Content-Type: multipart/mixed; boundary="-------------0306030517132" This is a multi-part message in MIME format. ---------------0306030517132 Content-Type: text/plain; name="03-252.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-252.comments" Dedicated to Giovanni Jona-Lasinio on his 70-th birthday ---------------0306030517132 Content-Type: text/plain; name="03-252.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-252.keywords" 2D Euler equation, Navier-Stokes, 2D turbulence, Reynolds number, Yudovich theorem, stationary measure, white noise ---------------0306030517132 Content-Type: application/postscript; name="paper.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="paper.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: euler.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) 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Fq(E)22 b Fs(k)p Fv(U)2126 2034 y Fu(\027)2173 2022 y Fs(k)2215 1988 y Fr(2)2215 2043 y(2)2274 2022 y Fw(=)2372 1966 y(1)p 2372 2003 V 2372 2079 a(2)2424 2022 y Fv(L)2481 1988 y Fo(\000)p Fr(2)2569 2022 y Fv(B)2632 2034 y Fr(1)2669 2022 y Fv(;)516 b Fw(\(1.3\))515 2220 y(where)27 b Fv(B)818 2232 y Fr(0)882 2220 y Fw(and)h Fv(B)1107 2232 y Fr(1)1171 2220 y Fw(are)f(explicit)g (constan)n(ts,)g(calculated)g(in)g(terms)h(of)f(the)h(force)k(^)-48 b Fv(\021)t Fw(,)27 b(and)515 2320 y(for)d Fv(m)f Fw(=)g(0)p Fv(;)14 b Fw(1)p Fv(;)g Fw(2)p Fv(;)g(:)g(:)g(:)37 b Fw(w)n(e)25 b(denote)g(b)n(y)g Fs(k)14 b(\001)g(k)1827 2332 y Fu(m)1914 2320 y Fw(the)26 b(norm)e(in)i(the)g(Sob)r(olev)e (space)h Fv(H)3107 2290 y Fu(m)3193 2320 y Fs(\032)d Fv(H)7 b Fw(.)515 2419 y(F)-7 b(or)27 b Fv(m)c Fw(=)f(0)27 b Fs(k)18 b(\001)h(k)1060 2431 y Fr(0)1124 2419 y Fw(is)28 b(the)g Fv(L)1408 2431 y Fr(2)1445 2419 y Fw(-norm,)e(it)i(will)g(b)r (e)g(denoted)g Fs(j)18 b(\001)h(j)p Fw(.)639 2519 y(The)30 b(estimates)f(\(1.3\))g(imply)g(that)h(the)f(a)n(v)n(eraged)e(energy) 2552 2489 y Fr(1)2618 2519 y Fq(E)c Fs(j)p Fv(U)2762 2531 y Fu(\027)2809 2519 y Fw(\()p Fv(t)p Fw(\))p Fs(j)2926 2489 y Fr(2)2993 2519 y Fw(of)29 b(the)h(\015o)n(w)515 2619 y Fv(U)572 2631 y Fu(\027)640 2619 y Fw(is)e(of)f(order)g Fv(L)1093 2588 y Fr(2)1157 2619 y Fw(\(see)h(b)r(elo)n(w)g(\(2.15\))o (\).)37 b(Hence,)28 b(its)g(Reynolds)f(n)n(um)n(b)r(er)g(is)1407 2852 y Fv(R)q Fw(\()p Fv(U)1560 2864 y Fu(\027)1601 2852 y Fw(\))c(=)1754 2796 y Fv(L)p Fw(\()p Fq(E)f Fs(j)p Fv(U)1986 2808 y Fu(\027)2033 2796 y Fs(j)2056 2766 y Fr(2)2093 2796 y Fw(\))2125 2766 y Fr(1)p Fu(=)p Fr(2)p 1754 2833 476 4 v 1968 2909 a Fv(\027)2263 2852 y Fs(\030)2360 2796 y Fv(L)2417 2766 y Fr(2)p 2360 2833 94 4 v 2384 2909 a Fv(\027)2464 2852 y(:)515 3060 y Fw(The)31 b(fact)g(that)g(the)g (energy)e(of)i(the)g(\015o)n(w)g(is)f(of)h(order)e Fv(L)2335 3030 y Fr(2)2403 3060 y Fw(while)i(its)g(Reynolds)f(n)n(um)n(b)r(er)515 3160 y(is)g(of)g(order)f Fv(L)975 3130 y Fr(2)1012 3160 y Fv(\027)1058 3130 y Fo(\000)p Fr(1)1177 3160 y Fw(mak)n(es)h(our)f (scaling)g(of)i(the)f(NS)h(system)f(con)n(v)n(enien)n(t)f(to)h(study)h (the)515 3260 y(2D)40 b(turbulence.)74 b(W)-7 b(e)40 b(note)g(that)g(the)g(substitution)h Fv(u)p Fw(\()p Fv(t)p Fw(\))i(=)h Fv(\027)2654 3229 y Fr(1)p Fu(=)p Fr(5)2758 3260 y Fv(v)s Fw(\()p Fv(\034)9 b Fw(\),)45 b Fv(\034)53 b Fw(=)43 b Fv(\027)3221 3229 y Fr(1)p Fu(=)p Fr(5)3326 3260 y Fv(t)p Fw(,)515 3359 y(transforms)26 b(\(1.1\))h(to)h(the)g (equation)1076 3534 y Fv(v)1116 3546 y Fu(\034)1176 3534 y Fs(\000)18 b Fv(\016)s Fw(\001)p Fv(v)k Fw(+)c(\()p Fv(v)k Fs(\001)c(r)p Fw(\))p Fv(v)k Fw(+)c Fs(r)p Fv(p)2005 3499 y Fo(0)2052 3534 y Fw(=)28 b(~)-48 b Fv(\021)t Fw(\()p Fv(\034)5 b(;)14 b(x)p Fw(\))p Fv(;)98 b(\016)26 b Fw(=)c Fv(\027)2690 3499 y Fr(4)p Fu(=)p Fr(5)2795 3534 y Fv(;)390 b Fw(\(1.4\))515 3708 y(where)35 b(~)-48 b Fv(\021)34 b Fw(is)c(a)f(new)h(random)f(\014eld,)i(distributed)g(as)k(^)-48 b Fv(\021)s Fw(.)45 b(Stationary)29 b(solutions)g(for)g(\(1.4\))515 3808 y(ha)n(v)n(e)21 b(energy)f(of)i(order)f Fv(L)1318 3778 y Fr(2)1355 3808 y Fv(\016)1395 3778 y Fo(\000)p Fr(1)1506 3808 y Fw(and)h(the)g(Reynolds)g(n)n(um)n(b)r(er)g(of)g (order)e Fv(L)2807 3778 y Fr(2)2844 3808 y Fv(\016)2884 3778 y Fo(\000)p Fr(3)p Fu(=)p Fr(2)3040 3808 y Fw(.)36 b(So)21 b(\(1.1\))515 3908 y(is)k(equiv)-5 b(alen)n(t)25 b(to)h(the)g(NS)g(system)f(\(1.4\),)h(p)r(erturb)r(ed)g(b)n(y)f(a)g (random)g(force)g(of)g(order)f(one,)515 4007 y(but)k(the)g(former)f (scaling)f(is)i(more)e(con)n(v)n(enien)n(t)h(than)h(the)g(latter.)639 4107 y(Our)c(goal)e(in)j(this)f(w)n(ork)e(is)i(to)g(examine)g(the)g(in) n(viscous)f(limit)i(of)f(the)g(pro)r(cess)f Fv(U)3184 4119 y Fu(\027)3225 4107 y Fw(,)i(i.e.)515 4206 y(its)k(limit)g(as)g Fv(\027)h Fs(!)25 b Fw(0.)41 b(The)29 b(p)r(erio)r(d)g Fv(L)f Fw(is)h(a)f(parameter)g(of)h(the)g(problem)f(whic)n(h)h(w)n(e)g (k)n(eep)515 4306 y(\014xed.)639 4406 y(In)21 b(Theorem)f(3.1)g(and)g (Lemma)h(3.2)e(w)n(e)i(sho)n(w)e(that)i(an)n(y)f(sequence)25 b(~)-47 b Fv(\027)2792 4418 y Fu(j)2850 4406 y Fs(!)24 b Fw(0)c(con)n(tains)f(a)515 4505 y(subsequence)f Fv(\027)1016 4517 y Fu(j)1074 4505 y Fs(!)23 b Fw(0)18 b(suc)n(h)g(that)h(the)g(pro) r(cess)f Fv(U)2061 4517 y Fu(\027)2094 4525 y Ft(j)2147 4505 y Fw(a.s.)34 b(con)n(v)n(erges)16 b(to)i(a)g(limiting)h(pro)r (cess)515 4605 y Fv(U)9 b Fw(\()p Fs(\001)p Fw(\).)56 b(Its)33 b(energy)g Fs(j)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(j)1353 4575 y Fr(2)1425 4605 y Fw(and)33 b(enstroph)n(y)g Fs(k)p Fv(U)9 b Fs(k)2132 4575 y Fr(2)2132 4626 y(1)2201 4605 y Fw(=)33 b Fs(j)p Fw(rot)14 b Fv(U)9 b Fs(j)2532 4575 y Fr(2)2602 4605 y Fw(are)33 b(time-indep)r(enden)n(t)515 4705 y(random)26 b(constan)n(ts:)1312 4879 y Fs(j)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(j)1518 4845 y Fr(2)1578 4879 y Fw(=)23 b Fv(E)5 b(;)97 b Fs(k)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(k)2096 4845 y Fr(2)2096 4900 y(1)2156 4879 y Fw(=)22 b(\012)166 b Fs(8)14 b Fv(t:)p 515 4929 1146 4 v 607 4983 a Fn(1)642 5006 y Fm(more)22 b(exactly)-6 b(,)25 b(the)g(doubled)f(energy)h(p)r(er)e(unit)h(area)g(since)g Fl(j)p Fk(U)2355 5014 y Fj(\027)2393 5006 y Fl(j)2413 4983 y Fn(2)2467 5006 y Fm(=)19 b(\(2)p Fk(\031)r(L)p Fm(\))2721 4983 y Fi(\000)p Fn(2)2817 4953 y Fh(R)2875 5006 y Fl(j)p Fk(U)2943 5014 y Fj(\027)2981 5006 y Fl(j)3001 4983 y Fn(2)3047 5006 y Fk(dx)p Fm(.)1926 5255 y Fw(2)p eop %%Page: 3 3 3 2 bop 515 523 a Fw(The)25 b(exp)r(ectation)g(of)h(the)f(enstroph)n(y) g(is)g(kno)n(wn,)g(while)g(the)h(exp)r(ectation)f(of)h(the)f(energy)515 623 y(is)i(b)r(ounded)h(from)f(b)r(elo)n(w)h(and)f(from)g(ab)r(o)n(v)n (e:)1232 838 y Fq(E)22 b Fw(\012)29 b(=)1482 782 y(1)p 1482 819 42 4 v 1482 895 a(2)1547 838 y Fv(B)1610 850 y Fr(0)1647 838 y Fv(;)1860 782 y(L)1917 751 y Fr(2)1954 782 y Fv(B)2021 751 y Fr(2)2017 802 y(0)p 1860 819 199 4 v 1888 895 a Fw(2)p Fv(B)1993 907 y Fr(1)2091 838 y Fs(\024)23 b Fq(E)f Fv(E)34 b Fs(\024)2435 782 y Fv(L)2492 751 y Fr(2)2529 782 y Fv(B)2592 794 y Fr(0)p 2435 819 194 4 v 2511 895 a Fw(2)2639 838 y Fv(:)546 b Fw(\(1.5\))515 1056 y(Moreo)n(v)n(er,)25 b(certain)i(exp)r(onen)n(tial)g(momen)n(ts)g (of)h Fv(E)k Fw(and)c(\012)f(are)g(\014nite,)h(and)1472 1273 y Fq(E)22 b Fs(k)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(k)1779 1239 y Fr(2)1779 1294 y(2)1845 1273 y Fs(\024)1942 1217 y Fw(1)p 1942 1254 42 4 v 1942 1330 a(2)1994 1273 y Fv(L)2051 1239 y Fo(\000)p Fr(2)2139 1273 y Fv(B)2202 1285 y Fr(1)2323 1273 y Fs(8)p Fv(t:)785 b Fw(\(1.6\))639 1479 y(Crucial)36 b(prop)r(ert)n(y)g(of)h(the)h(pro)r(cess)e Fv(U)9 b Fw(,)39 b(established)d(in)i(Theorem)e(3.4,)i(is)f(that)h(it) 515 1579 y(satis\014es)27 b(the)h(Euler)e(equation)1028 1778 y(_)1004 1799 y Fv(U)h Fw(+)18 b(\()p Fv(U)28 b Fs(\001)18 b(r)p Fw(\))p Fv(U)28 b Fw(+)18 b Fs(r)p Fv(p)23 b Fw(=)g(0)p Fv(;)96 b Fw(div)15 b Fv(U)32 b Fw(=)23 b(0)p Fv(;)2447 1686 y Fp(Z)2544 1799 y Fv(U)g(dx)g Fw(=)g(0)p Fv(:)318 b Fw(\(1.7\))515 2024 y(Note)25 b(that)g(although)g Fv(U)34 b Fw(solv)n(es)24 b(the)h(free)g(equation)g(\(1.7\))o(,)h (relations)e(\(1.5\))h(sho)n(w)f(that)i(it)515 2124 y(remem)n(b)r(ers) 21 b(some)h(c)n(haracteristics)e(of)i(the)h(force)28 b(^)-48 b Fv(\021)s Fw(.)35 b(In)23 b(particular,)f Fv(U)31 b Fw(is)22 b(not)g(iden)n(tically)515 2224 y(zero.)39 b(In)29 b(section)f(3.5)g(it)h(is)f(pro)n(v)n(ed)f(that)i(if)h(the)f (force)34 b(^)-48 b Fv(\021)s Fw(\()p Fv(t;)14 b(x)p Fw(\))29 b(is)g(stationary)e(in)i Fv(x)p Fw(,)h(then)515 2323 y(the)e(pro)r(cess)e Fv(U)37 b Fw(has)27 b(zero)f(mean-v)-5 b(alue:)1525 2506 y Fq(E)23 b Fv(U)9 b Fw(\()p Fv(t;)14 b(x)p Fw(\))29 b(=)23 b(0)165 b Fs(8)14 b Fv(t;)g(x)g(:)515 2689 y Fw(Due)28 b(to)f(\(1.5\))h(this)g(implies)f(that)h(the)g(pro)r (cess)f Fv(U)36 b Fw(is)28 b(`gen)n(uinely)f(random'.)639 2788 y(Due)k(to)f(\(1.6\),)g Fv(U)9 b Fw(\(0\))27 b Fs(2)h Fv(H)1502 2758 y Fr(2)1569 2788 y Fw(a.s.)44 b(No)30 b(existence)g(and)g(uniqueness)g(theorem)f(for)h(the)515 2888 y(Euler)18 b(equation)g(with)i(Cauc)n(h)n(y)e(data)g(in)i Fv(H)1874 2858 y Fr(2)1930 2888 y Fw(is)f(kno)n(wn.)33 b(Still,)22 b(as)c(w)n(e)h(sho)n(w)f(in)h(section)g(3.4,)515 2987 y(estimates)38 b(\(1.5\))f(and)h(\(1.6\))g(imply)h(that)f(with)h (probabilit)n(y)e(one)g(tra)5 b(jectories)37 b(of)h(the)515 3087 y(pro)r(cess)32 b Fv(U)42 b Fw(b)r(elong)33 b(to)h(a)f(uniqueness) g(class)f(for)h(the)h(Euler)f(equation,)h(and)f(that)h(they)515 3187 y(are)f(tra)5 b(jectories)33 b(of)i(a)f(con)n(tin)n(uous)g (dynamical)g(system)g(whic)n(h)h(the)g(equation)f(de\014nes)515 3286 y(in)g(a)f(certain)g(functional)h(space)f Fv(H)1676 3256 y Fu(E)1732 3286 y Fw(.)55 b(Accordingly)-7 b(,)34 b(the)g(distribution)g Fs(D)r Fv(U)9 b Fw(\()p Fv(t)p Fw(\))34 b(of)g(the)515 3386 y(pro)r(cess)28 b Fv(U)39 b Fw(at)30 b(an)n(y)f(p)r(oin)n(t)h Fv(t)g Fw(is)g(an)f(in)n(v)-5 b(arian)n(t)29 b(measure)g(for)g(the)h(Euler)f(equation)h(in)g(the)515 3486 y(space)d Fv(H)813 3455 y Fu(E)896 3486 y Fw(\(see)h(Theorem)e (3.6\).)639 3585 y(Join)n(tly)h(with)h(\(1.2\))g(the)g(established)f (results)g(mean)g(that)h(the)g(double)g(limit)1464 3768 y(lim)1439 3819 y Fu(\027)1472 3827 y Ft(j)1504 3819 y Fo(!)p Fr(0)1663 3768 y Fw(lim)1631 3821 y Fu(T)9 b Fo(!1)1825 3768 y Fv(u)1873 3780 y Fu(\027)1906 3788 y Ft(j)1941 3768 y Fw(\()p Fv(T)30 b Fw(+)18 b Fs(\001)p Fw(\))23 b(=)g Fv(U)9 b Fw(\()p Fs(\001)p Fw(\))754 b(\(1.8\))515 3997 y(exists)24 b(\(the)h(con)n(v)n(ergences)d(are)h(understo)r(o)r(d) h(as)g Fs(\003)p Fw(-w)n(eak)f(con)n(v)n(ergences)f(of)i(distributions) 