs(and)c Fl(\024)p Fs(\).)23 b(Then,)16 b(w)o(e)g(get)g(that)g Fl(!)r Fs(\(1\))d Fk(2)i Fl(U)1735 695 y Fq(\024)1755 685 y Fe(0)1768 687 y Fs(,)i(using)-47 743 y(the)e(upp)q(er)h(b)q(ound) g(in)g(\(7\))e(and)i(the)f(fact)g(that,)f(for)g Fl(\027)19 b Fs(small,)c Fl(\024)1062 727 y Ff(0)1087 743 y Fk(\025)e Fl(\020)s(\024)g Fk(\025)g Fl(C)s(\020)s(\027)1329 727 y Ff(\000)p Ft(2)p Fq(\013)1414 743 y Fs(is)j(m)o(uc)o(h)f(larger)g (than)g(2)p Fl(\024)1865 750 y Fq(\015)1887 743 y Fs(.)23 800 y(Hence,)f(the)f(probabilit)o(y)i(in)f(\(46\))e(is)i(b)q(ounded)g (b)o(y)f(the)h(probabilit)o(y)g(that)e(at)h(least)g(one)h(of)e(the)i (inequalities)-47 856 y(in)g(\(58\))e(is)j(violated.)20 b(Since)15 b(the)f Fl(g)549 863 y Fg(k)572 856 y Fs('s)f(are)g (Gaussian)h(random)f(v)m(ariables)i(with)f(co)o(v)m(ariance)g Fl(\015)1580 863 y Fg(k)1603 856 y Fs(,)g(this)g(ev)o(en)o(t)f(has)-47 913 y(a)h(probabilit)o(y)j(less)f(than:)493 1020 y(1)9 b Fk(\000)571 979 y Fh(Y)587 1072 y Fg(k)624 1020 y Fs(\(1)g Fk(\000)i Fl(C)f Fs(exp)q(\()p Fk(\000)p Fl(C)s(\017)940 1001 y Ft(2)940 1031 y(1)960 1020 y Fl(\027)985 1001 y Ft(4)1005 1020 y Fs(\()p Fl(\024)1049 1001 y Ff(0)1060 1020 y Fs(\))1085 987 y Fj(2)p 1083 993 21 2 v 1083 1014 a Fd(\013)1110 1020 y Fl(b)1130 1001 y Ff(\000)p Ft(1)1177 1020 y Fk(j)p Fm(k)p Fk(j)1231 1001 y Ft(2)1249 1020 y Fs(\)\))p Fl(;)523 b Fs(\(59\))-47 1173 y(F)l(or)15 b Fl(\027)60 1157 y Ft(4)80 1173 y Fs(\()p Fl(\024)124 1157 y Ff(0)135 1173 y Fs(\))160 1142 y Fj(2)p 158 1148 V 158 1168 a Fd(\013)201 1173 y Fs(large,)g(eac)o(h)h(exp)q(onen)o (tial)h(in)f(the)g(pro)q(duct)g(is)g(small.)23 b(The)15 b(factor)g Fk(j)p Fm(k)p Fk(j)1438 1157 y Ft(2)1472 1173 y Fs(con)o(trols)h(the)f(sum)h(o)o(v)o(er)-47 1236 y Fm(k)f Fs(of)g(exp)q(\()p Fk(\000)p Fl(C)s(\017)225 1220 y Ft(2)225 1248 y(1)245 1236 y Fl(\027)270 1220 y Ft(4)290 1236 y Fs(\()p Fl(\024)334 1220 y Ff(0)345 1236 y Fs(\))370 1205 y Fj(2)p 368 1211 V 368 1231 a Fd(\013)395 1236 y Fl(b)415 1220 y Ff(\000)p Ft(1)462 1236 y Fk(j)p Fm(k)p Fk(j)516 1220 y Ft(2)534 1236 y Fs(\),)h(sum)g(whic)o(h)h(is)f(small)h (for)e Fl(\027)1070 1220 y Ft(4)1090 1236 y Fs(\()p Fl(\024)1134 1220 y Ff(0)1146 1236 y Fs(\))1171 1205 y Fj(2)p 1168 1211 V 1168 1231 a Fd(\013)1211 1236 y Fs(large)h(enough,)g(and)g (therefore)g(\(59\))f(is)-47 1299 y(b)q(ounded)h(from)f(ab)q(o)o(v)o(e) f(b)o(y)h(1)10 b Fk(\000)h Fs(\(1)e Fk(\000)i Fs(exp\()p Fk(\000)p Fl(c\027)779 1283 y Ft(4)799 1299 y Fl(\024)825 1281 y Ff(0)842 1268 y Fj(2)p 840 1274 V 840 1294 a Fd(\013)867 1299 y Fs(\)\),)j(i.e.)20 b(b)o(y)15 b(the)h(RHS)g(of)e(\(46\).)463 b Fc(2)-47 1421 y Fu(3.2)55 b(Pro)r(ofs)19 b(of)g(Lemm)o(as)d(2)j(and)g (3.)-47 1507 y Fm(Pro)q(of)e(of)g(Lemma)h(2.)i Fs(Let)15 b Fl(!)r Fs(\(0\))d Fk(2)h Fl(U)659 1514 y Fq(\024)694 1507 y Fk(\022)g Fl(U)20 b Fs(and)15 b(consider)h(\(10\))e(for)h Fl(n)e Fs(=)g(0.)19 b(Cho)q(ose)c Fl(g)r Fs(\(1\))f(suc)o(h)h(that,)g Fk(8)p Fm(k)p Fs(,)604 1617 y Fk(j)p Fl(g)639 1624 y Fg(k)662 1617 y Fs(\(1\))p Fk(j)d(\024)h Fl(\017)812 1624 y Ft(1)832 1617 y Fl(\027)857 1598 y Ft(2)877 1617 y Fl(e)898 1598 y Fq(\017)912 1603 y Fj(2)929 1598 y Fq(\027)r Ff(j)p Fg(k)p Ff(j)992 1617 y Fl(b)1012 1595 y Ff(\000)1044 1584 y Fj(1)p 1044 1587 18 2 v 1044 1605 a(2)1068 1617 y Fs(\()p Fl(\015)1110 1624 y Fg(k)1133 1617 y Fs(\))1156 1584 y Fj(1)p 1156 1587 V 1156 1605 a(2)1180 1617 y Fl(:)628 b Fs(\(60\))-47 1719 