515 4097 y(of)29 b(random)g(pro)r(cesses)g Fs(f)p Fv(u)p Fw(\()p Fv(t)p Fw(\),)h Fv(t)c Fs(\025)h Fw(0)p Fs(g)14 b Fw(\),)29 b(and)h(the)g(limiting)g(stationary)f(pro)r(cess)f Fv(U)39 b Fw(is)29 b(a)515 4197 y(random)h(solution)h(of)g(the)h (deterministic)f(Euler)g(equation)f(\(1.7\).)48 b(The)31 b(limits)h(in)f(\(1.8\))515 4296 y(do)37 b(not)g(comm)n(ute.)66 b(Indeed,)39 b(when)f Fv(\027)1788 4308 y Fu(j)1862 4296 y Fs(!)h Fw(0,)g Fv(u)2136 4308 y Fu(\027)2169 4316 y Ft(j)2204 4296 y Fw(\()p Fv(t)p Fw(\))f(con)n(v)n(erges)d(to)i(a)g (deterministic)515 4396 y(solution)26 b Fv(u)876 4408 y Fr(0)913 4396 y Fw(\()p Fv(t)p Fw(\))h(of)33 b(\(1.7\))o(.)k(When)27 b Fv(t)c Fs(!)g(1)k Fw(this)g(solution)f(has)g(no)g(reason)f(to)h(con)n (v)n(erge)e(to)515 4496 y(a)j(limit)h(and)g(to)f(remem)n(b)r(er)g(c)n (haracteristics)e(of)j(the)g(force)33 b(~)-48 b Fv(\021)31 b Fw(\(cf.)37 b(\(1.5\)\).)639 4595 y(Our)19 b(results)g(sho)n(w)f (that)i(the)g(scaling)f(\(1.1\))h(of)f(the)h(2D)f(NS)h(system)f(is)h (the)f(righ)n(t)g(one)g(to)515 4695 y(study)25 b(the)h(2D)f(turbulence) h(since)f(under)g(the)h(limit)g Fv(\027)i Fs(!)23 b Fw(0)i(stationary)f (solutions)g(of)i(the)515 4794 y(equation)f(con)n(v)n(erge)f(\(at)i (least,)g(along)f(a)g(subsequence\))h(to)g(a)g(regular)e(limit,)j (while)f(their)515 4894 y(Reynolds)18 b(n)n(um)n(b)r(ers)h(div)n(erge)e (to)i(in\014nit)n(y)-7 b(.)34 b(In)19 b(the)h(3D)f(case)f(the)h(righ)n (t)f(scaling)g(is)h(di\013eren)n(t,)515 4994 y(if)26 b(w)n(e)f(b)r(eliev)n(e)h(that)g(predictions)f(of)h(the)g(Kolmogoro)n (v)c(theory)j(apply)h(to)f(space-p)r(erio)r(dic)1926 5255 y(3)p eop %%Page: 4 4 4 3 bop 515 523 a Fw(turbulen)n(t)28 b(\015o)n(ws.)38 b(Indeed,)29 b(let)f Fv(u)1584 535 y Fu(\027)1653 523 y Fw(b)r(e)h(a)f(stationary)e(\(in)j(time\))g(solution)f(of)g(the)g(3D) h(NS)515 623 y(system)24 b(\(1.1\))h(with)1169 563 y Fs(p)p 1238 563 47 4 v 60 x Fv(\027)g Fw(^)-48 b Fv(\021)28 b Fw(replaced)c(b)n(y)31 b(^)-48 b Fv(\021)s Fw(.)1908 593 y Fr(2)1981 623 y Fw(Then,)26 b(applying)e(the)h(Ito)f(form)n(ula,) h(w)n(e)f(get)515 722 y(that)f Fv(\027)5 b Fq(E)22 b Fs(k)p Fv(u)889 734 y Fu(\027)936 722 y Fs(k)978 692 y Fr(2)978 743 y(1)1037 722 y Fw(=)1135 690 y Fr(1)p 1135 704 34 4 v 1135 751 a(2)1178 722 y Fv(B)1241 734 y Fr(0)1301 722 y Fw(\(cf.)36 b(\(1.3\)\).)f(In)23 b(particular,)g Fv(\027)5 b Fq(E)23 b Fs(k)p Fv(u)2417 734 y Fu(\027)2463 722 y Fs(k)2505 692 y Fr(2)2505 743 y(1)2564 722 y Fw(\(trivially\))g (con)n(v)n(erges)d(to)515 822 y(a)h(p)r(ositiv)n(e)h(\014nite)g(limit)h (when)f Fv(\027)28 b Fs(!)c Fw(0,)e(as)g(predicted)f(b)n(y)h(the)g (Kolmogoro)n(v)d(theory)-7 b(.)34 b(So)22 b(in)515 922 y(the)27 b(3D)g(case)f(exactly)g(this)h(scaling)f(\(without)h(the)g (factor)2401 862 y Fs(p)p 2470 862 47 4 v 60 x Fv(\027)32 b Fw(in)27 b(the)h(r.h.s.\))36 b(should)27 b(b)r(e)515 1021 y(correct.)41 b(W)-7 b(e)30 b(also)f(note)g(that)h(our)f(results)g (apparen)n(tly)g(disagree)f(with)i(the)g(Kraic)n(hnan)515 1121 y(theory)f(of)g(2D)h(turbulence,)g(see)g([Kra67)m(])g(and)g([KM80) n(,)h(Section)f(4.2].)42 b(Indeed,)30 b(if)h Fv(u)3253 1133 y Fu(\027)3323 1121 y Fw(is)515 1220 y(a)g(stationary)f(solution)h (of)38 b(\(1.1\),)33 b(then)f Fq(E)22 b Fs(k)p Fv(u)1983 1232 y Fu(\027)2029 1220 y Fw(\()p Fv(t)p Fw(\))p Fs(k)2165 1190 y Fr(2)2165 1241 y(2)2232 1220 y Fw(=)30 b Fv(O)r Fw(\(1\))i(as)f Fv(\027)k Fs(!)30 b Fw(0,)i(see)f(\(1.3\).)49 b(In)515 1320 y(particular)43 b(lim)14 b Fv(\027)5 b Fq(E)23 b Fs(k)p Fv(u)1244 1332 y Fu(\027)1290 1320 y Fs(k)1332 1290 y Fr(2)1332 1341 y(2)1397 1320 y Fw(=)k(0.)45 b(But)31 b(it)g(is)g(p)r(ostulated)f(in)h(the)g(theory)f(that)h(this)f (limit)515 1420 y(is)d(p)r(ositiv)n(e)g(and)h(\014nite.)1311 1390 y Fr(3)639 1560 y Fw(Due)f(to)f(\(1.8\))o(,)h(solutions)e(of)h (the)g(Euler)f(equation)g(appro)n(ximate)g(the)h(v)n(elo)r(cit)n(y)f (\014elds)515 1660 y(of)42 b(2D)h(turbulen)n(t)g(\015o)n(ws)f(with)h (high)n(t)f(Reynolds)g(n)n(um)n(b)r(ers.)82 b(This)43 b(prop)r(ert)n(y)e(of)i(the)515 1760 y(2D)e(turbulence)h(w)n(as)f (predicted)h(b)n(y)f(man)n(y)g(ph)n(ysicists,)k(cf.)80 b([Ons49)n(,)42 b(KM80)o(].)79 b(Ac-)515 1859 y(cordingly)-7 b(,)41 b(one)e(can)g(try)g(to)g(use)g(the)h(Euler)f(equation)f(to)i (calculate)e(c)n(haracteristics)515 1959 y(of)j(solutions)f Fv(u)1031 1971 y Fu(\027)1113 1959 y Fw(for)g(\(1.1\))h(with)g Fv(\027)50 b Fs(\034)c Fw(1.)76 b(F)-7 b(or)40 b(example,)k(let)d(us)g (tak)n(e)f(a)h(con)n(tin)n(u-)515 2059 y(ous)f(functional)g Fv(f)49 b Fw(on)40 b Fv(H)7 b Fw(,)43 b(consider)d(an)g(observ)-5 b(able)39 b(v)-5 b(ariable)39 b Fv(f)9 b Fw(\()p Fv(u)2774 2071 y Fu(\027)2815 2059 y Fw(\()p Fv(t)p Fw(\)\))41 b(and)f(try)g(to)515 2158 y(\014nd)f(its)g(a)n(v)n(erage)d(in)j(ensem)n (ble.)71 b(If)39 b(w)n(e)g(can)f(use)h(some)g(ergo)r(dic)e(argumen)n (ts)h(to)h(cal-)515 2258 y(culate)34 b(the)h(a)n(v)n(erages)c Fs(h)p Fv(f)9 b Fs(i)35 b Fw(of)g Fv(f)43 b Fw(along)33 b(tra)5 b(jectories)33 b(of)h(the)h(Euler)f(\015o)n(w,)h(then)g(due)g (to)515 2357 y(\(1.8\))30 b(w)n(e)g(will)h(ha)n(v)n(e)61 b(lim)1342 2369 y Fu(\027)1375 2377 y Ft(j)1406 2369 y Fo(!)p Fr(0)1523 2357 y Fw(lim)1639 2369 y Fu(t)p Fo(!1)1814 2357 y Fq(E)22 b Fv(f)9 b Fw(\()p Fv(u)2007 2369 y Fu(\027)2040 2377 y Ft(j)2081 2357 y Fw(\()p Fv(t)p Fw(\)\))29 b(=)f Fs(h)p Fv(f)9 b Fs(i)p Fw(,)32 b(indep)r(enden)n(tly)f(of)g(the)g(se-) 515 2457 y(quence)j Fv(\027)837 2469 y Fu(j)872 2457 y Fw(.)57 b(So)33 b Fs(h)p Fv(f)9 b Fs(i)35 b Fw(w)n(ould)f(appro)n (ximate)e(asymptotic)i(in)g(time)h(v)-5 b(alue)34 b(of)g Fq(E)23 b Fv(f)9 b Fw(\()p Fv(u)3206 2469 y Fu(\027)3252 2457 y Fw(\()p Fv(t)p Fw(\)\))515 2557 y(for)30 b(solutions)f(with)i (small)f(viscosit)n(y)f Fv(\027)5 b Fw(.)46 b(As)31 b(w)n(e)f(sho)n(w)f (in)i(Prop)r(osition)e(3.7,)h(the)h(quan-)515 2656 y(tities)g Fv(I)762 2668 y Fu(g)829 2656 y Fw(=)922 2589 y Fp(R)991 2656 y Fv(g)s Fw(\(rot)14 b Fv(U)9 b Fw(\()p Fv(t;)14 b(x)p Fw(\)\))g Fv(dx)31 b Fw(are)e(in)n(tegrals)h(of)g(motion)h(for)f (the)h(limiting)g(pro)r(cesses)515 2756 y Fv(U)9 b Fw(.)38 b(Hence,)28 b(their)g(tra)5 b(jectories)26 b(cannot)i(b)r(e)g(ergo)r (dic)f(in)h(the)h(co)r(dimension-t)n(w)n(o)d(surfaces)515 2856 y Fv(H)584 2868 y Fu(a;b)697 2856 y Fw(=)d Fs(fj)p Fv(u)p Fs(j)h Fw(=)f Fv(a;)14 b Fs(k)p Fv(u)p Fs(k)1246 2868 y Fr(1)1306 2856 y Fw(=)23 b Fv(b)p Fs(g)p Fw(.)38 b(So)28 b(it)h(is)f(unlik)n(ely)g(that)g(the)h(`righ)n(t')e(a)n(v)n (erage)e Fs(h)p Fv(f)9 b Fs(i)29 b Fw(can)f(b)r(e)515 2955 y(found,)22 b(assuming)e(equidistribution)g(of)g(tra)5 b(jectories)19 b(on)h(the)h(surfaces)e Fv(H)2849 2967 y Fu(a;b)2959 2955 y Fw(\(concerning)515 3055 y(suc)n(h)29 b(calculations)f(see)h([KM80)o(]\).)42 b(On)29 b(the)h(con)n(trary)-7 b(,)28 b(it)i(is)f(plausible)g(that)h(the)f(energy)515 3154 y Fs(j)p Fv(U)9 b Fs(j)627 3124 y Fr(2)700 3154 y Fw(and)36 b(the)h(quan)n(tities)e Fv(I)1449 3166 y Fu(g)1525 3154 y Fw(giv)n(e)g(a)h(complete)g(list)g(of)g(in)n(tegrals)f (of)h(motion)g(for)g(the)515 3254 y(pro)r(cess)27 b Fv(U)9 b Fw(.)38 b(If)28 b(so,)g(then)h(the)f(theory)g(of)g(Rob)r(ert)g (\(e.g.,)g(see)g(in)g([RS94)o(]\))h(can)f(b)r(e)g(used)g(to)515 3354 y(calculate)23 b(the)h(a)n(v)n(erages)c(of)k Fv(f)32 b Fw(along)23 b(the)h(corresp)r(onding)e(surfaces)g(of)i(in\014nite)g (co)r(dimen-)515 3453 y(sion.)37 b(Next)28 b(one)f(can)g(hop)r(e)h(to)g (in)n(tegrate)f(the)h(resulting)f(quan)n(tities)g(in)h(the)g (parameter,)515 3553 y(whic)n(h)34 b(parameterizes)e(the)i(surfaces)e (\(at)i(least,)i(to)d(in)n(tegrate)g(appro)n(ximately\).)54 b(This)515 3653 y(w)n(ould)26 b(giv)n(e)g(us)g(appro)n(ximate)f(v)-5 b(alue)27 b(of)g(the)g(exp)r(ectations)f Fq(E)c Fv(f)9 b Fw(\()p Fv(u)2632 3665 y Fu(\027)2679 3653 y Fw(\()p Fv(t)p Fw(\)\))27 b(for)g(large)e(time)i Fv(t)515 3752 y Fw(and)g(small)g(viscosit)n(y)g Fv(\027)5 b Fw(.)639 3893 y(The)34 b(approac)n(h)e(to)i(study)g(limiting)g(b)r(eha)n(viour)f (of)h(the)g(NS)h(system,)g(suggested)e(in)515 3993 y(this)23 b(w)n(ork,)g(applies)g(to)h(other)e(damp)r(ed/driv)n(en)h(Hamiltonian)g (PDEs)g(with)h(t)n(w)n(o)e(or)h(more)515 4092 y(in)n(tegrals)i(of)h (motion.)36 b(In)27 b(particular,)f(to)g(the)h(damp)r(ed/driv)n(en)e (nonlinear)h(Sc)n(hr\177)-42 b(odinger)515 4192 y(equation)1031 4291 y(_)k Fv(u)18 b Fs(\000)g Fv(\027)5 b Fw(\001)p Fv(u)19 b Fw(+)f Fv(i)p Fw(\(\001)p Fv(u)g Fs(\000)g(j)p Fv(u)p Fs(j)1803 4257 y Fr(2)1840 4291 y Fv(u)p Fw(\))23 b(=)2030 4228 y Fs(p)p 2099 4228 V 63 x Fv(\027)d(\021)s Fw(\()p Fv(t;)14 b(x)p Fw(\))p Fv(;)98 b Fw(dim)14 b Fv(x)23 b Fs(\024)g Fw(4)p Fv(:)515 4428 y Fw(No)n(w)18 b(the)i(limiting)g(pro)r(cesses)d Fv(U)28 b Fw(are)18 b(non-trivial)g(stationary)g(solutions)g(of)i(the)f(nonlinear)515 4527 y(Sc)n(hr\177)-42 b(odinger)30 b(equation,)i(and)f(the)h (distributions)g Fs(D)r Fv(U)9 b Fw(\(0\))31 b(are)g(its)h(in)n(v)-5 b(arian)n(t)30 b(measures.)515 4627 y(Details)25 b(will)h(b)r(e)f(giv)n (en)g(in)g(a)g(join)n(t)g(pap)r(er)g(with)h(Armen)f(Shiriky)n(an)f (whic)n(h)h(is)h(no)n(w)e(under)515 4727 y(preparation.)p 515 4771 1146 4 v 607 4825 a Fn(2)642 4848 y Fm(W)-6 b(e)24 b(b)r(oldly)g(b)r(eliev)n(e)g(that,)h(as)e(in)h(the)g(2D)g (case,)g(suc)n(h)g(solutions)g(exist.)607 4904 y Fn(3)642 4927 y Fm(Note)d(that)h(for)d(an)n(y)i(other)g(scaling)g(of)f(the)h(2D) g(NS)f(equation)i(\(e.g.,)f(for)f(\(1.4\))q(\))g(the)i(limit)c(lim)3158 4936 y Fj(\027)s Fi(!)p Fn(0)3300 4927 y Fk(u)3341 4935 y Fj(\027)515 5006 y Fm(do)r(es)24 b(not)g(exist.)1926 5255 y Fw(4)p eop %%Page: 5 5 5 4 bop 639 523 a Fx(Notations)p Fw(.)76 b Fs(P)7 b Fw(\()p Fv(X)g Fw(\))40 b(is)h(the)g(set)g(of)g(probabilit)n(y)f(Borel)g (measures)f(on)i(a)f(metric)515 623 y(space)33 b Fv(X)7 b Fw(,)34 b(pro)n(vided)f(with)h(the)g Fs(\003)p Fw(-w)n(eak)d(top)r (ology)-7 b(.)54 b Fv(C)2294 635 y Fu(b)2327 623 y Fw(\()p Fv(X)7 b Fw(\))34 b(is)f(the)h(space)f(of)h(b)r(ounded)515 722 y(con)n(tin)n(uous)g(functions)h(on)g Fv(X)7 b Fw(.)59 b(By)35 b Fv(C)q(;)14 b(C)1878 734 y Fr(1)1916 722 y Fw(,)38 b(etc.)59 b(w)n(e)35 b(denote)g(v)-5 b(arious)34 b Fv(\027)5 b Fw(-indep)r(enden)n(t)515 822 y(\014nite)28 b(constan)n(ts.)639 962 y Fx(Ac)m(kno)m(wledgemen)m(ts.)34 b Fw(I)23 b(thank)h(Oliv)n(er)e(P)n(enrose)f(and)i(P)n(eter)g(Constan)n (tin)g(for)f(dis-)515 1061 y(cussions,)28 b(related)g(to)h(the)g (statistics)g(of)f(the)i(2D)e(turbulence)h(and)g(the)g(2D)g(Euler)f (equa-)515 1161 y(tion,)f(resp)r(ectiv)n(ely)-7 b(.)37 b(This)27 