y(>F)l(rom)14 b(Prop)q(osition)i(1,)f(\(60\))f(and)h(\(7\),)f(one)h(obtains,)378 1835 y Fk(j)p Fl(!)419 1842 y Fg(k)443 1835 y Fs(\(1\))p Fk(j)c(\024)i Fl(\027)s(a\024)p Fs(\(1\))p Fk(j)p Fm(k)p Fk(j)762 1816 y Ff(\000)p Fq(r)807 1835 y Fl(e)828 1816 y Ff(\000)p Fq(\024)p Ft(\(1\))921 1804 y Fe(\000)p Fj(1)962 1816 y Ff(j)p Fg(k)p Ff(j)1015 1835 y Fs(+)e Fl(\017)1079 1842 y Ft(1)1099 1835 y Fl(\027)1124 1816 y Ft(2)1144 1835 y Fl(e)1165 1816 y Fq(\017)1179 1821 y Fj(2)1197 1816 y Fq(\027)r Ff(j)p Fg(k)p Ff(j)1260 1835 y Fl(e)1281 1816 y Ff(\000)p Fq(\024)1328 1801 y Fe(\000)p Fj(1)1328 1822 y Fd(\015)1375 1799 y Fe(j)p Fb(k)p Fe(j)p 1375 1809 38 2 v 1386 1829 a Fj(2)1821 1835 y Fs(\(61\))-47 1937 y(Then,)j(from)g(\(17\),)f(one)i(gets,)f(for)g(an)o(y)g Fl(\032)e(>)h Fs(0,)h(b)o(y)h(c)o(ho)q(osing)g Fl(\017)1040 1944 y Ft(1)1060 1937 y Fl(;)8 b(\017)1099 1944 y Ft(2)1133 1937 y Fs(small)16 b(enough,)e(that)g Fk(9)1545 1925 y Fs(\026)1543 1937 y Fl(\025)f(<)g(e)1652 1921 y Ff(\000)p Fq(c\027)1729 1937 y Fl(<)g Fs(1)h(suc)o(h)-47 1994 y(that,)g Fk(8)p Fm(k)p Fs(,)611 2096 y Fk(j)p Fl(!)652 2103 y Fg(k)676 2096 y Fs(\(1\))p Fk(j)d(\024)i Fl(\027)s(a\024)882 2077 y Ff(0)894 2096 y Fk(j)p Fm(k)p Fk(j)948 2077 y Ff(\000)p Fq(r)994 2096 y Fl(e)1015 2077 y Ff(\000)p Ft(\()p Fq(\024)1076 2065 y Fe(0)1088 2077 y Ft(\))1102 2065 y Fe(\000)p Fj(1)1143 2077 y Ff(j)p Fg(k)p Ff(j)1821 2096 y Fs(\(62\))-47 2198 y(with)i Fl(\024)82 2181 y Ff(0)107 2198 y Fs(=)d(max\()259 2186 y(\026)257 2198 y Fl(\025)o(\024;)c Fs(2)p Fl(\024)379 2205 y Fq(\015)411 2198 y Fs(+)i Fl(\032\027)s Fs(\).)23 2254 y(>F)l(rom)15 b(\(30\),)f(\(57\),)f(\(58\),)h(one)h(also)g(easily)h(obtains)g(that) 761 2364 y(\010\(1\))c Fk(\024)h Fl(\027)938 2345 y Ft(2)958 2364 y Fl(')p Fs(\()p Fl(\024)1032 2345 y Ff(0)1043 2364 y Fs(\))1068 2331 y Fj(2)p 1066 2337 21 2 v 1066 2358 a Fd(\013)-47 2466 y Fs(and)i(th)o(us)g(that)f Fl(!)r Fs(\(1\))e Fk(2)h Fl(U)414 2474 y Fq(\024)434 2464 y Fe(0)448 2466 y Fs(.)19 b(Th)o(us,)444 2576 y Fl(P)6 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2808 y Fj(2)663 2803 y Fq(\027)r Ff(j)p Fg(k)p Ff(j)725 2821 y Fl(b)745 2799 y Ff(\000)777 2788 y Fj(1)p 777 2792 V 777 2809 a(2)802 2821 y Fs(\()p Fl(\015)844 2828 y Fg(k)866 2821 y Fs(\))889 2788 y Fj(1)p 889 2792 V 889 2809 a(2)913 2821 y Fs(\))g Fk(\025)g Fs(1)d Fk(\000)g Fs(exp\()p Fk(\000)p Fl(c\017)1230 2803 y Ft(2)1230 2833 y(1)1250 2821 y Fl(\027)1275 2803 y Ft(4)1295 2821 y Fl(e)1316 2803 y Ft(2)p Fq(\017)1348 2808 y Fj(2)1366 2803 y Fq(\027)r Ff(j)p Fg(k)p Ff(j)1428 2821 y Fl(b)1448 2803 y Ff(\000)p Ft(1)1495 2821 y Fs(\))308 b(\(63\))904 3006 y(10)p eop %%Page: 11 11 11 10 bop -47 -2 a Fs(for)13 b Fk(j)p Fm(k)p Fk(j)f(\025)h Fl(C)s(\027)196 -18 y Ff(\000)p Ft(1)250 -2 y Fs(log)c Fl(\027)342 -18 y Ff(\000)p Ft(1)389 -2 y Fs(,)14 b(if)h Fl(C)i Fs(is)e(c)o(hosen)g(so)f(that)f Fl(b\027)894 -18 y Ff(\000)p Ft(4)954 -2 y Fk(\024)g Fl(e)1023 -18 y Fq(\017)1037 -13 y Fj(2)1055 -18 y Fq(\027)r Ff(j)p Fg(k)p Ff(j)1132 -2 y Fs(\(note)g(that)h(the)g(pro)q(duct)h(o)o(v)o(er)f(suc)o(h)g 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Fs(\))13 b(denotes)h(the)f(v)m(alue)i(at)e(time)g(1)g(of)g(the)h (solution)g(of)f(\(18\))f(with)i(initial)-47 1433 y(condition)g Fl(!)178 1440 y Ft(0)198 1433 y Fs(.)20 b(In)14 b(view)g(of)f(Prop)q (osition)i(1,)e(and)h(the)f(de\014nition)j(of)1154 1421 y(\026)1146 1433 y Fl(U)i Fs(=)13 b Fl(U)1274 1440 y Ft(2)p Fq(\024)1312 1444 y Fd(\015)1332 1440 y Ft(+)p Fq(\032\027)1399 1433 y Fs(,)g(w)o(e)h(can)f(b)q(ound,)i Fk(8)p Fl(!)1777 1440 y Ft(0)1810 1433 y Fk(2)1861 1421 y Fs(\026)1853 1433 y Fl(U)5 b Fs(,)562 1552 y Fk(j)p Fl(F)604 1559 y Fg(k)627 1552 y Fs(\()p Fl(!)673 1559 y Ft(0)693 1552 y Fs(\))p Fk(j)12 b(\024)h Fl(C)s(\027)s(ae)890 1525 y Ff(\000)931 1508 y Fe(j)p Fb(k)p Fe(j)p 923 1518 54 2 v 923 1538 a Fj(2)p Fd(\024)956 1542 y(\015)983 1552 y Fl(e)1004 1533 y Ff(\000)p Fq(\032\027)r Ff(j)p Fg(k)p Ff(j)1125 1552 y Fk(\021)g Fl(\017)1191 1559 y Fg(k)1215 1552 y Fl(;)593 b Fs(\(67\))-47 1645 y(pro)o(vided)16 b(w)o(e)e(c)o(ho)q(ose)i Fl(\032)e Fs(su\016cien)o(tly)j(small)f(so)f 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Fs(=)e Fl(!)r(;)8 b(!)1614 1923 y Ff(0)1625 1940 y Fs(.)24 b(W)l(e)16 b(get)g(that)-47 1996 y(the)f(LHS)h(of)e(\(50\))g(is)i(b)q(ounded)h(from)d(b)q(elo)o(w)i (b)o(y)596 2049 y Fh(Z)625 2134 y Ft(\026)619 2143 y Fq(U)656 2066 y Fh(Y)672 2158 y Fg(k)722 2082 y Fq(d)s Ft(\026)-21 b Fq(!)762 2088 y Fb(k)782 2082 y Ff(^)p Fq(d!)846 2088 y Fb(k)p 722 2096 147 2 v 749 2124 a Ft(2)p Fq(\031)q(i\015)818 2130 y Fb(k)881 2106 y Fs(exp\()p Fk(\000)1008 2081 y Ft(\()p Ff(j)p Fq(!)1054 2087 y Fb(k)1074 2081 y Ff(j)p Ft(+)p Fq(\017)1125 2087 y Fb(k)1146 2081 y Ft(\))1160 2071 y Fj(2)p 1008 2096 171 2 v 1073 2124 a Fq(\015)1091 2130 y Fb(k)1184 2106 y Fs(\))619 b(\(68\))-47 2240 y(In)15 b(order)g(to)f(estimate)h(that)f(latter)g(in)o(tegral,)h (observ)o(e)g(that,)f(b)o(y)h(\(7\),)e Fl(!)i Fk(2)1279 2229 y Fs(\026)1271 2240 y Fl(U)i Fs(=)c Fl(U)1398 2247 y Ft(2)p Fq(\024)1436 2251 y Fd(\015)1456 2247 y Ft(+)p Fq(\032\027)1538 2240 y Fs(pro)o(vided)j(that,)e Fk(8)p Fm(k)p Fs(,)595 2346 y Fk(j)p Fl(!)636 2353 y Fg(k)659 2346 y Fk(j)f(\024)g Fl(\017)751 2353 y Ft(1)771 2346 y Fl(\027)s(e)817 2327 y Fq(\017)831 2332 y Fj(2)849 2327 y Fq(\027)r Ff(j)p Fg(k)p Ff(j)912 2346 y Fl(b)932 2324 y Ff(\000)964 2313 y Fj(1)p 963 2316 18 2 v 963 2334 a(2)988 2346 y Fs(\()p Fl(\015)1030 2353 y Fg(k)1053 2346 y Fs(\))1076 2313 y Fj(1)p 1075 2316 V 1075 2334 a(2)1112 2346 y Fk(\021)h Fs(\026)-24 b Fl(\017)1178 2353 y Fg(k)1821 2346 y Fs(\(69\))-47 2439 y(if)13 b(w)o(e)g(tak)o(e)g Fl(\017)172 2446 y Ft(1)192 2439 y Fl(;)8 b(\017)231 2446 y Ft(2)263 2439 y Fs(small)14 b(enough.)20 b(Th)o(us,)13 b(b)o(y)g(restricting)h(the)f(domain)h(of)f(in)o(tegration,)g(w)o(e)g (get)g(a)f(lo)o(w)o(er)h(b)q(ound)-47 2496 y(on)i(\(68\):)634 2571 y Fh(Y)650 2663 y Fg(k)687 2612 y Fs([)700 2554 y Fh(Z)742 2567 y Ft(\026)-19 b Fq(\017)755 2573 y Fb(k)723 2648 y Ft(0)790 2588 y(2)p Fq(r)q(dr)p 790 2601 72 2 v 806 2630 a(\015)824 2636 y Fb(k)875 2612 y Fs(exp\()p Fk(\000)1002 2586 y Ft(\()p Fq(r)q Ft(+)p Fq(\017)1074 2592 y Fb(k)1095 2586 y Ft(\))1109 2577 y Fj(2)p 1002 2601 126 2 v 1045 2630 a Fq(\015)1063 2636 y Fb(k)1133 2612 y Fs(\)])657 b(\(70\))-47 2740 y(Eac)o(h)15 b(factor)f(is)h(b)q (ounded)i(from)d(b)q(elo)o(w)i(b)o(y)556 2859 y(1)10 b Fk(\000)g Fl(C)678 2828 y(\017)696 2811 y Ft(2)696 2841 y Fg(k)p 675 2848 48 2 v 675 2890 a Fl(\015)699 2897 y Fg(k)737 2859 y Fk(\000)h Fs(exp\()p Fk(\000)p Fl(c\017)943 2840 y Ft(2)943 2870 y(1)963 2859 y Fl(\027)988 2840 y Ft(2)1008 2859 y Fl(e)1029 2840 y Ft(2)p Fq(\017)1061 2845 y Fj(2)1078 2840 y Ff(j)p Fg(k)p Ff(j)1121 2859 y Fs(\()p Fl(b)p Fs(\))1177 2840 y Ff(\000)p Ft(1)1224 2859 y