b(researc)n(h)f(w)n(as)g(supp)r(orted)i(b)n(y)f(EPSR)n(C,)g (gran)n(t)f(N63055/01.)515 1431 y Fy(2)134 b(Preliminaries)515 1613 y Fw(Let)27 b Fv(H)33 b Fw(b)r(e)28 b(the)f(Hilb)r(ert)g(space)f (of)h(square-in)n(tegrable)d(div)n(ergence-free)h(v)n(ector)g(\014elds) i(on)515 1713 y Fv(T)576 1682 y Fr(2)645 1713 y Fw(with)33 b(zero)e(mean-v)-5 b(alue,)34 b(giv)n(en)e(the)h(scalar)e(pro)r(duct)h (\()p Fv(u;)14 b(v)s Fw(\))32 b(=)f(\(2)p Fv(\031)s(L)p Fw(\))2945 1682 y Fo(\000)p Fr(2)3048 1646 y Fp(R)3117 1713 y Fv(u)21 b Fs(\001)h Fv(v)17 b(dx)515 1812 y Fw(and)23 b(the)h(norm)f Fs(j)p Fv(u)p Fs(j)g Fw(=)g(\()p Fv(u;)14 b(u)p Fw(\))1426 1782 y Fr(1)p Fu(=)p Fr(2)1530 1812 y Fw(.)35 b(Let)24 b(\005)g(b)r(e)g Fv(L)1985 1824 y Fr(2)2022 1812 y Fw(-orthogonal)d(pro)5 b(jector)22 b(to)h Fv(H)7 b Fw(.)36 b(Applying)515 1912 y(\005)28 b(to)f(\(1.1\))g(w)n(e)h (get)1266 2070 y(_)-38 b Fv(u)p Fw(\()p Fv(t)p Fw(\))19 b(+)f Fv(\027)5 b(Au)p Fw(\()p Fv(t)p Fw(\))19 b(+)f Fv(B)t Fw(\()p Fv(u)p Fw(\()p Fv(t)p Fw(\))p Fv(;)28 b(u)p Fw(\()p Fv(t)p Fw(\)\))c(=)e Fv(\027)2470 2036 y Fr(1)p Fu(=)p Fr(2)2575 2070 y Fv(\021)s(;)566 b Fw(\(2.1\))515 2228 y(where)25 b Fv(A)e Fw(=)g Fs(\000)p Fw(\005\001,)j Fv(B)t Fw(\()p Fv(u;)14 b(u)p Fw(\))23 b(=)f(\005\()p Fv(u)15 b Fs(\001)f(r)p Fw(\))p Fv(u)26 b Fw(and)g Fv(\021)g Fw(=)d(\005)6 b(^)-48 b Fv(\021)s Fw(.)36 b(In)26 b(\(2.1\))f(w)n(e)h (view)f Fv(u)g Fw(and)h Fv(\021)j Fw(as)515 2328 y(curv)n(es)d(in)i Fv(H)7 b Fw(.)37 b(See)27 b(e.g.)37 b([CF88)o(,)28 b(Gal01)n(].)639 2427 y(F)-7 b(or)30 b Fv(\027)j Fw(=)28 b(0)i(equation)g(\(1.1\))g(b)r (ecomes)g(the)h(Euler)f(equation)g(\(1.7\).)45 b(Application)31 b(of)515 2527 y(the)d(pro)5 b(jector)26 b(\005)i(transforms)e(it)i(to)f (the)h(equation)f(2.1)g(with)h Fv(\027)g Fw(=)23 b(0:)1519 2685 y(_)-38 b Fv(u)p Fw(\()p Fv(t)p Fw(\))19 b(+)f Fv(B)t Fw(\()p Fv(u)p Fw(\()p Fv(t)p Fw(\))p Fv(;)28 b(u)p Fw(\()p Fv(t)p Fw(\)\))23 b(=)g(0)p Fv(:)639 2843 y Fw(Let)34 b Fs(f)p Fv(e)875 2855 y Fu(s)909 2843 y Fv(;)14 b(s)33 b Fs(2)g Fq(Z)1167 2813 y Fr(2)1167 2864 y(0)1231 2843 y Fw(:=)f Fq(Z)1412 2813 y Fr(2)1443 2843 y Fs(nf)p Fw(0)p Fs(gg)f Fw(b)r(e)j(the)g(Hilb)r(ert)g(basis)e(of)h Fv(H)7 b Fw(,)35 b(formed)e(b)n(y)g(trigono-)515 2943 y(metric)27 b(v)n(ector)g(\014elds:)866 3101 y Fs(f)p Fv(e)947 3113 y Fu(s)981 3101 y Fv(;)h(e)1071 3113 y Fo(\000)p Fu(s)1158 3101 y Fs(g)23 b Fw(=)f Fs(f)p Fv(c)1388 3113 y Fu(s)1423 3101 y Fv(s)1462 3066 y Fo(?)1532 3101 y Fw(cos)13 b Fv(L)1714 3066 y Fo(\000)p Fr(1)1803 3101 y Fv(s)18 b Fs(\001)g Fv(x;)29 b(c)2036 3113 y Fu(s)2071 3101 y Fv(s)2110 3066 y Fo(?)2180 3101 y Fw(sin)14 b Fv(L)2353 3066 y Fo(\000)p Fr(1)2441 3101 y Fv(s)19 b Fs(\001)f Fv(x)p Fs(g)83 b(8)14 b Fv(s)22 b Fs(2)h Fq(Z)2974 3066 y Fr(2)2974 3121 y(0)3005 3101 y Fv(;)515 3309 y Fw(where)34 b Fv(s)801 3278 y Fo(?)892 3309 y Fw(=)991 3191 y Fp(\022)1052 3258 y Fs(\000)p Fv(s)1156 3270 y Fr(2)1096 3358 y Fv(s)1135 3370 y Fr(1)1193 3191 y Fp(\023)1289 3309 y Fw(for)g(an)n(y)g(v)n (ector)1844 3191 y Fp(\022)1905 3258 y Fv(s)1944 3270 y Fr(1)1905 3358 y Fv(s)1944 3370 y Fr(2)1981 3191 y Fp(\023)2077 3309 y Fs(2)h Fq(Z)2229 3278 y Fr(2)2229 3329 y(0)2260 3309 y Fw(,)i(and)d Fv(c)2524 3321 y Fu(s)2594 3309 y Fw(=)2694 3240 y Fs(p)p 2763 3240 42 4 v 69 x Fw(2)13 b Fs(j)p Fv(s)p Fs(j)2903 3278 y Fo(\000)p Fr(1)2993 3309 y Fv(L)3050 3278 y Fo(\000)p Fr(1)3173 3309 y Fw(is)35 b(the)515 3453 y Fv(L)572 3465 y Fr(2)608 3453 y Fw(-normalising)26 b(factor.)36 b(Then)1346 3611 y Fv(Ae)1447 3623 y Fu(s)1505 3611 y Fw(=)23 b Fv(\013)1646 3623 y Fu(s)1682 3611 y Fv(e)1721 3623 y Fu(s)1756 3611 y Fv(;)97 b(\013)1929 3623 y Fu(s)1987 3611 y Fw(=)23 b Fv(L)2132 3577 y Fo(\000)p Fr(2)2221 3611 y Fs(j)p Fv(s)p Fs(j)2306 3577 y Fr(2)2426 3611 y Fs(8)14 b Fv(s:)515 3769 y Fw(F)-7 b(or)37 b(an)n(y)f Fv(r)43 b Fs(\025)c Fw(0)e(w)n(e)g(de\014ne)h(the)f(Sob)r(olev)g(space) g Fv(H)2264 3739 y Fu(r)2338 3769 y Fw(as)g Fv(H)2526 3739 y Fu(r)2602 3769 y Fw(=)i Fs(f)p Fv(u)g Fs(2)h Fv(H)46 b Fs(j)40 b(k)p Fv(u)p Fs(k)3240 3781 y Fu(r)3314 3769 y Fw(=)515 3869 y Fs(j)p Fv(A)600 3839 y Fu(r)r(=)p Fr(2)704 3869 y Fv(u)p Fs(j)24 b Fv(<)f Fs(1g)p Fw(.)39 b(In)28 b(particular,)f Fv(H)1661 3839 y Fr(0)1722 3869 y Fw(=)d Fv(H)35 b Fw(and)28 b Fs(k)18 b(\001)h(k)2221 3881 y Fr(0)2282 3869 y Fw(=)24 b Fs(j)19 b(\001)g(j)p Fw(.)38 b(F)-7 b(or)28 b Fv(r)f(<)d Fw(0)j(w)n(e)h(set)g Fv(H)3240 3839 y Fu(r)3305 3869 y Fw(to)515 3969 y(b)r(e)g(equal)f(to)g(the)h (completion)g(of)f Fv(H)35 b Fw(in)27 b(the)h(norm)f Fs(k)18 b(\001)h(k)2314 3981 y Fu(r)2350 3969 y Fw(,)28 b(de\014ned)g(as)f(ab)r(o)n(v)n(e.)639 4068 y(Since)k(for)g Fv(u)d Fw(=)1159 4006 y Fp(P)1260 4068 y Fv(u)1308 4080 y Fu(s)1343 4068 y Fv(e)1382 4080 y Fu(s)1448 4068 y Fw(w)n(e)j(ha)n(v)n(e)e Fs(k)p Fv(u)p Fs(k)1900 4038 y Fr(2)1900 4089 y(1)1965 4068 y Fw(=)2058 4006 y Fp(P)2159 4068 y Fv(\013)2212 4080 y Fu(s)2248 4068 y Fs(j)p Fv(u)2319 4080 y Fu(s)2354 4068 y Fs(j)2377 4038 y Fr(2)2445 4068 y Fw(and)i Fv(\013)2663 4080 y Fu(s)2727 4068 y Fs(\025)d Fv(L)2877 4038 y Fo(\000)p Fr(2)2997 4068 y Fw(for)i(eac)n(h)g Fv(s)p Fw(,)515 4168 y(then)1676 4268 y Fs(k)p Fv(u)p Fs(k)1808 4280 y Fr(1)1867 4268 y Fs(\025)23 b Fv(L)2012 4233 y Fo(\000)p Fr(1)2101 4268 y Fs(j)p Fv(u)p Fs(j)p Fv(:)990 b Fw(\(2.2\))515 4402 y(The)27 b(norms)g Fs(k)18 b(\001)h(k)1079 4414 y Fu(j)1141 4402 y Fw(satisfy)27 b(the)h(in)n(terp)r(olation)f(inequalities.)36 b(In)28 b(particular,)1635 4560 y Fs(k)p Fv(u)p Fs(k)1767 4526 y Fr(2)1767 4581 y(1)1826 4560 y Fs(\024)23 b(j)p Fv(u)p Fs(j)18 b(\001)g(k)p Fv(u)p Fs(k)2199 4572 y Fr(2)2235 4560 y Fv(:)950 b Fw(\(2.3\))515 4718 y(The)27 b(random)g(force)g Fv(\021)k Fw(is)c(assumed)g(to)h(b)r(e)1261 4915 y Fv(\021)s Fw(\()p Fv(t)p Fw(\))c(=)1536 4859 y Fv(d)p 1521 4896 74 4 v 1521 4972 a(dt)1604 4915 y(\020)6 b Fw(\()p Fv(t)p Fw(\))p Fv(;)98 b(\020)29 b Fw(=)2028 4836 y Fp(X)2014 5021 y Fu(s)p Fo(2)p Fg(Z)2135 5001 y Ff(2)2135 5039 y(0)2175 4915 y Fv(b)2211 4927 y Fu(s)2246 4915 y Fv(\014)2293 4927 y Fu(s)2329 4915 y Fw(\()p Fv(t)p Fw(\))p Fv(e)2462 4927 y Fu(s)2497 4915 y Fw(\()p Fv(x)p Fw(\))p Fv(;)577 b Fw(\(2.4\))1926 5255 y(5)p eop %%Page: 6 6 6 5 bop 515 523 a Fw(where)28 b Fs(f)p Fv(\014)845 535 y Fu(s)880 523 y Fs(g)g Fw(are)g(indep)r(enden)n(t)i(standard)d(Wiener) i(pro)r(cesses,)f(de\014ned)h(for)f Fv(t)d Fs(2)g Fq(R)p Fw(,)35 b(and)515 623 y Fs(f)p Fv(b)593 635 y Fu(s)650 623 y Fs(\025)23 b Fw(0)p Fs(g)k Fw(are)f(constan)n(ts.)36 b(F)-7 b(or)27 b Fv(j)h Fw(=)23 b(0)p Fv(;)14 b Fw(1)p Fv(;)g(:)g(:)g(:)40 b Fw(w)n(e)27 b(set)1541 800 y Fv(B)1604 812 y Fu(j)1662 800 y Fw(=)1763 721 y Fp(X)1749 899 y Fu(s)p Fo(2)p Fg(Z)1870 907 y Ff(0)1910 800 y Fs(j)p Fv(s)p Fs(j)1995 765 y Fr(2)p Fu(j)2063 800 y Fv(b)2099 765 y Fr(2)2099 820 y Fu(s)2159 800 y Fs(\024)c(1)p Fv(:)515 1042 y Fw(Ev)n(erywhere)i(b)r(elo)n(w)i(w)n(e)h(assume)e(that)1720 1203 y Fv(B)1783 1215 y Fr(0)1820 1203 y Fv(;)14 b(B)1920 1215 y Fr(1)1980 1203 y Fv(<)23 b Fs(1)p Fv(:)1034 b Fw(\(2.5\))639 1363 y(In)n(tegrating)28 b(\(2.1\))f(from)g Fv(\034)38 b Fw(to)27 b Fv(T)34 b(>)23 b(\034)37 b Fw(w)n(e)27 b(get)870 1585 y Fv(u)p Fw(\()p Fv(T)12 b Fw(\))17 b(+)1143 1472 y Fp(Z)1226 1493 y Fu(T)1189 1661 y(\034)1278 1585 y Fw(\()p Fv(\027)5 b(Au)19 b Fw(+)f Fv(B)t Fw(\()p Fv(u;)c(u)p Fw(\)\))g Fv(dt)23 b Fw(=)g Fv(u)p Fw(\()p Fv(\034)9 b Fw(\))19 b(+)2321 1521 y Fs(p)p 2390 1521 47 4 v 64 x Fv(\027)h(\020)6 b Fw(\()p Fv(T)12 b Fw(\))18 b Fs(\000)2719 1521 y(p)p 2788 1521 V 64 x Fv(\027)h(\020)6 b Fw(\()p Fv(\034)j Fw(\))p Fv(:)186 b Fw(\(2.6\))639 1792 y(An)37 b(a.s.)60 b(con)n(tin)n(uous)35 b(random)g(pro)r(cess)g Fv(u)p Fw(\()p Fv(t)p Fw(\))i Fs(2)g Fv(H)7 b Fw(,)37 b Fv(t)g Fs(\025)f Fv(\034)9 b Fw(,)39 b(suc)n(h)d(that)g(its)g(norm) 515 1892 y Fs(k)p Fv(u)p Fw(\()p Fv(t)p Fw(\))p Fs(k)741 1904 y Fr(1)806 1892 y Fw(is)28 b(square-in)n(tegrable)d(on)j(ev)n(ery) g(\014nite)h(time-in)n(terv)-5 b(al)27 b(is)i(called)f(a)g FB(solution)h Fw(for)515 1991 y(\(2.1\))o(,)k(if)e(the)h(relation)e (\(2.6\))h(holds)g(a.s.)48 b(for)30 b(eac)n(h)h Fv(T)40 b(>)29 b(\034)9 b Fw(.)49 b(Here)30 b(b)r(oth)i(parts)f(of)37 b(\(2.6\))515 2091 y(are)28 b(understo)r(o)r(d)g(as)h(elemen)n(ts)f(of) h Fv(H)1706 2061 y Fo(\000)p Fr(1)1795 2091 y Fw(.)41 b(It)30 b(is)f(kno)n(wn)f(\(see)h(e.g.)40 b([VF88]\))29 b(that)g(for)g(an)n(y)515 2190 y Fv(u)563 2202 y Fr(0)623 2190 y Fs(2)23 b Fv(H)34 b Fw(the)28 b(equation)e(\(2.1\))h(has)g(a)g (unique)g(solution)g Fv(u)p Fw(\()p Fv(t)p Fw(\))p Fv(;)g(t)c Fs(\025)g Fv(\034)9 b Fw(,)28 b(equal)f Fv(u)2979 2202 y Fr(0)3043 2190 y Fw(for)g Fv(t)c Fw(=)f Fv(\034)9 b Fw(.)639 2290 y(A)28 b(random)e(pro)r(cess)31 b(~)-48 b Fv(u)p Fw(\()p Fv(t)p Fw(\))24 b Fs(2)f Fv(H)1634 2302 y Fr(0)1671 2290 y Fv(;)28 b(t)23 b Fs(\025)g Fv(\034)9 b Fw(,)28 b(is)e(called)h(a)g FB(we)l(ak)i(solution)f Fw(for)e(\(2.1\))h(if)g(one)515 2390 y(can)20 b(\014nd)h(a)g(pro)r (cess)1172 2368 y(~)1164 2390 y Fv(\020)7 b Fw(\()p Fv(t)p Fw(\))24 b Fs(2)f Fv(H)7 b Fw(,)22 b(distributed)f(as)f Fv(\020)6 b Fw(\()p Fv(t)p Fw(\),)24 b(suc)n(h)c(that)27 b(~)-48 b Fv(u)21 b Fw(is)f(a)h(solution)f(for)g(\(2.1\))515 2489 y(with)28 b Fv(\020)35 b Fw(replaced)28 b(b)n(y)1227 2467 y(~)1220 2489 y Fv(\020)6 b Fw(.)39 b(Its)28 b(distribution)h Fs(D)r Fw(\()5 b(~)-47 b Fv(u)p Fw(\()p Fs(\001)p Fw(\)\))29 b(\(i.e.)39 b(a)28 b(measure)f(in)i Fv(C)6 b Fw(\([)p Fv(\034)f(;)14 b Fs(1)p Fw(\);)39 b Fv(H)7 b Fw(\)\))515 2589 y(also)28 b(is)h(called)g(a)g(w)n(eek)f(solution,)i(cf.)42 b([VF88)o(,)30 b(Chapter)f(X].)h(A)f(solution)g Fv(u)p Fw(\()p Fv(t)p Fw(\))g(\(w)n(eak)g(or)515 2689 y(strong\),)20 b(de\014ned)h(for)f(all)f Fv(t)p Fw(,)j(is)e(called)g(a)g FB(stationary)j(solution)e Fw(if)f Fv(u)g Fw(is)g(a)g(stationary)e(pro) r(cess)515 2788 y(in)29 b Fv(H)7 b Fw(.)41 b(F)-7 b(or)28 b(a)h(random)f(v)-5 b(ariable)28 b Fv(v)g Fs(2)e Fv(H)36 b Fw(with)29 b(a)g(\014nite)g(second)g(momen)n(t,)g(indep)r(enden)n(t) 515 2888 y(of)23 b(the)g(pro)r(cess)g Fv(\021)s Fw(\()p Fv(t)p Fw(\))p Fv(;)28 b(t)23 b Fs(\025)g Fw(0,)g(w)n(e)g(denote)g(b)n (y)g Fv(u)p Fw(\()p Fv(t)p Fw(;)14 b Fv(v)s Fw(\),)25 b Fv(t)e Fs(\025)f Fw(0,)i(a)f(strong)f(solution)h(for)f(\(2.1\),)515 2987 y(equal)31 b Fv(v)j Fw(at)d Fv(t)f Fw(=)f(0)i(\(suc)n(h)g(a)g (solution)g(exists,)h(see)f([VF88)o(]\).)49 b(A)32 b(measure)e Fv(\026)f Fs(2)h(P)7 b Fw(\()p Fv(H)g Fw(\))31 b(is)515 3087 y(called)c(a)g FB(stationary)k(me)l(asur)l(e)c