Fs(\))579 b(\(71\))904 3006 y(11)p eop %%Page: 12 12 12 11 bop -47 -2 a Fs(for)18 b Fk(j)p Fm(k)p Fk(j)h(\025)h Fl(C)s(\027)215 -18 y Ff(\000)p Ft(1)270 -2 y Fs(log)8 b Fl(\027)361 -18 y Ff(\000)p Ft(1)409 -2 y Fs(.)32 b(T)l(o)19 b(b)q(ound)i(the)e(pro)q(duct)h(o)o(v)o(er)f(those)g Fm(k)p Fs('s)g(of)g(the)g(factors)g(giv)o(en)h(b)o(y)f(\(71\))f(b)o(y)i (a)-47 55 y(strictly)e(p)q(ositiv)o(e)h(constan)o(t,)e(indep)q(enden)o (t)j(of)e Fl(\027)s Fs(,)g(observ)o(e)g(\014rst)f(that)g(the)h(last)g (term)f(is)i(summable)f(o)o(v)o(er)f Fm(k)p Fs(,)-47 111 y(for)12 b Fk(j)p Fm(k)p Fk(j)g(\025)h Fl(C)s(\027)195 95 y Ff(\000)p Ft(1)250 111 y Fs(log)8 b Fl(\027)341 95 y Ff(\000)p Ft(1)402 111 y Fs(and)14 b(that)f(the)g(sum)h(is)g (small.)20 b(Moreo)o(v)o(er,)12 b(using)j(\(67\))d(and)i(the)f(lo)o(w)o (er)h(b)q(ound)g(in)h(\(7\),)-47 168 y(w)o(e)f(get,)643 263 y Fl(\017)661 247 y Ft(2)661 277 y Fg(k)p 640 284 48 2 v 640 325 a Fl(\015)664 332 y Fg(k)705 294 y Fk(\024)f Fl(C)s(ab\027)858 275 y Ft(2)885 294 y Fs(exp\()p Fk(\000)p Fs(2)p Fl(\032\027)s Fk(j)p Fm(k)p Fk(j)p Fs(\))p Fl(:)657 b Fs(\(72\))-47 421 y(Then,)15 b(\(72\))g(is)h(also)g(summable)g(o)o(v) o(er)f Fm(k)p Fs(,)g(for)g Fk(j)p Fm(k)p Fk(j)d(\025)i Fl(C)s(\027)934 404 y Ff(\000)p Ft(1)988 421 y Fs(log)9 b Fl(\027)1080 404 y Ff(\000)p Ft(1)1143 421 y Fs(and)16 b(the)f(sum)h(is)g(also)g(small.)22 b(Finally)l(,)17 b(for)-47 477 y Fk(j)p Fm(k)p Fk(j)12 b(\024)h Fl(C)s(\027)128 461 y Ff(\000)p Ft(1)183 477 y Fs(log)8 b Fl(\027)274 461 y Ff(\000)p Ft(1)322 477 y Fs(,)15 b(eac)o(h)h(factor)e(in)i (\(70\))f(is)h(b)q(ounded)h(from)e(b)q(elo)o(w,)g(using)i(\(7,)d(69\),) h(b)o(y)g Fl(C)1600 443 y Fh(R)1628 456 y Ft(\026)-18 b Fq(\017)1642 462 y Fb(k)1620 491 y Ft(0)1677 464 y Fj(2)p Fd(r)q(dr)p 1677 467 64 2 v 1690 481 a(\015)1706 489 y Fb(k)1759 477 y Fk(\025)14 b Fl(C)s(\027)1869 461 y Ft(2)1889 477 y Fs(,)-47 534 y(whic)o(h)i(yields)g(the)g(claim)g(of)f (the)g(Lemma.)1167 b Fc(2)-47 677 y Fn(4)66 b(Pro)r(of)22 b(of)f(the)i(Theorem.)-47 778 y Fs(W)l(e)15 b(deduce)h(the)g(Theorem)f (from)f(Prop)q(osition)i(2.)23 835 y(Let)h(us)h(c)o(ho)q(ose)f(a)g(n)o (um)o(b)q(er)g(\026)-23 b Fl(\017)17 b Fs(small)h(enough)g(and)f(a)g (time)g Fl(\034)5 b Fs(,)17 b(large)h(enough,)f(i.e)h Fl(\034)j Fs(=)16 b Fl(cm)1625 818 y Ff(\000)p Ft(1)1671 835 y Fl(\027)1696 818 y Ff(\000)p Fq(q)1760 835 y Fs(so)h(that)-47 891 y(\(45\))c(is)j(less)g(than)291 873 y Ft(\026)-18 b Fq(\017)p 289 880 18 2 v 289 907 a Ft(2)312 891 y Fs(.)20 b(Then,)15 b(for)f Fl(T)21 b Fs(an)16 b(in)o(teger)f(m)o(ultiple)i(of)e Fl(\034)5 b Fs(,)15 b(write)645 1005 y Fl(P)680 986 y Fq(T)708 1005 y Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))j(=)i(\()p Fl(P)944 986 y Fq(\034)965 1005 y Fs(\))988 973 y Fd(T)p 988 979 23 2 v 990 999 a(\034)1017 1005 y Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))p Fl(:)667 b Fs(\(73\))-47 1106 y(Next,)14 b(let)628 1207 y Fl(\031)r Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))i Fk(\021)j Fl(\031)r Fs(\()p Fl(E)s Fs(\))d(=)j Fl(Q)1033 1188 y Fq(\034)1055 1207 y Fs(\(0)p Fl(;)8 b(E)s Fs(\))649 b(\(74\))-47 1308 y(and)418 1398 y(\026)408 1409 y Fl(R)p Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))18 b(=)j Fl(P)676 1390 y Fq(\034)698 1409 y Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))g Fk(\000)j Fl(Q)912 1390 y Fq(\034)933 1409 y Fs(\()p Fl(!)r(;)d(E)s Fs(\))396 1478 y Fl(R)431 1459 y Ff(0)443 1478 y Fs(\()p Fl(!)r(;)g(E)s Fs(\))18 b(=)j Fl(Q)677 1459 y Fq(\034)699 1478 y Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))g Fk(\000)i Fl(Q)912 1459 y Fq(\034)934 1478 y Fs(\(0)p Fl(;)e(E)s Fs(\))i(=)j Fl(Q)1145 1459 y Fq(\034)1166 1478 y Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))g Fk(\000)i Fl(\031)r Fs(\()p Fl(E)s Fs(\))377 b(\(75\))421 1547 y Fl(r)q Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))18 b(=)651 1536 y(\026)641 1547 y Fl(R)p Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))g(+)i Fl(R)888 1528 y Ff(0)900 1547 y Fs(\()p Fl(!)r(;)e(E)s Fs(\))p Fl(:)784 b Fs(\(76\))-47 1648 y(One)15 b(ma)o(y)g(then)g(write)622 1755 y Fl(P)657 1736 y Fq(T)685 1755 y Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))i(=)j(\()p Fl(\031)e Fs(+)g Fl(r)q Fs(\))1013 1723 y Fd(T)p 1012 1729 V 1014 1749 a(\034)1041 1755 y Fs(\()p Fl(!)r(;)d(E)s Fs(\))p Fl(:)643 b Fs(\(77\))23 1864 y(W)l(e)15 b(can)h(expand)g(\()p Fl(\031)11 b Fs(+)f Fl(r)q Fs(\))488 1832 y Fd(T)p 488 1838 V 490 1859 a(\034)532 1864 y Fs(in)16 b(p)q(o)o(w)o(ers)f(of)g Fl(r)q Fs(:)453 1973 y(\()p Fl(\031)c Fs(+)f Fl(r)q Fs(\))598 1941 y Fd(T)p 598 1947 V 600 1967 a(\034)640 1973 y Fs(=)688 1933 y Fh(X)702 2025 y Fq(k)720 2030 y Fd(i)755 1973 y Fl(\031)783 1954 y Fq(k)801 1959 y Fj(1)820 1973 y Fl(r)842 1954 y Fq(k)860 1959 y Fj(2)888 1973 y Fl(:)e(:)g(:)d(\031)976 1954 y Fq(k)994 1960 y Fd(l)1018 1973 y Fs(+)11 b Fl(r)1091 1941 y Fd(T)p 1091 1947 V 1093 1967 a(\034)1132 1973 y Fk(\021)i Fs(\006)1213 1954 y Ft(1)1243 1973 y Fs(+)e Fl(r)1316 1941 y Fd(T)p 1315 1947 V 1317 1967 a(\034)1821 1973 y Fs(\(78\))-47 2124 y(where)k(the)f(sum)h(\006)290 2107 y Ft(1)324 2124 y Fs(runs)g(o)o(v)o(er)f Fl(k)546 2131 y Fq(i)573 2124 y Fk(\025)f Fs(0,)671 2091 y Fh(P)722 2124 y Fl(k)746 2131 y Fq(i)773 2124 y Fs(=)826 2106 y Fq(T)p 826 2113 26 2 v 829 2139 a(\034)871 2124 y Fs(and)i(collects)h (all)f(the)g(terms)f(with)h(at)g(least)f(one)h(factor)f Fl(\031)r Fs(.)-47 2180 y(No)o(w)g(observ)o(e)h(that,)f(using)i(\(77\)) e(with)i Fl(T)i Fs(=)13 b Fl(\034)5 b Fs(,)15 b(w)o(e)g(ha)o(v)o(e)g (that)391 2281 y Fl(r)q Fs(\()p Fl(!)r(;)8 b Fs(\012\))i(=)j Fl(P)626 2262 y Fq(\034)648 2281 y Fs(\()p Fl(!)r(;)8 b Fs(\012\))h Fk(\000)h Fl(\031)r Fs(\(\012\))i(=)h(1)c Fk(\000)i Fl(\031)r Fs(\(\012\))g(=)i(1)d Fk(\000)g Fl(Q)1327 2262 y Fq(\034)1349 2281 y Fs(\(0)p Fl(;)e(U)d Fs(\))-47 2382 y(is)15 b(indep)q(enden)o(t)j(of)d Fl(!)r Fs(;)f(hence,)i(b)o(y)f (\(74\),)124 2500 y(\()p Fl(r)q(\031)r Fs(\)\()p Fl(!)r(;)8 b(d!)331 2507 y Ft(2)348 2500 y Fs(\))13 b(=)426 2442 y Fh(Z)476 2500 y Fl(r)q Fs(\()p Fl(!)r(;)8 b(d!)619 2507 y Ft(1)637 2500 y Fs(\))p Fl(\031)r Fs(\()p Fl(!)729 2507 y Ft(1)748 2500 y Fl(;)g(d!)821 2507 y Ft(2)840 2500 y Fs(\))k(=)h Fl(r)q Fs(\()p Fl(!)r(;)8 b Fs(\012\))p Fl(\031)r Fs(\()p Fl(d!)1158 2507 y Ft(2)1175 2500 y Fs(\))k(=)h(\(1)d Fk(\000)g Fl(Q)1385 2481 y Fq(\034)1407 2500 y Fs(\(0)p Fl(;)e(U)d Fs(\)\))p Fl(\031)r Fs(\()p Fl(d)o(!)1638 2507 y Ft(2)1655 2500 y Fs(\))148 b(\(79\))-47 2615 y(is)16 b(also)g(indep)q(enden)o(t)j(of)d Fl(!)r Fs(.)23 b(F)l(rom)15 b(this,)i(w)o(e)f(conclude)i(that,)d(since)j (there)e(is)h(at)e(least)i(one)f(factor)f Fl(\031)j Fs(in)f(eac)o(h)-47 2671 y(term)d(of)h(\006)146 2655 y Ft(1)166 2671 y Fs(,)f(\006)226 2655 y Ft(1)246 2671 y Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))i(=)j(\006)461 2655 y Ft(1)481 