Fw(for)g(\(2.1\))h (if)1316 3247 y Fs(D)r Fw(\()p Fv(v)s Fw(\))c(=)f Fv(\026)g Fs(\))g(D)r Fv(u)p Fw(\()p Fv(t)p Fw(;)14 b Fv(v)s Fw(\))24 b(=)e Fv(\026)83 b Fs(8)14 b Fv(t)22 b Fs(\025)h Fw(0)p Fv(:)515 3408 y Fw(If)40 b Fv(u)p Fw(\()p Fs(\001)p Fw(\))g(is)f(a)g (stationary)f(solution,)k(then)e(the)g(distribution)g Fs(D)r Fv(u)p Fw(\()p Fv(t)p Fw(\),)j Fv(t)g Fs(2)h Fq(R)p Fw(,)k(is)40 b(a)f(\()p Fv(t)p Fw(-)515 3507 y(indep)r(enden)n(t\))24 b(stationary)d(measure)h(in)i Fv(H)7 b Fw(.)35 b(Con)n(v)n(ersely)-7 b(,)22 b(if)h Fv(\026)g Fw(is)g(a)g(stationary)e(measure,)515 3607 y(then)30 b(there)f(exists)g(a)g(\(w)n(eak\))g(stationary)f (solution)h Fv(u)g Fw(suc)n(h)h(that)f Fs(D)r Fv(u)p Fw(\()p Fv(t)p Fw(\))e Fs(\021)f Fv(\026)k Fw(\(e.g.,)f(see)515 3707 y(in)f([KS02)n(]\).)639 3806 y(It)42 b(is)f(kno)n(wn)f(that)h(ev)n (ery)f(equation)h(\(2.1\))f(\(satisfying)i(\(2.5\)\))f(has)g(a)g (stationary)515 3906 y(measure)19 b Fv(\026)p Fw(,)i(see)f([VF88].)34 b(Its)20 b(uniqueness)g(is)g(pro)n(v)n(en)f(under)h(the)g(additional)g (assumption)515 4005 y(that)1600 4105 y Fv(b)1636 4117 y Fu(s)1694 4105 y Fs(6)p Fw(=)j(0)83 b Fs(8)14 b(j)p Fv(s)p Fs(j)22 b(\024)g Fv(N)2229 4117 y Fu(\027)2270 4105 y Fv(;)915 b Fw(\(2.7\))515 4241 y(where)33 b Fv(N)828 4253 y Fu(\027)902 4241 y Fw(is)h(a)f(su\016cien)n(tly)g(large)f(n)n (um)n(b)r(er.)54 b(See)34 b([KS00)n(,)g(KS02)o(])f(\(also)g(see)g ([EMS01)o(,)515 4341 y(BKL02)n(])28 b(and)f(the)h(review)f([Kuk02)o (]\).)37 b(Moreo)n(v)n(er,)25 b(under)i(the)h(assumption)f(\(2.7\),)892 4544 y Fs(j)p Fq(E)9 b Fv(f)g Fw(\()p Fv(u)p Fw(\()p Fv(t)p Fw(;)42 b Fv(v)s Fw(\)\))20 b Fs(\000)1432 4431 y Fp(Z)1529 4544 y Fv(f)9 b Fw(\()p Fv(u)p Fw(\))14 b Fv(\026du)p Fs(j)22 b(\024)h Fv(C)2038 4556 y Fu(\027)2079 4544 y Fw(\(1)18 b(+)g Fq(E)23 b Fs(j)p Fv(v)s Fs(j)2407 4509 y Fr(2)2450 4544 y Fw(\))p Fv(e)2521 4509 y Fo(\000)p Fu(\033)2611 4517 y Ft(\027)2648 4509 y Fu(t)2677 4544 y Fv(;)37 b(\033)2784 4556 y Fu(\027)2849 4544 y Fv(>)23 b Fw(0)p Fv(;)206 b Fw(\(2.8\))515 4746 y(for)26 b(an)n(y)f(b)r(ounded) i(Lipsc)n(hitz)g(functional)f Fv(f)35 b Fw(on)27 b Fv(H)33 b Fw(suc)n(h)26 b(that)h Fs(j)p Fv(f)9 b Fs(j)22 b(\024)h Fw(1)j(and)g(Lip)q(\()p Fv(f)9 b Fw(\))23 b Fs(\024)f Fw(1.)515 4846 y(Let)27 b(us)h(in)n(tro)r(duce)f(in)h Fs(P)7 b Fw(\()p Fv(H)g Fw(\))28 b(the)g(distance)1385 5006 y(dist\()p Fv(\026)1601 5018 y Fr(1)1639 5006 y Fv(;)g(\026)1740 5018 y Fr(2)1777 5006 y Fw(\))23 b(=)g(sup\()p Fv(\026)2127 5018 y Fr(1)2183 5006 y Fs(\000)18 b Fv(\026)2316 5018 y Fr(2)2353 5006 y Fv(;)28 b(f)9 b Fw(\))p Fv(;)1926 5255 y Fw(6)p eop %%Page: 7 7 7 6 bop 515 523 a Fw(where)24 b(the)g(suprem)n(um)g(is)h(tak)n(en)e(o)n (v)n(er)g(all)h Fv(f)33 b Fw(as)24 b(ab)r(o)n(v)n(e.)34 b(It)25 b(is)f(kno)n(wn)g(that)h(\()p Fs(P)7 b Fw(\()p Fv(H)g Fw(\))p Fv(;)27 b Fw(dist)q(\))515 623 y(is)c(a)f(complete)h (metric)g(space,)g(and)g(that)g(con)n(v)n(ergence)e(with)j(resp)r(ect)e (to)h(this)g(distance)g(is)515 722 y(equiv)-5 b(alen)n(t)26 b(to)g(the)h Fs(\003)p Fw(-w)n(eak)d(con)n(v)n(ergence)g(of)i (measures,)g(see)g([Dud89].)36 b(W)-7 b(e)27 b(can)f(rewrite)515 822 y(\(2.8\))h(as)1225 922 y(dist\()p Fs(D)r Fv(u)p Fw(\()p Fv(t)p Fw(;)38 b Fv(v)s Fw(\))p Fv(;)28 b(\026)p Fw(\))23 b Fs(\024)g Fv(C)2006 934 y Fu(\027)2047 922 y Fw(\(1)c(+)f Fq(E)k Fs(j)p Fv(v)s Fs(j)2375 887 y Fr(2)2419 922 y Fw(\))p Fv(e)2490 887 y Fo(\000)p Fu(\033)2580 895 y Ft(\027)2616 887 y Fu(t)2646 922 y Fv(:)539 b Fw(\(2.9\))639 1059 y(W)-7 b(e)28 b(note)g(that)g(if)1733 1159 y Fv(b)1769 1171 y Fu(s)1827 1159 y Fs(6)p Fw(=)23 b(0)82 b Fs(8)14 b Fv(s;)1005 b Fw(\(2.10\))515 1296 y(then)27 b(the)h(assumption)e (\(2.7\))h(holds)g(for)f(eac)n(h)g Fv(\027)j(>)22 b Fw(0,)27 b(so)g(for)f(an)n(y)g Fv(\027)j(>)22 b Fw(0)27 b(equation)f(\(2.1\))515 1396 y(has)h(a)g(unique)h(stationary)e(measure)g Fv(\026)1770 1408 y Fu(\027)1812 1396 y Fw(.)515 1545 y Fx(Lemma)19 b(2.1.)33 b FB(If)39 b Fw(\(2.10\))21 b FB(holds,)26 b(then)21 b(the)h(me)l(asur)l(e)g Fv(\026)2258 1557 y Fu(\027)2322 1545 y Fs(2)i(P)7 b Fw(\()p Fv(H)g Fw(\))21 b FB(c)l(ontinuously)h(dep)l(ends)515 1645 y(on)30 b Fv(\027)e(>)23 b Fw(0)p FB(.)515 1795 y Fx(Pro)s(of.)43 b Fw(Due)30 b(to)g(\(2.9\))f(it)i(is)e(su\016cien)n(t)h(to)g(c)n(hec)n (k)f(that)h Fs(D)r Fv(u)2445 1807 y Fu(\027)2486 1795 y Fw(\()p Fv(t)p Fw(\))d Fs(2)g(P)7 b Fw(\()p Fv(H)g Fw(\))30 b(con)n(tin)n(uously)515 1895 y(dep)r(ends)21 b(on)f Fv(\027)26 b Fw(for)21 b(eac)n(h)f Fv(t)j Fs(\025)f Fw(0,)g(where)e Fv(u)1813 1907 y Fu(\027)1854 1895 y Fw(\()p Fs(\001)p Fw(\))i(is)e(a)h(solution)f(of)27 b(\(2.1\),)22 b(v)-5 b(anishing)20 b(at)h Fv(t)i Fw(=)f(0.)515 1994 y(T)-7 b(o)24 b(v)n(erify)f(this)i(prop)r(ert)n(y)e(w)n(e)h(mak)n(e)f (in)i(\(2.1\))o(,)h(\(2.4\))e(the)g(substitution)h Fv(u)e Fw(=)f Fv(\027)5 b(U)k Fw(\()p Fv(\034)g(\027)c Fw(\))26 b(and)515 2094 y(get)h(for)g Fv(U)37 b Fw(the)28 b(equation)1037 2295 y Fv(U)1094 2307 y Fu(\034)1154 2295 y Fw(+)18 b Fv(AU)27 b Fw(+)18 b Fv(B)t Fw(\()p Fv(U;)28 b(U)9 b Fw(\))23 b(=)g Fv(\027)1928 2261 y Fo(\000)p Fr(5)p Fu(=)p Fr(2)2117 2239 y Fv(@)p 2094 2276 95 4 v 2094 2352 a(@)5 b(\034)2199 2295 y(\020)h Fw(\()p Fv(\027)f(\034)k Fw(\))25 b(=:)2564 2239 y Fv(@)p 2541 2276 V 2541 2352 a(@)5 b(\034)2646 2295 y(\020)2682 2307 y Fu(\027)2723 2295 y Fw(\()p Fv(\034)k Fw(\))p Fv(:)515 2485 y Fw(T)-7 b(o)30 b(pro)n(v)n(e)e(the)j(con)n(tin) n(uit)n(y)f(w)n(e)g(ma)n(y)g(assume)f(that)i Fv(\027)i(>)27 b(\016)k(>)c Fw(0.)45 b(Then)30 b(0)d Fs(\024)g Fv(\034)38 b Fs(\024)27 b Fv(t=\027)32 b(<)515 2585 y(T)i Fw(=)22 b Fv(t=\033)s Fw(.)36 b(It)25 b(is)g(easy)e(to)i(c)n(hec)n(k)f(that)g (a)h(solution)f Fv(U)32 b Fs(2)23 b Fv(C)6 b Fw(\([0)p Fv(;)28 b(T)12 b Fw(];)36 b Fv(H)7 b Fw(\))23 b(=)g Fs(H)i Fw(con)n(tin)n(uously)515 2685 y(dep)r(ends)f(on)f Fv(\020)979 2697 y Fu(\027)1044 2685 y Fs(2)g Fv(C)6 b Fw(\([0)p Fv(;)28 b(T)12 b Fw(];)36 b Fv(H)1554 2655 y Fr(1)1591 2685 y Fw(\))23 b(=)g Fs(H)1805 2655 y Fr(1)1866 2685 y Fw(\(e.g.,)h(see)f(Lemma)h(4.2)e(in)i([KS03)o(]\).)36 b(Since)24 b(a.s.)515 2784 y Fv(\020)39 b Fs(2)33 b Fv(C)6 b Fw(\([0)p Fv(;)28 b Fs(1)p Fw(\))p Fv(;)g(H)1133 2754 y Fr(1)1170 2784 y Fw(\),)35 b(then)f(a.s.)54 b Fv(\020)1666 2796 y Fu(\027)1740 2784 y Fs(2)33 b(H)1899 2754 y Fr(1)1970 2784 y Fw(con)n(tin)n(uously)f(dep)r(ends)i(on)f Fv(\027)38 b Fs(2)33 b Fw([)p Fv(\016)o(;)27 b Fw(1],)35 b(as)515 2884 y(w)n(ell)24 b(as)g Fv(U)31 b Fs(2)24 b(H)q Fw(.)36 b(As)24 b Fv(u)1243 2896 y Fu(\027)1284 2884 y Fw(\()p Fv(t)p Fw(\))f(=)g Fv(\027)5 b(U)k Fw(\()p Fv(t\027)1709 2854 y Fo(\000)p Fr(1)1799 2884 y Fw(\),)25 b(then)g(a.s.)35 b Fv(u)2269 2896 y Fu(\027)2310 2884 y Fw(\()p Fv(t)p Fw(\))24 b Fs(2)f Fv(H)31 b Fw(con)n(tin)n(uously)24 b(dep)r(ends)515 2984 y(on)j Fv(\027)5 b Fw(.)37 b(Hence,)28 b Fs(D)r Fv(u)1120 2996 y Fu(\027)1157 2979 y Fe(0)1184 2984 y Fw(\()p Fv(t)p Fw(\))23 b Fs(!)g(D)r Fv(u)1521 2996 y Fu(\027)1563 2984 y Fw(\()p Fv(t)p Fw(\))28 b(as)f Fv(\027)1833 2953 y Fo(0)1879 2984 y Fs(!)d Fv(\027)5 b Fw(,)28 b(and)f(the)h(lemma)g(is)f(pro)n(v)n(en.)p 3318 2984 4 57 v 3322 2931 50 4 v 3322 2984 V 3372 2984 4 57 v 639 3125 a(Let)g Fv(u)p Fw(\()p Fv(t)p Fw(\))f(b)r(e)g(a)g (stationary)f(solution)g(of)33 b(\(2.1\))26 b(\(w)n(eak)f(or)h (strong\).)35 b(Due)27 b(to)f(\(2.5\))g(the)515 3225 y(Ito)e(form)n(ula)f(applies)g(to)h(the)h(functional)f Fs(j)p Fv(u)p Fw(\()p Fv(t)p Fw(\))p Fs(j)2029 3194 y Fr(2)2066 3225 y Fw(.)36 b(Since)25 b(\()p Fv(B)t Fw(\()p Fv(u;)14 b(u)p Fw(\))p Fv(;)27 b(u)p Fw(\))c(=)g(0,)h(then)h(taking)515 3324 y(exp)r(ectation)i(of)h(the)g(form)n(ula)e(w)n(e)i(get)f(that)785 3540 y Fq(E)c Fs(j)p Fv(u)p Fw(\()p Fv(t)p Fw(\))p Fs(j)1037 3505 y Fr(2)1099 3540 y Fs(\000)18 b Fq(E)k Fs(j)p Fv(u)p Fw(\(0\))p Fs(j)1445 3505 y Fr(2)1511 3540 y Fw(=)g Fs(\000)p Fw(2)p Fv(\027)1765 3427 y Fp(Z)1848 3447 y Fu(t)1811 3615 y Fr(0)1891 3540 y Fq(E)g Fw(\()p Fv(Au;)34 b(u)p Fw(\))14 b Fv(d\034)28 b Fw(+)18 b Fv(\027)2497 3461 y Fp(X)2541 3635 y Fu(s)2631 3427 y Fp(Z)2714 3447 y Fu(t)2677 3615 y Fr(0)2757 3540 y Fv(b)2793 3505 y Fr(2)2793 3560 y Fu(s)2830 3540 y Fs(j)p Fv(e)2892 3552 y Fu(s)2927 3540 y Fs(j)2950 3505 y Fr(2)3001 3540 y Fv(d\034)5 b(;)515 3777 y Fw(\(see)35 b([VF88,)g(EMS01)o(,)h(Shi02)o(]\).)61 b(As)35 b Fv(u)h Fw(is)f(a)g(stationary)f(pro)r(cess,)i(then)g(this)g (relation)515 3877 y(implies)28 b(that)1540 4011 y Fq(E)9 b Fs(k)p Fv(u)p Fw(\()p Fv(\034)g Fw(\))p Fs(k)1831 3976 y Fr(2)1831 4031 y(1)1897 4011 y Fw(=)1994 3955 y(1)p 1994 3992 42 4 v 1994 4068 a(2)2046 4011 y Fv(B)2109 4023 y Fr(0)2229 4011 y Fs(8)14 b Fv(\034)5 b(:)813 b Fw(\(2.11\))515 4176 y(Since)28 b Fv(u)22 b Fw(=)h Fv(u)p Fw(\()p Fv(x)p Fw(\))28 b(is)f(a)h(div)n(ergence-free)d(v)n(ector)h (\014eld,)i(then)1388 4381 y Fs(k)p Fv(u)p Fs(k)1520 4347 y Fr(2)1520 4401 y(1)1579 4381 y Fw(=)23 b(\(2)p Fv(\031)s(L)p Fw(\))1880 4347 y Fo(\000)p Fr(2)1982 4268 y Fp(Z)2029 4456 y Fu(T)2077 4440 y Ff(2)2127 4381 y Fs(j)14 b Fw(rot)f Fv(u)p Fs(j)2355 4347 y Fr(2)2392 4381 y Fv(dx:)515 4590 y Fw(So)27 b(the)h(l.h.s.)37 b(of)d(\(2.11\))27 b(is)g(the)h(a)n(v)n(eraged)d(enstroph)n(y)i(of)g Fv(u)g Fw(\(p)r(er)h(unit)h(area\).)639 4690 y(Similar,)k(applying)e(the)i (Ito)e(form)n(ula)g(to)h(the)g(enstroph)n(y)f(functional)h Fv(f)9 b Fw(\()p Fv(u)p Fw(\))30 b(=)g Fs(k)p Fv(u)p Fs(k)3343 4659 y Fr(2)3343 4710 y(1)515 4789 y Fw(and)d(using)g(that)h (\()p Fv(B)t Fw(\()p Fv(u;)14 b(u)p Fw(\))p Fv(;)28 b(Au)p Fw(\))23 b(=)g(0,)k(w)n(e)g(obtain)1468 4991 y Fq(E)8 b Fs(k)p Fv(u)p Fw(\()p Fv(\034)h Fw(\))p Fs(k)1758 4957 y Fr(2)1758 5011 y(2)1824 4991 y Fw(=)1922 4935 y(1)p 1922 4972 V 1922 5048 a(2)1973 4991 y Fv(L)2030 4957 y Fo(\000)p Fr(2)2119 4991 y Fv(B)2182 5003 y Fr(1)2302 4991 y Fs(8)14 b Fv(\034)5 b(:)740 b Fw(\(2.12\))1926 5255 y(7)p eop %%Page: 8 8 8 7 bop 639 523 a Fw(Tw)n(o)22 b(useful)i(inequalities)e(b)r(elo)n(w)h (follo)n(w)f(b)n(y)h(applying)f(the)h(Ito)g(form)n(ula)f(to)h(the)g (func-)515 623 y(tionals)k(exp\()p Fv(\015)5 b Fs(j)p Fv(u)p Fs(j)1084 593 y Fr(2)1120 623 y Fw(\))28 b(and)g(exp\()p Fv(\015)5 b Fs(k)p Fv(u)p Fs(k)1681 593 y Fr(2)1681 643 y(1)1716 623 y Fw(\),)28 b(where)1128 780 y Fv(\015)g Fw(=)22 b(\(2)p Fv(L)1417 745 y Fr(2)1454 780 y Fv(b)1490 745 y Fr(2)1490 800 y Fu(max)1626 780 y Fw(\))1658 745 y Fo(\000)p Fr(1)1747 780 y Fv(;)97 b(b)1903 792 y Fu(max)2063 780 y Fw(=)22 b(max)p Fs(f)p Fv(b)2383 792 y Fu(s)2418 780 y Fv(;)27 b(s)c Fs(2)h Fq(Z)2670 745 y Fr(2)2670 