2671 y Fs(\()p Fl(!)529 2655 y Ff(0)540 2671 y Fl(;)8 b(E)s Fs(\),)13 b Fk(8)p Fl(!)r(;)8 b(!)748 2655 y Ff(0)759 2671 y Fs(,)14 b Fk(8)p Fl(E)s Fs(,)h(and,)g(using)h (\(77\),)d(that)401 2781 y Fk(j)p Fl(P)449 2762 y Fq(T)476 2781 y Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))g Fk(\000)j Fl(P)689 2762 y Fq(T)717 2781 y Fs(\()p Fl(!)765 2762 y Ff(0)776 2781 y Fl(;)d(E)s Fs(\))p Fk(j)i(\024)j(j)p Fl(r)963 2749 y Fd(T)p 963 2755 23 2 v 965 2775 a(\034)992 2781 y Fs(\()p Fl(!)r(;)8 b(E)s Fs(\))p Fk(j)f Fs(+)k Fk(j)p Fl(r)1222 2749 y Fd(T)p 1221 2755 V 1223 2775 a(\034)1250 2781 y Fs(\()p Fl(!)1298 2762 y Ff(0)1310 2781 y Fl(;)d(E)s Fs(\))p Fk(j)422 b Fs(\(80\))-47 2882 y(where)15 b(the)g(RHS)h(is)g (con)o(trolled)g(b)o(y:)904 3006 y(12)p eop %%Page: 13 13 13 12 bop -47 -2 a Fm(Lemma)17 b(4.)j Fr(F)m(or)c Fl(T)22 b Fr(an)16 b(inte)n(ger)f(multiple)h(of)h Fl(\034)5 b Fr(,)662 106 y Fk(j)p Fl(r)702 74 y Fd(T)p 701 80 23 2 v 703 100 a(\034)731 106 y Fs(\()p Fl(!)r(;)j(E)s Fs(\))p Fk(j)i(\024)j Fl(C)s Fs(\()p Fl(!)r Fs(\))p Fl(e)1049 87 y Ff(\000)p 1076 68 32 2 v Fq(m)o(T)1821 106 y Fs(\(81\))-47 214 y Fr(wher)n(e)p 81 189 40 2 v 16 w Fl(m)f Fs(=)p 181 189 V 13 w Fl(m)p Fs(\()p Fl(\027)s Fs(\))h Fk(\025)g Fs(exp\()p Fk(\000)p Fl(C)s(\027)526 197 y Ff(\000)p Ft(3)573 214 y Fs(\(log)8 b Fl(\027)682 197 y Ff(\000)p Ft(1)730 214 y Fs(\))748 197 y Fq(c)765 214 y Fs(\))16 b Fr(and)g(wher)n(e)g Fl(C)s Fs(\()p Fl(!)r Fs(\))c Fk(\024)h Fl(C)s Fs(\()1236 192 y Ff(jj)p Fq(!)q Ff(jj)p Ft(+1)p 1236 203 108 2 v 1279 229 a Fq(\027)1348 214 y Fs(\))1366 197 y Fq(c)1383 214 y Fr(,)j(for)h Fl(C)q(;)8 b(c)i(<)j Fk(1)p Fr(.)23 294 y Fs(T)l(o)h(conclude)j(the)d(pro)q(of)g(of)g (\(12\),)f(it)i(is)g(enough)g(to)f(sho)o(w)g(that)g(lim)1204 301 y Fq(T)5 b Ff(!1)1310 294 y Fl(P)1345 277 y Fq(T)1373 294 y Fs(\(0)p Fl(;)j(E)s Fs(\))i(=)j Fl(\026)p Fs(\()p Fl(E)s Fs(\))g(exists.)20 b(And,)-47 350 y(to)14 b(pro)o(v)o(e)h(that,) f(w)o(e)h(write,)g(for)f Fl(T)19 b(>)13 b(T)634 334 y Ff(0)645 350 y Fs(,)252 466 y Fk(j)p Fl(P)300 447 y Fq(T)327 466 y Fs(\(0)p Fl(;)8 b(E)s Fs(\))g Fk(\000)i Fl(P)532 447 y Fq(T)558 435 y Fe(0)572 466 y Fs(\(0)p Fl(;)e(E)s Fs(\))p Fk(j)h(\024)759 408 y Fh(Z)808 466 y Fl(P)843 447 y Fq(T)c Ff(\000)p Fq(T)922 435 y Fe(0)936 466 y Fs(\(0)p Fl(;)j(d!)r Fs(\))p Fk(j)p Fl(P)1118 447 y Fq(T)1144 435 y Fe(0)1154 466 y Fs(\()p Fl(!)r(;)g(E)s Fs(\))g Fk(\000)i Fl(P)1366 447 y Fq(T)1392 435 y Fe(0)1406 466 y Fs(\(0)p Fl(;)e(E)s Fs(\))p Fk(j)p Fl(:)272 b Fs(\(82\))-47 586 y(W)l(e)15 b(ma)o(y)f(write)h(the)g(in)o(tegral)g(as)g(an)g(in)o (tegral)g(o)o(v)o(er)f Fl(U)20 b Fs(plus)c(a)f(sum)g(o)o(v)o(er)f Fl(\024)f Fk(2)g Fm(N)p Fs(,)h Fl(\024)f Fk(\025)g Fl(\027)1509 570 y Ff(\000)p Fq(p)1557 586 y Fs(,)h(of)h(in)o(tegrals)g(o)o(v)o(er) -47 643 y Fl(U)-16 650 y Fq(\024)p Ft(+1)51 643 y Fk(n)p Fl(U)105 650 y Fq(\024)145 643 y Fs(and,)i(com)o(bining)i(\(80\),)d (Lemma)i(4,)f Fl(C)s Fs(\()p Fl(!)r Fs(\))f Fk(\024)g Fl(C)s Fs(\()1026 621 y Ff(jj)p Fq(!)q Ff(jj)p Ft(+1)p 1026 632 V 1070 659 a Fq(\027)1138 643 y Fs(\))1156 626 y Fq(c)1190 643 y Fk(\024)g Fl(C)s Fs(\()1300 621 y Ff(jj)p Fq(!)q Ff(jj)1363 625 y Fd(\024)1382 621 y Ft(+1)p 1300 632 128 2 v 1354 659 a Fq(\027)1432 643 y Fs(\))1450 626 y Fq(c)1467 643 y Fs(,)i Fk(8)p Fl(\024)p Fs(,)g(and)g(\(46\))e (\(whic)o(h)-47 705 y(implies)h(a)e(similar)h(b)q(ound)g(for)f Fl(P)539 688 y Fq(T)5 b Ff(\000)p Fq(T)618 677 y Fe(0)631 705 y Fs(\(0)p Fl(;)j(U)729 688 y Fq(c)724 718 y(\024)744 709 y Fe(0)756 705 y Fs(\)\),)14 b(w)o(e)h(b)q(ound)h(\(82\))e(b)o(y) 303 816 y Fl(C)s(e)360 797 y Ff(\000)p 387 778 32 2 v Fq(mT)444 785 y Fe(0)457 816 y Fs(\()p Fl(\027)500 797 y Ff(\000)p Ft(\()p Fq(p)p Ft(+1\))p Fq(c)645 816 y Fs(+)758 775 y Fh(X)691 870 y Fq(\024)p Ff(2)p Fg(N)p Fq(;\024)p Ff(\025)p 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Fl(k)h Fk(\000)g Fs(1)1344 771 y Fh(\023)1387 831 y Fk(\024)i Fs(4)1458 812 y Fq(k)1479 831 y Fs(2)1502 812 y Fq(m)-47 998 y Fs(with)i Fl(p)79 1005 y Fq(i)106 998 y Fs(=)e Fl(n)181 1005 y Fq(i)p Ff(\000)p Ft(1)250 998 y Fk(\000)e Fl(n)323 1005 y Fq(i)350 998 y Fk(\025)i Fs(1,)h Fl(n)475 1005 y Ft(0)508 998 y Fs(=)f Fl(m)p Fs(,)h(yielding)k(the)d(claim)h(since)h Fl(k)c Fk(\024)g Fl(m)p Fs(.)20 b(T)l(o)15 b(pro)o(v)o(e)g(\(88\),)e(write)670 1100 y(\()p Fl(u)714 1081 y Fq(k)735 1100 y Fl(d)759 1081 y Fq(l)771 1100 y Fs(\))789 1107 y Fq(mn)856 1100 y Fs(=)915 1060 y Fh(X)904 1152 y Fq(l)p Ff(\024)p Fq(m)973 1143 y Fe(0)993 1100 y Fl(u)1019 1081 y Fq(k)1019 1111 y(mm)1081 1102 y Fe(0)1095 1100 y Fl(d)1119 1081 y Fq(l)1119 1111 y(m)1150 1102 y Fe(0)1161 1111 y Fq(n)-47 1251 y Fs(where)j(the)h(constrain)o(t)f(in)h(the)g(sum)g(comes)f(from)g(the)h (second)g(inequalit)o(y)h(in)f(\(87\),)f(and)g(note)h(that)f Fl(u)1779 1235 y Fq(k)1779 1265 y(mm)1841 1255 y Fe(0)1871 1251 y Fs(is)-47 1308 y(b)q(ounded)g(b)o(y)f(\(\026)-23 b Fl(\017)p Fs(\))255 1291 y Fq(k)292 1308 y Fs(if)15 b Fl(m)e Fs(=)g Fl(m)474 1291 y Ff(0)498 1308 y Fs(=)g(0)i(and)g(b)o(y) 700 1410 y(exp\()p Fk(\000)p Fl(c\030)864 1391 y Fq(m)895 1379 y Fe(0)908 1410 y Fl(\027)933 1391 y Ff(\000)p Fq(q)980 1410 y Fs(\)\()p Fl(C)t Fs(\026)-24 b Fl(\017)p Fs(\))1088 1391 y Fq(k)q Ff(\000)p Ft(1)-47 1512 y Fs(otherwise)21 b(\(b)q(oth)h(b)q(ounds)h(follo)o(wing)f(from)f(\(84\))g(and)h(the)f (fact)h(that,)g(b)o(y)f(de\014nition,)k Fl(u)1586 1519 y Fq(mn)1665 1512 y Fs(=)e(0,)g(unless)-47 1569 y Fl(m)13 b Fk(\025)h Fl(n)p Fs(\).)21 b(The)16 b(b)q(ound)g(\(88\))f(follo)o(ws) g(b)o(y)h(com)o(bining)h(these)f(with)g(\(87\),)e(since)j Fl(l)d Fk(\024)f Fl(m)1455 1552 y Ff(0)1467 1569 y Fs(,)i(w)o(e)h(can)g (therefore)f(use)-47 1625 y(the)f(factor)g(exp\()p Fk(\000)p Fl(c\030)324 1608 y Fq(m)355 1597 y Fe(0)368 1625 y Fl(\027)393 1608 y Ff(\000)p Fq(q)440 1625 y Fs(\))g(and)h Fl(\027)j Fs(small)d(to)f(obtain)h(the)g(factor)f(\026)-24 b Fl(\017)1139 1608 y Fq(l)p Ft(+1)1212 1625 y Fs(in)16 b(\(88\))d(\(\026)-23 b Fl(\017)15 b Fs(is)g(a)f(\014xed)h(small)h(n)o(um)o(b)q(er\).)23 1681 y(Inserting)g(\(87\),)e(\(88\))g(in)o(to)539 1784 y Fl(\032)563 1765 y Fq(N)609 1784 y Fs(=)f(\()p Fl(u)d Fs(+)g Fl(d)p Fs(\))798 1765 y Fq(N)844 1784 y Fs(=)892 1743 y Fh(X)959 1784 y Fl(d)983 1765 y Fq(l)994 1770 y Fj(0)1013 1784 y Fl(u)1039 1765 y Fq(k)1057 1770 y Fj(1)1077 1784 y Fl(d)1101 1765 y Fq(l)1112 1770 y Fj(1)1138 1784 y Fl(:)e(:)g(:)d(u)1224 1765 y Fq(k)1242 1769 y Fd(s)1262 1784 y Fl(d)1286 1765 y Fq(l)1297 1769 y Fd(s)-47 1886 y Fs(where)15 b Fl(l)98 1893 y Fq(i)124 1886 y Fk(\025)e Fs(0,)i Fl(l)237 1893 y Fq(i)263 1886 y Fl(>)e Fs(0)i(for)f Fl(i)f Fk(6)p Fs(=)g(0)p Fl(;)8 b(s)p Fs(,)14 b(w)o(e)h(obtain)g(the)g (b)q(ound)720 1947 y Fh(X)739 2035 y Fq(n)787 1988 y Fl(\032)811 1969 y Fq(N)811 1999 y(n)832 2004 y Fj(0)850 1999 y Fq(n)886 1988 y Fk(\024)e Fl(C)970 1969 y Fq(N)1003 