800 y(0)2701 780 y Fs(g)p Fv(:)515 937 y Fw(Namely)-7 b(,)1168 1082 y Fq(E)22 b Fw(exp)6 b(\()p Fv(\015)f Fs(j)p Fv(u)p Fw(\()p Fv(\034)k Fw(\))p Fs(j)1647 1047 y Fr(2)1685 1082 y Fw(\))23 b Fs(\024)g Fd(B)1901 1094 y Fr(0)1961 1082 y Fw(=)g Fv(B)2112 1094 y Fr(0)2163 1082 y Fw(exp)2304 965 y Fp(\022)2375 1025 y Fv(B)2438 1037 y Fr(0)2494 1025 y Fw(+)18 b(1)p 2375 1062 244 4 v 2390 1139 a(2)p Fv(b)2468 1115 y Fr(2)2468 1159 y Fu(max)2628 965 y Fp(\023)2703 1082 y Fv(;)1004 1327 y Fq(E)k Fw(exp)6 b(\()p Fv(\015)f Fs(k)p Fv(u)p Fw(\()p Fv(\034)k Fw(\))p Fs(k)1521 1293 y Fr(2)1521 1347 y(1)1558 1327 y Fw(\))23 b Fs(\024)g Fd(B)1774 1339 y Fr(1)1834 1327 y Fw(=)g Fv(L)1979 1293 y Fo(\000)p Fr(2)2068 1327 y Fv(B)2131 1339 y Fr(1)2182 1327 y Fw(exp)2322 1210 y Fp(\022)2393 1271 y Fv(L)2450 1241 y Fo(\000)p Fr(2)2539 1271 y Fv(B)2602 1283 y Fr(1)2658 1271 y Fw(+)18 b(1)p 2393 1308 389 4 v 2481 1384 a(2)p Fv(b)2559 1360 y Fr(2)2559 1404 y Fu(max)2792 1210 y Fp(\023)2867 1327 y Fv(:)277 b Fw(\(2.13\))515 1511 y(See)27 b([VF88,)h(p.)f(395])g(and)g([Shi02].)37 b(Due)28 b(to)f(\(2.13\))o(,) 1518 1668 y Fq(E)8 b Fs(k)p Fv(u)p Fw(\()p Fv(\034)h Fw(\))p Fs(k)1808 1628 y Fu(p)1808 1690 y Fr(1)1875 1668 y Fs(\024)23 b Fv(C)2022 1680 y Fu(p)2061 1668 y Fv(\015)2109 1634 y Fo(\000)p Fu(p=)p Fr(2)2266 1668 y Fd(B)2339 1680 y Fr(1)3167 1668 y Fw(\(2.14\))515 1825 y(for)k(an)n(y)g Fv(p)22 b Fs(\025)h Fw(1.)639 1925 y(Equalities)38 b(\(2.11\))h(and)f (\(2.12\))h(imply)g(lo)n(w)n(er)e(and)i(upp)r(er)g(b)r(ounds)g(for)g Fq(E)22 b Fs(j)p Fv(u)p Fw(\()p Fv(\034)9 b Fw(\))q Fs(j)3313 1894 y Fr(2)3356 1925 y Fw(.)515 2024 y(Indeed,)28 b(taking)f(the)h (exp)r(ectation)f(of)34 b(\(2.3\))27 b(w)n(e)h(\014nd)g(that)1222 2181 y Fq(E)9 b Fs(k)p Fv(u)p Fw(\()p Fv(\034)g Fw(\))p Fs(k)1513 2147 y Fr(2)1513 2202 y(1)1579 2181 y Fs(\024)23 b Fw(\()p Fq(E)f Fs(j)p Fv(u)p Fw(\()p Fv(\034)10 b Fw(\))p Fs(j)1966 2147 y Fr(2)2009 2181 y Fw(\))2041 2147 y Fr(1)p Fu(=)p Fr(2)2146 2181 y Fw(\()p Fq(E)e Fs(k)p Fv(u)p Fw(\()p Fv(\034)i Fw(\))p Fs(k)2469 2147 y Fr(2)2469 2202 y(2)2512 2181 y Fw(\))2544 2147 y Fr(1)p Fu(=)p Fr(2)2648 2181 y Fv(:)515 2338 y Fw(No)n(w)36 b(\(2.11\))o(,)i (\(2.12\))d(imply)i(a)e(lo)n(w)n(er)f(b)r(ound)i(for)g Fq(E)22 b Fs(j)p Fv(u)p Fw(\()p Fv(\034)9 b Fw(\))q Fs(j)2445 2308 y Fr(2)2488 2338 y Fw(.)62 b(Com)n(bining)35 b(it)h(with)g(an)515 2438 y(upp)r(er)28 b(b)r(ound,)g(whic)n(h)f(follo)n(ws)g(from)g (\(2.2\))g(and)h(\(2.11\))o(,)g(w)n(e)f(get)1433 2592 y(1)p 1433 2629 42 4 v 1433 2705 a(2)1494 2592 y Fv(L)1551 2562 y Fr(2)1588 2592 y Fv(B)1655 2562 y Fr(2)1651 2612 y(0)p 1494 2629 199 4 v 1543 2705 a Fv(B)1606 2717 y Fr(1)1725 2648 y Fs(\024)c Fq(E)f Fs(j)p Fv(u)p Fw(\()p Fv(\034)10 b Fw(\))p Fs(j)2080 2614 y Fr(2)2146 2648 y Fs(\024)2244 2592 y Fv(L)2301 2562 y Fr(2)2338 2592 y Fv(B)2401 2604 y Fr(0)p 2244 2629 194 4 v 2320 2705 a Fw(2)2448 2648 y Fv(:)696 b Fw(\(2.15\))515 2929 y Fy(3)134 b(The)45 b(results)515 3128 y Fc(3.1)112 b(Tigh)m(tness)515 3281 y Fw(The)25 b(equalit)n(y)h(\(2.12\))e(and)h(the)h(Prokhoro)n(v)21 b(theorem)k(immediately)h(imply)f(that)g(the)h(set)515 3381 y Fs(f)p Fv(\026)607 3393 y Fu(\027)648 3381 y Fv(;)14 b(\027)38 b Fs(2)c Fw(\(0)p Fv(;)14 b Fw(1])p Fs(g)33 b Fw(of)h(stationary)f(measures)f(for)i(equation)f(\(2.1\))h(is)g(tigh) n(t)g(in)g Fv(H)3129 3351 y Fr(2)p Fo(\000)p Fu(\017)3280 3381 y Fw(for)515 3480 y(an)n(y)c Fv(\017)f(>)f Fw(0.)47 b(Join)n(tly)31 b(with)g(Lemma)g(3.2)f(it)i(means)e(that)i(if)f Fv(b)2497 3492 y Fu(s)2561 3480 y Fs(6)p Fw(=)e(0)i(for)f(all)h Fv(s)p Fw(,)h(then)g(the)515 3580 y(measures)26 b Fv(\026)921 3592 y Fu(\027)985 3580 y Fs(2)e(P)7 b Fw(\()p Fv(H)1237 3550 y Fr(2)p Fo(\000)p Fu(\017)1353 3580 y Fw(\))28 b(con)n(tin)n(uously)f(dep)r(end)h(on)f Fv(\027)i Fs(2)23 b Fw(\(0)p Fv(;)14 b Fw(1].)639 3680 y(What)29 b(is)e(m)n(uc)n(h)h (more)f(imp)r(ortan)n(t,)h(distributions)g Fv(m)2361 3692 y Fu(\027)2426 3680 y Fw(=)23 b Fs(D)r Fw(\()p Fv(u)2660 3692 y Fu(\027)2701 3680 y Fw(\()p Fs(\001)p Fw(\)\))i Fs(2)f(P)7 b Fw(\()p Fv(C)f Fw(\()p Fq(R)p Fw(;)44 b Fv(H)7 b Fw(\)\))515 3779 y(of)27 b(the)g(w)n(eak)f(stationary)f (solutions)h Fv(u)1746 3791 y Fu(\027)1814 3779 y Fw(for)g(\(2.1\))h (also)f(are)f(tigh)n(t.)37 b(Belo)n(w)26 b(w)n(e)g(pro)n(v)n(e)g(the) 515 3879 y(tigh)n(tness)h(for)g(the)h(most)f(imp)r(ortan)n(t)h(case)e Fv(\027)j Fs(!)23 b Fw(0.)639 3979 y(Let)28 b(us)g(\014x)f(an)n(y)g Fv(")c Fs(2)g Fw(\(0)p Fv(;)28 b Fw(1\))f(and)h(de\014ne)f(the)h(follo) n(wing)f(spaces)g(of)g(tra)5 b(jectories:)935 4136 y Fv(Z)88 b Fw(=)83 b Fv(L)1285 4148 y Fr(2)11 b(lo)r(c)1416 4136 y Fw(\()p Fq(R)p Fv(;)34 b(H)1635 4101 y Fr(2)p Fo(\000)p Fu(")1755 4136 y Fw(\))19 b Fs(\\)g Fv(C)6 b Fw(\()p Fq(R)p Fv(;)34 b(H)2164 4101 y Fo(\000)p Fr(1)p Fo(\000)p Fu(")2336 4136 y Fw(\))p Fv(;)895 4260 y(Z)952 4272 y Fu(n)1080 4260 y Fw(=)83 b Fv(L)1285 4272 y Fr(2)1322 4260 y Fw(\([)p Fs(\000)p Fv(n;)27 b(n)p Fw(])p Fv(;)h(H)1742 4226 y Fr(2)p Fo(\000)p Fu(")1862 4260 y Fw(\))19 b Fs(\\)g Fv(C)6 b Fw(\([)p Fs(\000)p Fv(n;)27 b(n)p Fw(])p Fv(;)h(H)2472 4226 y Fo(\000)p Fr(1)p Fo(\000)p Fu(")2644 4260 y Fw(\))p Fv(;)65 b(n)23 b Fs(2)g Fq(N)t Fv(:)515 4417 y Fw(The)k(norms)g(in)h (the)g(spaces)f Fv(Z)1487 4429 y Fu(n)1560 4417 y Fw(de\014ne)h(a)f (coun)n(table)g(system)g(of)h(semi-norms)e([)p Fs(\001)p Fw(])3123 4429 y Fu(n)3196 4417 y Fw(in)i Fv(Z)6 b Fw(,)515 4517 y(and)27 b(de\014ne)h(there)f(the)h(distance)1350 4701 y(dist)q(\()p Fv(u;)f(v)s Fw(\))d(=)1801 4622 y Fp(X)1935 4701 y Fw(2)1977 4667 y Fo(\000)p Fu(n)2155 4645 y Fw([)p Fv(u)18 b Fs(\000)g Fv(v)s Fw(])2393 4657 y Fu(n)p 2084 4682 427 4 v 2084 4758 a Fw([)p Fv(u)g Fs(\000)g Fv(v)s Fw(])2322 4770 y Fu(n)2386 4758 y Fw(+)g(1)2520 4701 y Fv(;)515 4907 y Fw(cf.)68 b([VF88)o(,)41 b(p.)d(340].)66 b(All)38 b(the)g(spaces)f Fv(Z)1920 4919 y Fu(n)2003 4907 y Fw(and)h Fv(Z)43 b Fw(are)37 b(P)n(olish)g(\(i.e.,)j(complete)e (and)515 5006 y(separable\).)1926 5255 y(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fx(Theorem)22 b(3.1.)33 b FB(A)n(ny)23 b(se)l(quenc)l(e)f Fs(f)p Fv(m)1714 535 y Fr(~)-37 b Fu(\027)1743 543 y Ft(j)1778 523 y Fs(g)p FB(,)29 b Fw(~)-47 b Fv(\027)1910 535 y Fu(j)1968 523 y Fs(!)23 b Fw(0)p FB(,)i(c)l(ontains)e(a)g(c)l(onver)l(ging)h(subse)l(quenc)l(e)1617 706 y Fv(m)1690 718 y Fu(\027)1723 726 y Ft(j)1781 706 y Fs(!)g Fv(m)e Fs(2)i(P)7 b Fw(\()p Fv(Z)f Fw(\))p Fv(:)931 b Fw(\(3.1\))639 888 y(The)28 b(theorem)f(is)h(pro)n(v)n(en)e(in)h (section)h(4.1.)639 1038 y(The)j(measures)f Fv(m)1246 1050 y Fu(\027)1279 1058 y Ft(j)1345 1038 y Fw(are)g(stationary)g (since)h(they)g(are)f(distributions)h(of)g(stationary)515 1137 y(pro)r(cesses,)h(so)f(the)i(limiting)g(measure)e Fv(m)h Fw(is)g(stationary)f(as)h(w)n(ell.)50 b(By)32 b(the)h(Sk)n(orokho)r(d)515 1237 y(theorem)e(\(see)h([IW89,)h(section)f (I.2],)h(w)n(e)f(can)f(\014nd)i(random)e(pro)r(cesses)g Fv(U)2931 1249 y Fu(\027)2964 1257 y Ft(j)2999 1237 y Fw(\()p Fv(t)p Fw(\),)j Fv(j)h Fs(\025)30 b Fw(1,)515 1337 y(and)d Fv(U)9 b Fw(\()p Fv(t)p Fw(\),)28 b(de\014ned)g(on)f(the)h (same)f(probabilit)n(y)g(space,)g(suc)n(h)g(that)1407 1519 y Fs(D)r Fv(U)1530 1531 y Fu(\027)1563 1539 y Ft(j)1622 1519 y Fw(=)22 b Fv(m)1782 1531 y Fu(\027)1815 1539 y Ft(j)1934 1519 y Fs(8)14 b Fv(j;)96 b Fs(D)r Fv(U)32 b Fw(=)23 b Fv(m;)515 1702 y Fw(and)1515 1802 y Fv(U)1572 1814 y Fu(\027)1605 1822 y Ft(j)1664 1802 y Fs(!)g Fv(U)105 b Fw(in)97 b Fv(Z)103 b Fw(a)p Fv(:)p Fw(s)p Fv(:)829 b Fw(\(3.2\))639 1951 y(Belo)n(w)27 b(w)n(e)g(study)h(prop)r(erties)f (of)g(the)h(limiting)g(random)f(pro)r(cess)f Fv(U)9 b Fw(.)515 2183 y Fc(3.2)112 b(Estimates)36 b(for)h Fb(U)1526 2198 y Fk(\027)1561 2208 y Fj(j)1635 2183 y Fc(and)h Fb(U)515 2337 y Fw(F)-7 b(or)27 b(an)n(y)g Fv(N)k Fs(\025)23 b Fw(1)k(w)n(e)g(consider)g(the)h(pro)5 b(jection)1245 2531 y Fv(P)1298 2543 y Fu(N)1384 2531 y Fw(:)23 b Fv(H)30 b Fs(!)23 b Fv(H)r(;)1840 2452 y Fp(X)1826 2638 y Fu(s)p Fo(2)p Fg(Z)1947 2618 y Ff(2)1947 2656 y(0)1987 2531 y Fv(u)2035 2543 y Fu(s)2070 2531 y Fv(e)2109 2543 y Fu(s)2167 2531 y Fs(7!)2304 2452 y Fp(X)2273 2634 y Fo(j)p Fu(s)p Fo(j\024)p Fu(N)2469 2531 y Fv(u)2517 2543 y Fu(s)2552 2531 y Fv(e)2591 2543 y Fu(s)2626 2531 y Fv(;)515 2813 y Fw(and)k(de\014ne)h(the)g(functionals)1082 2996 y Fv(f)9 b Fw(\()p Fv(u)p Fw(\))22 b(=)h Fv(e)1393 2961 y Fu(\015)t Fo(k)p Fu(u)p Fo(k)1539 2936 y Ff(2)1539 2978 y(1)1575 2996 y Fv(;)97 b(f)1736 3008 y Fu(N)1798 2996 y Fw(\()p Fv(u)p Fw(\))24 b(=)e Fv(e)2060 2961 y Fu(\015)t Fo(k)p Fu(P)2175 2969 y Ft(N)2228 2961 y Fu(u)p Fo(k)2301 2936 y Ff(2)2301 2978 y(1)2334 2961 y Fo(^)p Fu(N)2441 2996 y Fv(;)97 b(N)32 b Fs(\025)23 b Fw(1)p Fv(:)515 3178 y Fw(Clearly)30 b Fv(f)39 b Fw(is)31 b(con)n(tin)n(uous)g(on)f Fv(H)1588 3148 y Fr(1)1625 3178 y Fw(,)i(the)g(functionals)f Fv(f)2294 3190 y Fu(N)2388 3178 y Fw(are)f(b)r(ounded)h(con)n(tin)n (uous)f(on)515 3278 y Fv(H)591 3248 y Fo(\000)p Fr(2)707 3278 y Fw(and)1281 3377 y(0)22 b Fv(<)h(f)1474 3389 y Fu(N)1536 3377 y Fw(\()p Fv(u)p Fw(\))h Fs(\045)f Fv(f)9 b Fw(\()p Fv(u)p Fw(\))22 b Fs(\024)h(1)83 b(8)14 b Fv(u)22 b Fs(2)h Fv(H)2501 3343 y Fo(\000)p Fr(2)2590 3377 y Fv(:)595 b Fw(\(3.3\))639 3527 y(Ev)n(erywhere)26 b(b)r(elo)n(w)h(in)h (this)g(section)1649 3710 y Fv(\027)h Fs(2)23 b(f)p Fv(\027)1880 3722 y Fr(1)1917 3710 y Fv(;)14 b(\027)1995 3722 y Fr(2)2032 3710 y Fv(;)g(:)g(:)g(:)g Fs(g)p Fv(;)515 3892 y Fw(where)27 b Fv(\027)796 3904 y Fr(1)833 3892 y Fv(;)14 b(\027)911 3904 y Fr(2)948 3892 y Fv(;)g(:)g(:)g(:)41 b Fw(is)28 b(the)g(sequence)f(in)h(\(3.1\).)639 3992 y(Due)f(to)g(\(3.2\))o(,)g (for)f(eac)n(h)g Fv(\034)33 b Fs(2)23 b Fq(R)32 b Fw(w)n(e)27 b(ha)n(v)n(e)e Fq(E)8 b Fv(f)2080 4004 y Fu(N)2149 3992 y Fw(\()p Fv(U)2238 4004 y Fu(\027)2271 4012 y Ft(j)2306 3992 y Fw(\()p Fv(\034)h Fw(\)\))25 b Fs(!)e Fq(E)8 b Fv(f)2668 4004 y Fu(N)2737 3992 y Fw(\()p Fv(U)h Fw(\()p Fv(\034)g Fw(\)\))28 b(as)e Fv(j)i Fs(!)23 b(1)p Fw(.)515 4091 y(Hence,)k Fq(E)8 b Fv(f)874 4103 y Fu(N)943 4091 y Fw(\()p Fv(U)h Fw(\()p Fv(\034)g Fw(\)\))25 b Fs(\024)d Fd(B)1367 4103 y Fr(1)1432 4091 y Fw(for)27 b(eac)n(h)f Fv(N)9 b Fw(,)27 b(due)g(to)g(\(2.13\))o(.)37 b(Ev)n(oking)26 b(\(3.3\))h(and)g(the)g(Levi)515 4191 y(theorem)g(w)n(e)g(get)g(that) 1430 4291 y Fq(E)c Fw(exp)5 b(\()p Fv(\015)g Fs(k)p Fv(U)k Fw(\()p Fv(\034)g Fw(\))p Fs(k)1965 4256 y Fr(2)1965 4311 y(1)2003 4291 y Fw(\))23 b Fs(\024)g Fd(B)2219 4303 y Fr(1)2339 4291 y Fs(8)14 b Fv(\034)5 b(:)744 b Fw(\(3.4\))515 4440 y(Similar,)1449 4540 y Fq(E)22 b Fw(exp)6 b(\()p Fv(\015)f Fs(j)p Fv(U)k Fw(\()p Fv(\034)g Fw(\))p Fs(j)1946 4505 y Fr(2)1984 4540 y Fw(\))23 b Fs(\024)g Fd(B)2200 4552 y Fr(0)2321 4540 y Fs(8)14 b Fv(\034)5 b(:)762 b Fw(\(3.5\))639 4689 y(A)32 b(remark)-5 b(able)30 b(prop)r(ert)n(y)g(of) h(the)h(pro)r(cess)e Fv(U)40 b Fw(is)31 b(that)h(a.s.)48 b(its)31 b(energy)f Fs(j)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(j)3176 4659 y Fr(2)3245 4689 y Fw(and)515 4789 y(enstroph)n(y)26 b Fs(k)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(k)1142 4759 y Fr(2)1142 4809 y(1)1206 4789 y Fw(are)27 b(time-indep)r(enden)n(t)i (random)d(constan)n(ts:)1926 5255 y(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fx(Lemma)29 b(3.2.)40 b FB(A)n(lmost)30 b(sur)l(ely)f(we)h(have)996 678 y Fs(j)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(j)1202 644 y Fr(2)1262 678 y Fw(=)23 b Fv(E)104 b Fw(and)98 b Fs(k)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(k)1991 644 y Fr(2)1991 698 y(1)2051 678 y Fw(=)22 b(\012)99 b(for)29 b(a)p Fv(:)g Fw(a)p Fv(:)99 b(t)23 b Fs(2)g Fq(R)p Fv(:)316 b Fw(\(3.6\))515 833 y FB(The)30 b(exp)l(e)l(ctations)g(of)h(the)f(r)l(andom)g(c)l(onstants) f Fv(E)35 b FB(and)30 b Fw(\012)g FB(satisfy)1279 1036 y Fq(E)23 b Fw(\012)29 b(=)1529 980 y(1)p 1529 1017 42 4 v 1529 1093 a(2)1581 1036 y Fv(B)1644 1048 y Fr(0)1681 1036 y Fv(;)1813 980 y(L)1870 950 y Fr(2)1906 980 y Fv(B)1973 950 y Fr(2)1969 1001 y(0)p 1813 1017 199 4 v 1841 1093 a Fw(2)p Fv(B)1946 1105 y Fr(1)2044 1036 y Fs(\024)22 b Fq(E)h Fv(E)34 b Fs(\024)2387 980 y Fv(L)2444 950 y Fr(2)2481 980 y Fv(B)2544 992 y Fr(0)p 2387 1017 194 4 v 2464 1093 a Fw(2)2591 1036 y Fv(:)594 b Fw(\(3.7\))515 1227 y FB(Besides,)1477 1361 y Fq(E)22 b Fs(k)p Fv(U)9 b Fw(\()p Fv(\034)g Fw(\))p Fs(k)1799 1327 y Fr(2)1799 1381 y(2)1865 1361 y Fs(\024)1963 1305 y Fw(1)p 1963 1342 42 4 v 1963 1418 a(2)2014 1361 y Fv(L)2071 1327 y Fr(2)2108 1361 y Fv(B)2171 1373 y Fr(1)2293 1361 y Fs(8)14 b Fv(\034)5 b(:)790 b Fw(\(3.8\))639 1528 y(The)28 b(lemma)f(is)h(pro)n(v)n(en)e(b)r(elo)n(w)h(in)h(section)f(4.2.)515 1756 y Fc(3.3)112 b(The)38 b(equation)f(for)h Fb(U)515 1909 y Fw(Since)29 b Fv(U)790 1921 y Fu(\027)831 1909 y Fw(,)h Fv(\027)i Fs(2)26 b(f)p Fv(\027)1121 1921 y Fu(j)1156 1909 y Fs(g)p Fw(,)k(is)f(a)g(w)n(eak)f(solution)h(for)g (\(2.1\),)h(then)g(for)f(an)n(y)g Fv(T)37 b(>)26 b(\034)39 b Fw(it)30 b(satis\014es)515 2018 y(\(2.6\))o(,)25 b(where)f Fv(\020)31 b Fw(is)24 b(replaced)f(b)n(y)h(a)g(pro)r(cess)1912 1996 y(~)1905 2018 y Fv(\020)31 b Fw(with)24 b(the)h(same)e (distribution.)36 b(W)-7 b(e)25 b(rewrite)515 2118 y(\(2.6\))i(as)g (follo)n(ws:)907 2334 y(\011\()p Fv(U)1061 2346 y Fu(\027)1101 2334 y Fw(\))84 b(:=)e Fv(U)1444 2346 y Fu(\027)1485 2334 y Fw(\()p Fv(T)12 b Fw(\))18 b Fs(\000)g Fv(U)1768 2346 y Fu(\027)1809 2334 y Fw(\()p Fv(\034)9 b Fw(\))20 b(+)2021 2221 y Fp(Z)2104 2242 y Fu(T)2067 2410 y(\034)2170 2334 y Fv(B)t Fw(\()p Fv(U)2326 2346 y Fu(\027)2368 2334 y Fv(;)28 b(U)2476 2346 y Fu(\027)2517 2334 y Fw(\))p Fv(dt)23 b Fw(=)1228 2575 y(=)94 b Fs(\000)p Fv(\027)1512 2462 y Fp(Z)1595 2483 y Fu(T)1558 2651 y(\034)1661 2575 y Fv(AU)1780 2587 y Fu(\027)1822 2575 y Fw(\()p Fv(s)p Fw(\))14 b Fv(ds)k Fw(+)2122 2511 y Fs(p)p 2192 2511 47 4 v 2192 2575 a Fv(\027)5 b Fw(\()2277 2553 y(~)2270 2575 y Fv(\020)i Fw(\()p Fv(T)12 b Fw(\))18 b Fs(\000)2546 2553 y Fw(~)2539 2575 y Fv(\020)6 b Fw(\()p Fv(\034)j Fw(\)\))25 b(=:)e Fv(G)2923 2587 y Fu(\027)2964 2575 y Fv(:)221 b Fw(\(3.9\))515 2782 y Fx(Lemma)26 b(3.3.)38 b FB(The)29 b(nonline)l(arity)f Fv(B)j FB(de\014nes)d(c)l(ontinuous)e (quadr)l(atic)j(maps)f Fv(H)3100 2752 y Fr(2)3160 2782 y Fs(!)23 b Fv(H)3342 2752 y Fr(1)515 2882 y FB(and)30 b Fv(H)752 2852 y Fr(1)812 2882 y Fs(!)23 b Fv(L)975 2894 y Fu(p)1036 2882 y Fw(=)g Fv(L)1181 2894 y Fu(p)1219 2882 y Fw(\()p Fv(T)1312 2852 y Fr(2)1348 2882 y Fv(;)14 b Fq(R)1439 2852 y Fr(2)1482 2882 y Fw(\))19 b Fs(\\)g Fv(H)7 b FB(,)30 b(for)g(any)g Fv(p)23 b(<)g Fw(2)p FB(.)639 3025 y Fw(The)35 b(pro)r(of)g(of)g(the)g(lemma's)g(assertions)e(is)i (ob)n(vious)f(since)h(the)g(m)n(ultiplication)g(of)515 3125 y(functions)e(de\014nes)g(con)n(tin)n(uous)f(bi-linear)g(maps)g Fv(H)2208 3095 y Fr(2)2245 3125 y Fw(\()p Fv(T)2338 3095 y Fr(2)2375 3125 y Fw(\))22 b Fs(\002)g Fv(H)2592 3095 y Fr(1)2629 3125 y Fw(\()p Fv(T)2722 3095 y Fr(2)2758 3125 y Fw(\))32 b Fs(!)g Fv(H)3013 3095 y Fr(1)3050 3125 y Fw(\()p Fv(T)3143 3095 y Fr(2)3180 3125 y Fw(\))h(and)515 3225 y Fv(H)591 3195 y Fr(1)628 3225 y Fw(\()p Fv(T)721 3195 y Fr(2)757 3225 y Fw(\))23 b Fs(\002)g Fv(L)957 3237 y Fr(2)994 3225 y Fw(\()p Fv(T)1087 3195 y Fr(2)1123 3225 y Fw(\))35 b Fs(!)f Fv(L)1364 3237 y Fu(p)1402 3225 y Fw(\()p Fv(T)1495 3195 y Fr(2)1531 3225 y Fw(\),)i(and)f(since)f(the) g(pro)5 b(jection)33 b(\005)i(is)f(con)n(tin)n(uous)f(in)i(the)515 3324 y Fv(L)572 3336 y Fu(p)610 3324 y Fw(-spaces)26 b(with)i(1)23 b Fv(<)f(p)h(<)g Fs(1)p Fw(.)639 3424 y(Due)30 b(to)f(the)g(lemma,)h(\011)f(de\014nes)g(a)f(con)n(tin)n(uous)h(map)g Fv(Z)i Fs(!)25 b Fv(H)2662 3436 y Fo(\000)p Fr(1)p Fo(\000)p Fu(")2835 3424 y Fw(.)41 b(So)29 b(\011\()p Fv(U)3170 3436 y Fu(\027)3203 3444 y Ft(j)3238 3424 y Fw(\))d Fs(!)515 3524 y Fw(\011\()p Fv(U)9 b Fw(\))27 b(in)h Fv(H)903 3536 y Fo(\000)p Fr(1)p Fo(\000)p Fu(")1103 3524 y Fw(as)f Fv(\027)1246 3536 y Fu(j)1304 3524 y Fs(!)c Fw(0,)28 b(almost)f(surely)-7 b(.)36 b(F)-7 b(rom)27 b(the)h(other)f(hand,)h (since)1690 3678 y Fq(E)23 b Fs(j)p Fv(G)1842 3690 y Fu(\027)1889 3678 y Fs(j)g(\024)g Fv(C)2088 3615 y Fs(p)p 2157 3615 V 63 x Fv(\027)515 3833 y Fw(due)j(to)g(\(2.12\))f(and)h (\(2.4\),)h(then)f Fv(G)1641 3845 y Fu(\027)1674 3853 y Ft(j)1733 3833 y Fs(!)d Fw(0)j(in)g Fv(H)2071 3845 y Fo(\000)p Fr(1)p Fo(\000)p Fu(")2270 3833 y Fw(in)g(probabilit)n(y)-7 b(.)36 b(P)n(assing)24 b(in)i(\(3.9\))515 3933 y(to)h(the)h(limit)g(in) f(probabilit)n(y)g(as)f Fv(\027)1620 3945 y Fu(j)1679 3933 y Fs(!)d Fw(0)k(w)n(e)g(get)g(that)g(\011\()p Fv(U)9 b Fw(\))23 b(=)g(0)k(a.s.)36 b(That)27 b(is,)h(almost)515 4033 y(surely)1500 4111 y(_)1477 4132 y Fv(U)9 b Fw(\()p Fv(t)p Fw(\))19 b(+)f Fv(B)t Fw(\()p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fv(;)28 b(U)9 b Fw(\()p Fv(t)p Fw(\)\))24 b(=)e(0)p Fv(:)750 b Fw(\(3.10\))515 4265 y(So,)28 b(a.s.)40 b Fv(U)9 b Fw(\()p Fs(\001)p Fw(\))25 b Fs(2)g Fv(Z)34 b Fw(is)29 b(a)f(w)n(eak)f(solution)h(of)h(the)g(Euler)f(equation)g (\(3.10\))o(,)h(equiv)-5 b(alen)n(t)28 b(to)515 4365 y(\(1.7\))o(.)44 b(Due)30 b(to)g(\(3.6\),)g(a.s.)43 b Fs(k)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(k)1664 4377 y Fr(1)1728 4365 y Fw(=)26 b(const.)44 b(Using)30 b(\(3.8\))g(and)g (the)g(F)-7 b(ubini)31 b(theorem)515 4464 y(w)n(e)c(get)g(that)1444 4564 y Fv(U)9 b Fw(\()p Fs(\001)p Fw(\))23 b Fs(2)h Fv(L)1756 4576 y Fr(2)e(lo)r(c)1898 4564 y Fw(\()p Fq(R)q Fw(;)42 b Fv(H)2126 4530 y Fr(2)2163 4564 y Fw(\))98 b(a)p Fv(:)p Fw(s)p Fv(:)13 b(:)717 b Fw(\(3.11\))515 4697 y(Therefore)26 b(due)i(to)f(Lemma)h(3.3)e(and)i(\(3.10\))o(,)1082 4830 y(_)1058 4851 y Fv(U)k Fs(2)24 b Fv(L)1283 4863 y Fr(1)e(lo)r(c)1425 4851 y Fw(\()p Fq(R)p Fw(;)43 b Fv(H)1653 4817 y Fr(1)1690 4851 y Fw(\))19 b Fs(\\)g Fv(L)1872 4863 y Fo(1)1942 4851 y Fw(\()p Fq(R)p Fw(;)43 b Fv(L)2151 4863 y Fu(p)2189 4851 y Fw(\))97 b(a)p Fv(:)p Fw(s)p Fv(:)o(;)g Fs(8)14 b Fv(p)22 b(<)h Fw(2)p Fv(:)331 b Fw(\(3.12\))639 5006 y(W)-7 b(e)28 b(ha)n(v)n(e)f(pro)n(v)n(en)f(the)i(follo)n(wing)e (result:)1905 5255 y(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fx(Theorem)23 b(3.4.)34 b FB(A)n(ny)24 b(limiting)g(me)l(asur)l(e)g Fv(m)f Fs(2)g(P)7 b Fw(\()p Fv(Z)f Fw(\))24 b FB(as)g(in)30 b Fw(\(3.1\))24 b FB(is)g(a)h(we)l(ak)f (stationary)515 623 y(solution)39 b(for)h(the)f(Euler)g(e)l(quation)46 b Fw(\(3.10\))38 b FB(in)h(the)g(fol)t(lowing)j(sense:)57 b(Ther)l(e)40 b(exists)f(a)515 722 y(stationary)j(r)l(andom)g(pr)l(o)l (c)l(ess)f Fv(U)9 b Fw(\()p Fv(t)p Fw(\))44 b Fs(2)g Fv(H)7 b FB(,)45 b(distribute)l(d)c(as)h Fv(m)p FB(,)i(which)f(a.s.)74 b(satis\014es)515 822 y Fw(\(3.10\))o FB(.)39 b(Mor)l(e)l(over,)639 922 y(1\))28 b(ener)l(gy)h Fs(j)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(j)1210 891 y Fr(2)1275 922 y FB(of)29 b(this)f(pr)l(o)l(c)l (ess)g(a.s.)39 b(is)29 b(a)f(time-indep)l(endent)g(r)l(andom)h(c)l (onstant)515 1021 y Fv(E)35 b FB(such)29 b(that)1551 1097 y Fv(L)1608 1067 y Fr(2)1645 1097 y Fv(B)1712 1067 y Fr(2)1708 1118 y(0)p 1551 1134 199 4 v 1579 1210 a Fw(2)p Fv(B)1684 1222 y Fr(1)1782 1153 y Fs(\024)23 b Fq(E)f Fv(E)34 b Fs(\024)2126 1097 y Fv(L)2183 1067 y Fr(2)2219 1097 y Fv(B)2282 1109 y Fr(0)p 2126 1134 194 4 v 2202 1210 a Fw(2)2330 1153 y Fv(;)639 1348 y FB(2\))28 b(its)f(enstr)l(ophy)h Fs(k)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(k)1473 1318 y Fr(2)1473 1369 y(1)1537 1348 y FB(a.s.)38 b(is)28 b(a)g(time-indep)l(endent)f(r)l(andom)h(c)l (onstant)f Fw(\012)g FB(such)515 1448 y(that)i Fq(E)23 b Fw(\012)29 b(=)934 1415 y Fr(1)p 934 1429 34 4 v 934 1477 a(2)977 1448 y Fv(B)1040 1460 y Fr(0)1078 1448 y FB(;)639 1548 y(3\))39 b(the)f(pr)l(o)l(c)l(ess)h(satis\014es)f(the)g (estimate)45 b Fw(\(3.8\))p FB(,)40 b(the)f(exp)l(onential)f(estimates) 45 b Fw(\(3.4\))515 1647 y FB(and)39 b Fw(\(3.5\))o FB(,)30 b(and)h(its)e(time-derivative)j(satis\014es)37 b Fw(\(3.12\))o FB(.)515 1880 y Fc(3.4)112 b(A)37 b(phase-space)j(for)d(the)h(Euler)e (equation)515 2033 y Fw(By)29 b(\(3.11\))e(and)h(\(3.12\),)g(with)h (probabilit)n(y)e(one)h(tra)5 b(jectories)26 b(of)j(the)f(pro)r(cess)f 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Fw(whic)n(h)e(satisfy)f(the)i(Euler)e (equa-)515 3623 y(tion.)41 b(Clearly)28 b Fs(K)1076 3593 y Fu(E)1162 3623 y Fw(is)h(a)f(closed)h(subset)g(of)g Fs(K)q Fw(,)h(in)n(v)-5 b(arian)n(t)28 b(under)h(the)g (time-translations)515 3723 y Fv(u)p Fw(\()p Fs(\001)p Fw(\))23 b Fs(!)g Fv(u)p Fw(\()p Fv(T)29 b Fw(+)18 b Fs(\001)p Fw(\),)28 b Fv(T)35 b Fs(2)23 b Fq(R)p Fw(.)43 b(Due)28 b(to)f(Lemma)h(3.5)f(the)h(map)1497 3905 y Fv(\031)f Fw(:)c Fs(K)h(!)f Fv(H)1886 3871 y Fr(1)1923 3905 y Fv(;)97 b(u)23 b Fs(7!)g Fv(u)p Fw(\(0\))p Fv(;)515 4088 y Fw(restricted)g(to)h Fs(K)1043 4058 y Fu(E)1100 4088 y Fw(,)h(de\014nes)f(a)f(con)n(tin)n (uous)g(em)n(b)r(edding)h Fs(K)2374 4058 y Fu(E)2454 4088 y Fs(!)f Fv(H)2636 4058 y Fr(1)2673 4088 y Fw(.)36 b(W)-7 b(e)24 b(denote)g(b)n(y)g Fv(H)3323 4058 y