1988 y Fs(\026)-23 b Fl(\017)1021 1969 y Fq(N)t Ff(\000)p Fq(n)1101 1974 y Fj(0)1121 1988 y Fl(:)-47 2128 y Fs(where)15 b(\026)-23 b Fl(\017)102 2112 y Ff(\000)p Fq(n)150 2117 y Fj(0)186 2128 y Fs(comes)15 b(from)f(the)i(fact)e(that)h(w)o(e)g(ha)o (v)o(e)f(no)i(\026)-23 b Fl(\017)15 b Fs(b)q(ound)h(on)f Fl(d)1189 2112 y Fq(l)1200 2117 y Fj(0)1219 2128 y Fs(,)g(but)g(w)o(e)g (can)g(use)h Fl(l)1573 2135 y Ft(0)1605 2128 y Fk(\024)d Fl(n)1680 2135 y Ft(0)1700 2128 y Fs(.)23 2185 y(This)j(pro)o(v)o(es)e (the)i(Lemma,)f(if)g(w)o(e)g(c)o(ho)q(ose)g(in)h(\(81\))p 326 2281 40 2 v 326 2306 a Fl(m)c Fs(=)h Fk(\000)467 2275 y Fs(1)p 466 2295 26 2 v 466 2337 a Fl(\034)504 2306 y Fs(log)c Fl(C)t Fs(\026)-24 b Fl(\017)12 b Fs(=)h Fk(\000)p Fl(cm)780 2287 y Ff(\000)p Ft(1)828 2306 y Fl(\027)853 2287 y Ff(\000)p Fq(q)907 2306 y Fs(log)8 b Fl(C)t Fs(\026)-24 b Fl(\017)13 b Fk(\025)g Fs(exp\()p Fk(\000)p Fl(C)s(\027)1271 2287 y Ff(\000)p Ft(3)1319 2306 y Fs(\(log)8 b Fl(\027)1428 2287 y 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2517 y Fq(c)1662 2534 y Fs(.)29 b(Moreo)o(v)o(er,)-47 2596 y(from)16 b(part)h(\(b\))g(of)g(Prop)q(osition)g(1,)g(w)o(e)g(kno) o(w)g(that,)g Fk(8)p Fl(!)h Fk(2)f Fs(\012,)g Fl(F)6 b Fs(\()p Fl(!)r Fs(\))16 b Fk(2)g Fl(U)1273 2603 y Fq(\024)1296 2596 y Fs(,)h(with)h Fl(\024)e Fk(\024)h Fl(C)s Fs(\(\()1603 2574 y Ff(jj)p Fq(!)q Ff(jj)p 1603 2586 63 2 v 1623 2612 a Fq(\027)1669 2596 y Fs(\))1687 2580 y Fq(\013)1723 2596 y Fs(+)1776 2578 y Ft(1)p 1775 2585 20 2 v 1775 2612 a Fq(\027)1799 2596 y Fs(\);)h(so,)-47 2663 y(altogether,)c Fl(C)s Fs(\()p Fl(!)r Fs(\))e Fk(\024)h Fl(C)s Fs(\()401 2641 y Ff(jj)p Fq(!)q Ff(jj)p Ft(+1)p 401 2653 108 2 v 444 2679 a Fq(\027)512 2663 y Fs(\))530 2647 y Fq(c)547 2663 y Fs(.)1307 b Fc(2)904 3006 y Fs(14)p eop %%Page: 15 15 15 14 bop -47 -2 a Fn(References)-25 100 y Fs([1])22 b(J.)39 b(Bricmon)o(t,)46 b(A.)39 b(Kupiainen,)48 b(R.)40 b(Lefev)o(ere:)69 b Fr(Pr)n(ob)n(abilistic)37 b(estimates)i(for)g(the)g (two)g(di-)46 156 y(mensional)34 b(sto)n(chastic)h(Navier-Stokes)g(e)n (quations)p Fs(.)g(Preprin)o(t,)41 b(to)35 b(app)q(ear)h(in)h(J.)f (Stat.)f(Ph)o(ys.)46 213 y(\(h)o(ttp://mp)q(ej.unige.c)o (h/mp-arc/e/00-113.\).)-25 307 y([2])22 b(J.L.)15 b(Do)q(ob:)20 b Fr(Sto)n(chastic)15 b(Pr)n(o)n(c)n(esses)p Fs(,)e(John)j(Wiley)l(,)g (New-Y)l(ork,)f(1953.)-25 400 y([3])22 b(F.)17 b(Flandoli,)j(B.)d (Maslo)o(wski:)25 b Fr(Er)n(go)n(dicity)19 b(of)g(the)g(2-D)g (Navier-Stokes)f(e)n(quation)h(under)g(r)n(andom)g(p)n(er-)46 457 y(turb)n(ations.)c Fs(Comm)o(un.)f(Math.)g(Ph)o(ys.)h Fm(171)h Fs(\(1995\),)d(119-141.)-25 551 y([4])22 b(S.)12 b(Kuksin,)i(A.)e(Shiriky)o(an:)19 b Fr(Sto)n(chastic)13 b(dissip)n(ative)g(PDE's)g(and)h(Gibbs)f(me)n(asur)n(es)p Fs(.)e(Preprin)o(t,)i(to)e(app)q(ear)46 607 y(in)16 b(Comm)o(un.)e (Math.)g(Ph)o(ys.)-25 701 y([5])22 b(J.)c(C.)f(Mattingly)l(,)i(Y.)f (Sinai:)26 b Fr(A)o(n)18 b(elementary)h(pr)n(o)n(of)g(of)g(the)g (existenc)n(e)f(and)g(uniqueness)g(the)n(or)n(em)h(for)46 757 y(the)d(Navier-Stokes)g(e)n(quations)p Fs(.)f(Preprin)o(t.)-25 851 y([6])22 b(J.)g(C.)g(Mattingly:)33 b Fr(Er)n(go)n(dicity)23 b(of)g(2D)g(Navier-Stokes)f(e)n(quations)g(with)h(r)n(andom)g(for)n (cing)f(and)h(lar)n(ge)46 908 y(visc)n(osity.)14 b Fs(Comm)o(un.)g (Math.)g(Ph)o(ys.)h Fm(206)h Fs(\(1999\),)d(273-288.)904 3006 y(15)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0006130629344--