Fu(E)515 4188 y Fw(its)k(image)e Fv(\031)s Fw(\()p Fs(K)1016 4158 y Fu(E)1073 4188 y Fw(\),)j(and)e(pro)n(vide)f Fv(H)1689 4158 y Fu(E)1773 4188 y Fw(with)i(the)g(top)r(ology)-7 b(,)26 b(induced)i(from)g Fs(K)3032 4158 y Fu(E)3088 4188 y Fw(.)639 4287 y(Due)g(to)f([BKM84)o(],)g(for)g(an)n(y)f Fv(u)1624 4299 y Fr(0)1684 4287 y Fs(2)e Fv(H)1839 4257 y Fr(3)1903 4287 y Fw(the)k(Euler)e(equation)h(has)f(a)h(unique)h (solution)515 4387 y Fv(u)37 b Fs(2)h(K)26 b(\\)e Fv(L)929 4399 y Fo(1)12 b Fr(lo)r(c)1094 4387 y Fw(\()p Fq(R)p Fw(;)58 b Fv(H)1337 4357 y Fr(3)1374 4387 y Fw(\),)39 b(equal)d Fv(u)1744 4399 y Fr(0)1817 4387 y Fw(at)g Fv(t)i Fw(=)f(0.)63 b(This)36 b(solution,)j(as)c(a)h(p)r(oin)n(t)h(in)f Fs(K)q Fw(,)515 4487 y(con)n(tin)n(uously)26 b(dep)r(ends)i(on)g Fv(u)1482 4499 y Fr(0)1518 4487 y Fw(.)37 b(So)1646 4669 y Fv(H)1722 4635 y Fr(3)1782 4669 y Fs(\032)22 b Fv(H)1945 4635 y Fu(E)2024 4669 y Fs(\032)h Fv(H)2188 4635 y Fr(1)2225 4669 y Fv(;)919 b Fw(\(3.15\))515 4852 y(and)34 b(the)h(inclusions)g (are)e(con)n(tin)n(uous.)57 b(W)-7 b(e)36 b(p)r(oin)n(t)e(out)h(that)g (w)n(e)f(do)h(not)g(kno)n(w)e(if)j(the)515 4952 y(\(top)r(ological\))26 b(space)h Fv(H)1299 4921 y Fu(E)1383 4952 y Fw(is)g(linear,)g(or)g (not.)1905 5255 y(11)p eop %%Page: 12 12 12 11 bop 639 523 a Fw(F)-7 b(or)28 b(an)n(y)f Fv(u)994 535 y Fr(0)1055 523 y Fs(2)d Fv(H)1210 493 y Fu(E)1266 523 y Fw(,)k Fv(u)23 b Fw(=)h Fv(\031)1527 493 y Fo(\000)p Fr(1)1616 523 y Fw(\()p Fv(u)1696 535 y Fr(0)1733 523 y Fw(\))29 b(is)f(the)g(unique)g(solution)g(of)g(the)g(Euler)g (equation)515 623 y(in)36 b(then)h(space)f Fs(K)q Fw(,)j(equal)d Fv(u)1451 635 y Fr(0)1524 623 y Fw(at)g Fv(t)h Fw(=)h(0.)62 b(Since)37 b Fv(u)p Fw(\()p Fv(t)p Fw(\))g(=)h Fv(\031)s Fw(\()p Fv(u)p Fw(\()p Fv(t)24 b Fw(+)g Fs(\001)p Fw(\)\))37 b(and)f(the)h(time-)515 722 y(translation)28 b(maps)g(are)g(con)n(tin)n (uous)g(in)h Fs(K)q Fw(,)h(then)g(the)f(Euler)f(equation)h(de\014nes)g (a)f(group)515 822 y(of)h(con)n(tin)n(uous)g(automorphisms)g Fv(S)1665 834 y Fu(t)1720 822 y Fw(:)e Fv(H)1846 792 y Fu(E)1929 822 y Fs(!)f Fv(H)2114 792 y Fu(E)2170 822 y Fw(,)31 b Fv(t)26 b Fs(2)h Fq(R)p Fw(.)50 b(As)29 b Fv(S)2664 834 y Fu(t)2713 822 y Fs(\016)20 b Fv(U)9 b Fw(\()p Fv(\034)g Fw(\))27 b(=)g Fv(U)9 b Fw(\()p Fv(t)20 b Fw(+)f Fv(\034)9 b Fw(\))515 922 y(and)27 b(the)h(pro)r(cess)f Fv(U)36 b Fw(is)27 b(stationary)-7 b(,)27 b(then)h(w)n(e)f(ha)n(v)n(e:) 515 1088 y Fx(Theorem)22 b(3.6.)34 b FB(The)24 b(distribution)g Fs(D)h Fw(=)e Fs(D)16 b Fv(U)9 b Fw(\(0\))23 b FB(of)h(the)g(pr)l(o)l (c)l(ess)f Fv(U)32 b FB(as)24 b(in)f(The)l(or)l(em)i(3.4)515 1187 y(is)32 b(an)h(invariant)g(Bor)l(el)g(me)l(asur)l(e)f(for)h(the)g (\015ow)f(which)i(the)f(Euler)f(e)l(quation)h(de\014nes)f(in)515 1287 y Fv(H)591 1257 y Fu(E)647 1287 y FB(.)38 b(The)31 b(sp)l(ac)l(e)f Fv(H)1172 1257 y Fu(E)1258 1287 y FB(satis\014es)37 b Fw(\(3.15\))o FB(,)30 b(and)g Fs(D)r Fw(\()p Fv(H)2179 1257 y Fu(E)2254 1287 y Fs(\\)19 b Fv(H)2404 1257 y Fr(2)2441 1287 y Fw(\))k(=)g(1)p FB(.)639 1453 y Fw(Since)g(v)n(ector)e(\014elds) h(from)g Fv(H)1569 1423 y Fr(2)1628 1453 y Fw(ha)n(v)n(e)f(the)i(mo)r (dulus)f(of)g(con)n(tin)n(uit)n(y)2757 1423 y Fr(4)2816 1453 y Fv(g)s Fw(\()p Fv(r)r Fw(\))i(=)f Fv(r)17 b Fw(max)o(\(1,)515 1553 y(log)d Fv(r)675 1522 y Fo(\000)p Fr(1)765 1553 y Fw(\),)41 b(then)d(an)n(y)f Fv(u)p Fw(\()p Fv(t;)14 b(x)p Fw(\))41 b Fs(2)g(K)f Fw(has)d(the)i(mo)r(dulus)f(of)g(con)n(tin) n(uit)n(y)f(in)h Fv(x)p Fw(,)k(equal)37 b(to)515 1652 y Fv(\036)p Fw(\()p Fv(t)p Fw(\))p Fv(g)s Fw(\()p Fv(r)r Fw(\),)44 b(where)39 b Fv(\036)k Fs(2)g Fv(L)1370 1664 y Fr(2)22 b(lo)r(c)1512 1652 y Fw(.)72 b(Therefore)38 b(b)n(y)h(the)h(Osgo)r(o)r(d)e(criterion)g([Har64)n(],)43 b(the)515 1752 y(di\013eren)n(tial)27 b(equation)1484 1851 y(_)-38 b Fv(x)p Fw(\()p Fv(t)p Fw(\))24 b(=)f Fv(u)p Fw(\()p Fv(t;)14 b(x)p Fw(\))p Fv(;)97 b(x)p Fw(\()p Fv(t)2177 1863 y Fr(1)2215 1851 y Fw(\))23 b(=)g Fv(y)s(;)515 2001 y Fw(has)k(a)g(unique)h(solution.)36 b(So)28 b(the)g(\015o)n (w-maps)e Fv(S)2053 1964 y Fu(t)2078 1972 y Ff(2)2048 2021 y Fu(t)2073 2029 y Ff(1)2137 2001 y Fw(:)e Fv(T)2245 1971 y Fr(2)2304 2001 y Fs(!)f Fv(T)2471 1971 y Fr(2)2508 2001 y Fw(,)k Fv(y)f Fs(7!)d Fv(x)p Fw(\()p Fv(t)2840 2013 y Fr(2)2878 2001 y Fw(\),)28 b(are)f(w)n(ell)g(de-)515 2100 y(\014ned)c(and)f(con)n(tin)n(uous)g(\(see)h([Che98)o(]\).)36 b(Accordingly)-7 b(,)22 b(due)h(to)g(the)g(classical)e(argumen)n(ts,) 515 2200 y(for)32 b(an)n(y)g Fv(u)e Fs(2)i(K)1038 2170 y Fu(E)1127 2200 y Fw(w)n(e)g(ha)n(v)n(e)g Fv(!)s Fw(\()p Fv(t;)14 b(x)p Fw(\))31 b Fs(\021)g Fv(!)1863 2212 y Fr(0)1900 2200 y Fw(\()p Fv(S)1988 2170 y Fr(0)1983 2221 y Fu(t)2025 2200 y Fw(\()p Fv(x)p Fw(\)\),)36 b(where)c Fv(!)s Fw(\()p Fv(t;)14 b(x)p Fw(\))32 b(=)g(rot)14 b Fv(u)p Fw(\()p Fv(t;)g(x)p Fw(\))32 b(\(and)515 2300 y Fv(!)567 2312 y Fr(0)627 2300 y Fw(=rot)14 b Fv(u)861 2312 y Fr(0)896 2300 y Fw(\).)639 2399 y(Appro)n(ximating)30 b Fv(u)p Fw(\()p Fv(t;)14 b(x)p Fw(\))31 b(b)n(y)g(smo)r(oth)f(div)n (ergence-free)f(v)n(ector)g(\014elds)i Fv(u)2974 2411 y Fu(\017)3005 2399 y Fw(\()p Fv(t;)14 b(x)p Fw(\))32 b(\(sa)n(y)-7 b(,)515 2499 y(obtained)23 b(as)f(con)n(v)n(olutions)f (with)j(scalar)d(molli\014ers\),)j(w)n(e)e(appro)n(ximate)g(an)n(y)g (con)n(tin)n(uous)515 2599 y(\015o)n(w-map)i Fv(S)k Fw(=)23 b Fv(S)1093 2562 y Fu(t)1118 2570 y Ff(2)1088 2619 y Fu(t)1113 2627 y Ff(1)1180 2599 y Fw(b)n(y)i(measure-preserving)e(smo)r (oth)i(maps)g Fv(T)2582 2568 y Fr(2)2642 2599 y Fs(!)e Fv(T)2809 2568 y Fr(2)2845 2599 y Fw(,)j(con)n(v)n(erging)d(to)515 2698 y Fv(S)37 b Fw(at)c(eac)n(h)f(p)r(oin)n(t,)i(see)e([Har64)n(,)i (Theorem)e(2.1,)h(Section)g(V].)g(P)n(assing)e(to)h(the)h(limit)h(as) 515 2798 y Fv(\017)28 b Fs(!)g Fw(0,)j(w)n(e)g(\014nd)g(that)1262 2731 y Fp(R)1301 2827 y Fu(T)1349 2811 y Ff(2)1400 2798 y Fv(f)9 b Fw(\()p Fv(S)c Fw(\()p Fv(x)p Fw(\)\))14 b Fv(dx)30 b Fw(=)1908 2731 y Fp(R)1947 2827 y Fu(T)1995 2811 y Ff(2)2045 2798 y Fv(f)9 b Fw(\()p Fv(x)p Fw(\))14 b Fv(dx)32 b Fw(for)f(eac)n(h)f(con)n(tin)n(uous)f(function)515 2897 y Fv(f)9 b Fw(.)36 b(Hence,)28 b(the)g(\015o)n(w-maps)e(preserv)n (e)g(the)i(Leb)r(esgue)f(measure,)g(and)880 3005 y Fp(Z)926 3194 y Fu(T)974 3177 y Ff(2)1025 3118 y Fv(g)s Fw(\()p Fv(!)s Fw(\()p Fv(t;)14 b(x)p Fw(\)\))g Fv(dx)24 b Fw(=)1580 3005 y Fp(Z)1627 3194 y Fu(T)1675 3177 y Ff(2)1725 3118 y Fv(g)s Fw(\()p Fv(!)1852 3130 y Fr(0)1907 3118 y Fs(\016)18 b Fv(S)2023 3084 y Fr(0)2018 3138 y Fu(t)2060 3118 y Fw(\)\()p Fv(x)p Fw(\))c Fv(dx)25 b Fw(=)2420 3005 y Fp(Z)2466 3194 y Fu(T)2514 3177 y Ff(2)2564 3118 y Fv(g)s Fw(\()p Fv(!)2691 3130 y Fr(0)2728 3118 y Fw(\()p Fv(x)p Fw(\)\))14 b Fv(dx)g(;)515 3347 y Fw(for)27 b(an)n(y)g Fv(t)g Fw(and)h(an)n(y)f Fv(g)e Fs(2)f Fv(C)1378 3359 y Fu(b)1411 3347 y Fw(\()p Fq(R)q Fw(\).)43 b(That)27 b(is,)1251 3459 y Fp(Z)1297 3648 y Fu(T)1345 3631 y Ff(2)1395 3572 y Fv(g)s Fw(\(rot)13 b Fv(U)c Fw(\()p Fv(t;)14 b(x)p Fw(\)\))g Fv(dx)25 b Fw(=)d(const)97 b Fs(8)14 b Fv(t)22 b Fs(2)h Fq(R)p Fv(;)530 b Fw(\(3.16\))515 3809 y(for)27 b(eac)n(h)g Fv(U)k Fs(2)24 b(K)1060 3779 y Fu(E)1116 3809 y Fw(.)37 b(In)28 b(particular,)e(w)n(e)i(ha)n(v)n(e)e(the)i (follo)n(wing)f(result:)515 3975 y Fx(Prop)s(osition)e(3.7.)36 b FB(F)-6 b(or)26 b(a.a.)39 b(tr)l(aje)l(ctory)26 b(of)h(the)f(pr)l(o)l (c)l(ess)g Fv(U)35 b FB(and)26 b(any)g(b)l(ounde)l(d)h(c)l(ontin-)515 4075 y(uous)i(function)h Fv(g)s FB(,)f(the)h(inte)l(gr)l(al)g(in)36 b Fw(\(3.16\))29 b FB(is)h(a)h(time-indep)l(endent)f(r)l(andom)g(c)l (onstant.)515 4307 y Fc(3.5)112 b(Homogeneous)38 b(solution)515 4460 y Fw(Let)27 b(us)h(assume)f(that)1519 4560 y Fv(b)1555 4572 y Fu(s)1613 4560 y Fw(=)c Fv(b)1737 4572 y Fo(\000)p Fu(s)1847 4560 y Fs(6)p Fw(=)f(0)83 b Fs(8)14 b Fv(s)22 b Fs(2)h Fq(Z)2321 4526 y Fr(2)2321 4581 y(0)2352 4560 y Fv(:)792 b Fw(\(3.17\))p 515 4613 1146 4 v 607 4667 a Fn(4)642 4690 y Fm(Indeed,)29 b(b)n(y)f(the)g(classical)f(em)n(b)r (edding)h(theorem,)g(spaces)g Fk(H)2340 4667 y Fj(r)2402 4690 y Fm(with)g(non-in)n(teger)g Fk(r)g(>)d Fm(1)j(are)f(em-)515 4769 y(b)r(edded)k(in)e(H\177)-35 b(older)29 b(spaces)h Fk(C)1366 4746 y Fj(r)r Fi(\000)p Fn(1)1479 4769 y Fm(.)48 b(So)30 b(b)n(y)g(the)g(in)n(terp)r(olation,)i(the)e(space)g Fk(H)2713 4746 y Fn(2)2777 4769 y Fm(is)f(em)n(b)r(edded)h(in)f(the)515 4848 y(in)n(terp)r(olating)23 b(H\177)-35 b(older)22 b(space)h Fk(C)1403 4825 y Fn(1)1398 4865 y Fi(\003)1460 4848 y Fm(\(the)g(Sigm)n(und)f(space\).)32 b(The)23 b(latter)g(is)e (formed)g(b)n(y)i(functions,)g(whic)n(h)515 4927 y(ha)n(v)n(e)h(the)h (mo)r(dulus)e(of)g(con)n(tin)n(uit)n(y)i Fk(g)r Fm(\()p Fk(r)r Fm(\),)f(see)g([Che98)q(,)f(p.)g(31].)1905 5255 y Fw(12)p eop %%Page: 13 13 13 12 bop 515 523 a Fw(Then)25 b(the)h(random)e(\014eld)i Fv(\020)6 b Fw(\()p Fv(t;)28 b(x)p Fw(\))e(is)g(homogeneous)d(in)j Fv(x)p Fw(,)g(i.e.)36 b(translations)24 b(of)h Fv(x)h Fw(do)f(not)515 623 y(c)n(hange)d(its)h(distribution)g(\(see)g([Kuk02)n (]\),)i(and)e(\(2.1\))f(has)h(a)f(unique)i(stationary)d(measure)515 722 y Fv(\026)565 734 y Fu(\027)606 722 y Fw(.)37 b(As)27 b(a)g(consequence,)g(the)h(measure)e Fv(\026)1865 734 y Fu(\027)1934 722 y Fw(and)h(a)g(corresp)r(onding)f(stationary)g (solution)515 822 y Fv(U)572 834 y Fu(\027)613 822 y Fw(\()p Fv(t;)i(x)p Fw(\))23 b(b)r(oth)g(are)f(homogeneous)e(in)j Fv(x)p Fw(,)h(see)e([Kuk02)o(].)35 b(Since)23 b Fv(U)2581 834 y Fu(\027)2645 822 y Fw(a.s.)34 b(con)n(v)n(erges)20 b(to)j(the)515 922 y(\014eld)28 b Fv(U)9 b Fw(\()p Fv(t)p Fw(\))24 b(=)f Fv(U)9 b Fw(\()p Fv(t;)14 b(x)p Fw(\))28 b(as)g(in)g(Theorem)f(3.4,)g(then)i Fv(U)36 b Fw(is)28 b(homogeneous)e(\(i.e.,)j(stationary)515 1021 y(in)f Fv(x)p Fw(\).)639 1121 y(That)35 b(is,)i(under)e(the)g(assumption)f (\(3.17\))h(the)g(limiting)g(measures)f Fv(m)h Fw(as)f(in)h(\(3.1\))515 1220 y(corresp)r(ond)26 b(to)i(random)e(solutions)h(of)35 b(\(1.7\))27 b(whic)n(h)h(are)f(stationary)f(b)r(oth)i(in)g(time)g(and) 515 1320 y(space.)36 b(In)28 b(particular,)e(since)1482 1253 y Fp(R)1552 1320 y Fv(U)c(dx)i Fs(\021)f Fw(0,)k(then)h(in)g(this) g(case)e(w)n(e)i(ha)n(v)n(e)1714 1479 y Fq(E)22 b Fv(U)9 b Fw(\()p Fv(t;)14 b(x)p Fw(\))30 b Fs(\021)22 b Fw(0)515 1639 y(\(i.e.,)28 b Fq(E)22 b Fv(U)9 b Fw(\()p Fv(t)p Fw(\))29 b(=)23 b(0)g Fs(2)g Fv(H)1263 1608 y Fo(\000)p Fr(1)p Fo(\000)p Fu(")1463 1639 y Fw(for)k(all)g Fv(t)p Fw(\).)515 1867 y Fc(3.6)112 b(One)38 b(degenerate)g(example)515 2020 y Fw(Let)32 b(us)g(consider)f(a)h(force)f Fv(\021)36 b Fw(of)c(the)g(form)g(\(2.4\),)h(suc)n(h)f(that)g Fv(b)2549 2032 y Fu(s)2615 2020 y Fw(=)e(0)i(if)g Fs(j)p Fv(s)p Fs(j)f Fv(>)f Fw(1.)50 b(Then)515 2120 y Fv(B)578 2132 y Fr(0)638 2120 y Fw(=)23 b Fv(B)789 2132 y Fr(1)826 2120 y Fw(,)28 b(so)f(b)n(y)g(Theorem)g(3.4)f(w)n(e)i(ha)n(v)n(e)1539 2328 y Fq(E)22 b Fv(E)34 b Fw(=)1795 2272 y Fv(L)1852 2241 y Fr(2)1888 2272 y Fv(B)1951 2284 y Fr(0)p 1795 2309 194 4 v 1871 2385 a Fw(2)2022 2328 y(=)22 b Fv(L)2166 2293 y Fr(2)2203 2328 y Fq(E)g Fw(\012)p Fv(:)818 b Fw(\(3.18\))639 2520 y(Let)33 b(us)g(denote)g Fv(s)1217 2532 y Fr(1)1287 2520 y Fw(=)e(\(1)p Fv(;)d Fw(0\))1582 2490 y Fu(t)1644 2520 y Fw(and)33 b Fv(s)1850 2532 y Fr(2)1919 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2985 y Fv(a)1849 2997 y Fr(2)1905 2985 y Fw(+)1998 2929 y Fv(x)2045 2941 y Fr(1)p 1998 2966 V 2012 3042 a Fv(L)2093 2918 y Fp(\001)2154 2985 y Fs(j)g Fv(C)2259 2997 y Fr(1)2296 2985 y Fv(;)14 b(C)2392 2997 y Fr(2)2453 2985 y Fs(2)23 b Fq(R)p Fv(;)48 b(a)2700 2997 y Fr(1)2737 2985 y Fv(;)14 b(a)2818 2997 y Fr(2)2878 2985 y Fs(2)23 b Fq(R)p Fv(=)p Fw(2)p Fv(\031)s Fq(Z)3206 2893 y Fp(o)3260 2985 y Fv(:)515 3178 y Fw(So)h(\(3.18\))g(implies)h (that)f(a)h(limiting)g(solution)f Fv(U)9 b Fw(\()p Fv(t)p Fw(\))25 b(as)f(in)g(Theorem)g(3.4)g(b)r(elongs)g(to)g Fv(M)3342 3147 y Fr(4)515 3277 y Fw(for)j(all)g Fv(t)p Fw(.)37 b(F)-7 b(or)27 b(an)n(y)g Fv(u)22 b Fs(2)i Fv(M)1392 3247 y Fr(4)1456 3277 y Fw(w)n(e)k(ha)n(v)n(e)578 3482 y(\()p Fv(u)18 b Fs(\001)h(r)p Fw(\))p Fv(u)k Fw(=)f Fv(L)1034 3447 y Fo(\000)p Fr(1)1123 3482 y Fv(C)1182 3494 y Fr(1)1220 3482 y Fv(C)1279 3494 y Fr(2)1316 3365 y Fp(\022)1377 3482 y Fv(s)1416 3494 y Fr(1)1467 3482 y Fw(cos)1592 3414 y Fp(\000)1630 3482 y Fv(a)1674 3494 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Fr(2)3035 3714 y Fw(+)3128 3658 y Fv(x)3175 3670 y Fr(1)p 3128 3695 V 3142 3771 a Fv(L)3222 3714 y Fw(\))3254 3647 y Fp(\001)3293 3714 y Fv(;)3167 3871 y Fw(\(3.19\))515 4045 y(so)27 b Fv(B)t Fw(\()p Fv(u;)14 b(u)p Fw(\))22 b(=)h(0.)36 b(That)28 b(is,)1430 4024 y(_)1407 4045 y Fv(U)j Fs(\021)23 b Fw(0)k(and)h Fv(U)36 b Fw(is)28 b(a)f(constan)n(t)g(pro)r(cess)997 4226 y Fv(U)9 b Fw(\()p Fv(t)p Fw(\))23 b(=)g Fv(C)1327 4238 y Fr(1)1365 4226 y Fv(s)1404 4238 y Fr(1)1455 4226 y Fw(sin)1570 4159 y Fp(\000)1608 4226 y Fv(a)1652 4238 y Fr(1)1708 4226 y Fw(+)1801 4170 y Fv(x)1848 4182 y Fr(2)p 1801 4207 V 1815 4283 a Fv(L)1896 4159 y Fp(\001)1952 4226 y Fw(+)18 b Fv(C)2094 4238 y Fr(2)2132 4226 y Fv(s)2171 4238 y Fr(2)2222 4226 y Fw(sin)2337 4159 y Fp(\000)2375 4226 y Fv(a)2419 4238 y Fr(2)2475 4226 y Fw(+)2568 4170 y Fv(x)2615 4182 y Fr(1)p 2568 4207 V 2582 4283 a Fv(L)2663 4159 y Fp(\001)2784 4226 y Fs(8)c Fv(t;)269 b Fw(\(3.20\))515 4408 y(where)27 b Fv(C)814 4420 y Fr(1)851 4408 y Fv(;)14 b(C)947 4420 y Fr(2)1013 4408 y Fw(and)27 b Fv(a)1218 4420 y Fr(1)1255 4408 y Fv(;)14 b(a)1336 4420 y Fr(2)1401 4408 y Fw(are)26 b(random)h(v)-5 b(ariables.)639 4508 y(The)36 b(pro)r(cesses)f Fv(U)45 b Fw(ab)r(o)n(v)n(e)35 b(are)g(v)n(ery)g(degenerate)g(b)r(ecause)g(due)i(to)f(\(3.19\))o(,)i (the)f(4-)515 4608 y(dimensional)d(space)h Fv(M)1299 4578 y Fr(4)1371 4608 y Fw(is)g(in)n(v)-5 b(arian)n(t)34 b(for)h(equation)f(\(2.1\))h(with)h Fv(\021)j Fw(=)c(0.)60 b(The)35 b(space)515 4707 y Fv(M)605 4677 y Fr(4)673 4707 y Fw(is)c(a)f(maximal)h(one)f(with)i(this)f(prop)r(ert)n(y)f (since)h(if)g Fv(M)40 b Fw(is)31 b(a)g(linear)f(subspace)g(of)h Fv(H)7 b Fw(,)515 4807 y(in)n(v)-5 b(arian)n(t)27 b(for)h(\(2.1\))h (with)g Fv(\021)f Fw(=)c(0,)29 b(and)f Fv(M)1886 4777 y Fr(4)1948 4807 y Fs(\032)c Fv(M)9 b Fw(,)29 b Fv(M)2269 4777 y Fr(4)2331 4807 y Fs(6)p Fw(=)24 b Fv(M)9 b Fw(,)29 b(then)g Fv(M)37 b Fw(is)29 b(dense)f(in)h Fv(H)7 b Fw(,)515 4907 y(see)26 b([EM01)o(].)36 b(Accordingly)-7 b(,)26 b(if)h Fv(b)1557 4919 y Fu(s)1616 4907 y Fs(6)p Fw(=)22 b(0)k(for)h Fs(j)p Fv(s)p Fs(j)c(\024)f Fw(2,)27 b(then)g(the)g (corresp)r(onding)e(pro)r(cesses)515 5006 y Fv(U)36 b Fw(ha)n(v)n(e)27 b(no)g(c)n(hances)f(to)i(b)r(e)g(as)f(degenerate)f(as) h(\(3.20\))o(.)1905 5255 y(13)p eop %%Page: 14 14 14 13 bop 515 523 a Fy(4)134 b(Pro)t(ofs)515 721 y Fc(4.1)112 b(Pro)s(of)37 b(of)h(Theorem)f(3.1)515 875 y Fw(Our)27 b(argumen)n(ts)f(in)i(this)g(section)f(closely)g(follo)n(w)f([VF88],)i (Chapter)f(X.)639 974 y(First)g(w)n(e)f(note)g(that)h(to)f(pro)n(v)n(e) f(the)i(theorem)f(it)g(is)h(su\016cien)n(t)f(to)h(c)n(hec)n(k)e(that)i (for)f(an)n(y)515 1074 y Fv(n)d Fs(\025)f Fw(1)27 b(the)h(sequence)g (of)f(measures)f Fs(f)p Fv(m)1797 1044 y Fu(n)1801 1096 y Fr(~)-37 b Fu(\027)1830 1104 y Ft(j)1865 1074 y Fv(;)28 b(j)g Fw(=)23 b(1)p Fv(;)14 b Fw(2)p Fv(;)g(:)g(:)g(:)e Fs(g)27 b Fw(is)h(tigh)n(t)f(in)h Fv(Z)2846 1044 y Fu(n)2891 1074 y Fw(,)g(where)1187 1260 y Fv(m)1260 1225 y Fu(n)1264 1280 y Fr(~)-37 b Fu(\027)1293 1288 y Ft(j)1351 1260 y Fw(=)23 b Fs(D)r Fw(\()p Fv(u)1589 1272 y Fr(~)-37 b Fu(\027)1618 1280 y Ft(j)1653 1260 y Fs(j)1676 1275 y Fr([)p Fo(\000)p Fu(n;)11 b(n)p Fr(])1883 1260 y Fw(\))23 b Fs(2)h(P)2082 1192 y Fp(\000)2119 1260 y Fv(C)6 b Fw(\([)p Fs(\000)p Fv(n;)28 b(n)p Fw(];)37 b Fv(H)7 b Fw(\))2646 1192 y Fp(\001)2684 1260 y Fv(:)515 1427 y Fw(Indeed,)22 b(w)n(e)e(need)h(to)f(v)n(erify)g(that)g(the)h(sequence)f Fs(f)p Fv(m)2186 1439 y Fr(~)-37 b Fu(\027)2215 1447 y Ft(j)2250 1427 y Fs(g)20 b Fw(con)n(tains)g(a)g(subsequence)g Fs(f)p Fv(m)3270 1439 y Fu(\027)3303 1447 y Ft(j)3337 1427 y Fs(g)515 1527 y Fw(suc)n(h)27 b(that)1356 1627 y(\()p Fv(m)1461 1639 y Fu(\027)1494 1647 y Ft(j)1529 1627 y Fv(;)h(f)9 b Fw(\))23 b Fs(!)g Fw(\()p Fv(m;)28 b(f)9 b Fw(\))p Fv(;)97 b(m)23 b Fs(2)g(P)7 b Fw(\()p Fv(Z)f Fw(\))p Fv(;)670 b Fw(\(4.1\))515 1767 y(for)27 b(an)n(y)f Fv(f)32 b Fs(2)23 b Fv(C)1008 1779 y Fu(b)1042 1767 y Fw(\()p Fv(Z)6 b Fw(\).)37 b(It)28 b(is)f(kno)n(wn)g(that)g(it)h (su\016ces)f(to)g(c)n(hec)n(k)g(\(4.1\))g(for)g(b)r(ounded)g(Lips-)515 1867 y(c)n(hitz)18 b(functionals)h Fv(f)1161 1879 y Fu(L)1229 1867 y Fw(\(see)f([Dud89],)j(Theorem)c(11.3.3\).)33 b(It)19 b(is)f(easy)g(to)g(appro)n(ximate)f(an)n(y)515 1966 y(suc)n(h)k Fv(f)737 1978 y Fu(L)809 1966 y Fw(b)n(y)g(functionals)h Fv(f)1385 1936 y Fu(n)1376 1989 y(L)1453 1966 y Fw(=)h Fv(g)1584 1936 y Fu(n)1581 1989 y(L)1637 1966 y Fs(\016)7 b Fv(\031)1733 1978 y Fu(n)1778 1966 y Fw(,)23 b(where)f Fv(\031)2106 1978 y Fu(n)2173 1966 y Fw(is)g(the)g(natural)f(pro)5 b(jection)21 b Fv(Z)29 b Fs(!)23 b Fv(Z)3311 1978 y Fu(n)3356 1966 y Fw(,)515 2066 y(and)d Fv(g)712 2036 y Fu(n)709 2089 y(L)780 2066 y Fw(is)g(a)h(b)r(ounded)g(Lipsc)n(hitz)g(functional) g(on)f Fv(Z)2147 2078 y Fu(n)2192 2066 y Fw(.)35 b(Since)21 b(\()p Fv(m)2565 2078 y Fu(\027)2606 2066 y Fv(;)28 b(g)2700 2036 y Fu(n)2697 2089 y(L)2751 2066 y Fs(\016)5 b Fv(\031)2845 2078 y Fu(n)2890 2066 y Fw(\))24 b(=)e(\()p Fv(m)3138 2036 y Fu(n)3138 2087 y(\027)3184 2066 y Fv(;)27 b(g)3277 2036 y Fu(n)3274 2089 y(L)3324 2066 y Fw(\),)515 2166 y(then)h(\(4.1\))f(holds)g(if)1645 2265 y Fv(m)1718 2231 y Fu(n)1718 2286 y(\027)1751 2294 y Ft(j)1809 2265 y Fs(!)c Fv(m)1988 2231 y Fu(n)2116 2265 y Fs(8)14 b Fv(n;)958 b Fw(\(4.2\))515 2406 y(where)33 b Fv(m)834 2376 y Fu(n)913 2406 y Fw(=)g(\()p Fv(\031)1090 2418 y Fu(n)1136 2406 y Fw(\))1168 2418 y Fo(\003)1206 2406 y Fv(m)p Fw(.)56 b(If)35 b(w)n(e)e(kno)n(w)g(that)i(the)f(sequence)f Fs(f)p Fv(m)2600 2376 y Fu(n)2604 2428 y Fr(~)-37 b Fu(\027)2633 2436 y Ft(j)2668 2406 y Fs(g)34 b Fw(is)g(tigh)n(t)g(in)g Fv(Z)3201 2418 y Fu(n)3280 2406 y Fw(for)515 2515 y(eac)n(h)c Fv(n)p Fw(,)h(then)g(w)n(e)g(can)f(use)h(the)g(diagonal)e(pro)r(cess)h (to)g(construct)g(a)h(subsequence)f Fs(f)p Fv(\027)3303 2527 y Fu(j)3337 2515 y Fs(g)515 2615 y Fw(suc)n(h)23 b(that)h(\(4.2\))g(holds)g(for)f(ev)n(ery)g Fv(n)p Fw(.)35 b(The)24 b(measures)f Fv(m)2323 2585 y Fu(n)2391 2615 y Fs(2)g(P)7 b Fw(\()p Fv(Z)2623 2627 y Fu(n)2668 2615 y Fw(\))24 b(form)g(a)f(compatible)515 2714 y(family)k(and)h(de\014ne)g (a)f(measure)f Fv(m)d Fs(2)h(P)7 b Fw(\()p Fv(Z)f Fw(\))27 b(that)h(satis\014es)f(\(4.1\))639 2814 y(So)34 b(w)n(e)h(need)f(to)g (c)n(hec)n(k)g(that)h(the)g(sequence)e(of)i(measures)e Fs(f)p Fv(\026)2674 2784 y Fu(n)2678 2836 y Fr(~)-37 b Fu(\027)2707 2844 y Ft(j)2742 2814 y Fs(g)34 b Fw(is)g(tigh)n(t)h (for)f(an)n(y)515 2923 y(\014xed)41 b Fv(n)p Fw(.)78 b(Let)41 b Fv(u)1090 2935 y Fu(\027)1172 2923 y Fw(b)r(e)h(a)f (stationary)f(solution)g(for)h(\(2.1\))g(suc)n(h)g(that)h Fs(D)r Fw(\()p Fv(u)3012 2935 y Fu(\027)3053 2923 y Fw(\))k(=)g Fv(m)3315 2935 y Fu(\027)3356 2923 y Fw(.)515 3023 y(W)-7 b(e)36 b(abbreviate)f Fv(u)i Fw(=)g Fv(u)1318 3035 y Fu(\027)1395 3023 y Fw(and)f(tak)n(e)g(an)n(y)f Fv(t)1949 3035 y Fr(1)1986 3023 y Fv(;)28 b(t)2067 3035 y Fr(2)2142 3023 y Fs(2)37 b Fw([)p Fs(\000)p Fv(n;)27 b(n)p Fw(],)39 b(satisfying)c Fv(t)2965 3035 y Fr(1)3040 3023 y Fs(\024)i Fv(t)3172 3035 y Fr(2)3245 3023 y Fw(and)515 3122 y Fs(j)p Fv(t)568 3134 y Fr(1)623 3122 y Fs(\000)18 b Fv(t)736 3134 y Fr(2)774 3122 y Fs(j)23 b(\024)f Fw(1.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)556 3348 y Fv(u)p Fw(\()p Fv(t)666 3360 y Fr(2)703 3348 y Fw(\))q Fs(\000)q Fv(u)p Fw(\()p Fv(t)912 3360 y Fr(1)949 3348 y Fw(\))c(=)e Fs(\000)p Fv(\027)1217 3235 y Fp(Z)1300 3255 y Fu(t)1325 3263 y Ff(2)1263 3423 y Fu(t)1288 3431 y Ff(1)1376 3348 y Fv(Au)p Fw(\()p Fv(s)p Fw(\))14 b Fv(ds)q Fs(\000)1752 3235 y Fp(Z)1835 3255 y Fu(t)1860 3263 y Ff(2)1798 3423 y Fu(t)1823 3431 y Ff(1)1910 3348 y Fv(B)t Fw(\()p Fv(u)p Fw(\()p Fv(s)p Fw(\))p Fv(;)28 b(u)p Fw(\()p Fv(s)p Fw(\)\))14 b Fv(ds)q Fw(+)2557 3284 y Fs(p)p 2627 3284 47 4 v 2627 3348 a Fv(\027)19 b Fw(\()p Fv(\020)6 b Fw(\()p Fv(t)2823 3360 y Fr(2)2861 3348 y Fw(\))q Fs(\000)q Fv(\020)g Fw(\()p Fv(t)3064 3360 y Fr(1)3102 3348 y Fw(\)\))42 b(\(4.3\))515 3571 y(\(strictly)33 b(sp)r(eaking,)i(since)e Fv(u)g Fw(is)h(a)f(w)n(eak)f(solution,)j(then)f Fv(\020)40 b Fw(should)34 b(b)r(e)g(replaced)e(b)n(y)h(a)515 3670 y(suitable)i(pro)r(cess)1137 3648 y(~)1129 3670 y Fv(\020)7 b Fw(,)37 b(distributed)f(as)f Fv(\020)6 b Fw(.)61 b(This)36 b(inaccuracy)e(mak)n(es)g(no)h(di\013erence)h(for)515 3770 y(what)27 b(follo)n(ws\).)639 3869 y(The)h Fv(H)7 b Fw(-norm)26 b(of)i(the)g(\014rst)f(in)n(tegral)g(in)g(the)h(r.h.s.)37 b(of)d(\(4.3\))27 b(is)h(b)r(ounded)g(b)n(y)564 4095 y Fv(\027)624 3978 y Fp(\022)685 3982 y(Z)768 4002 y Fu(t)793 4010 y Ff(2)731 4170 y Fu(t)756 4178 y Ff(1)844 4095 y Fs(k)p Fv(u)p Fs(k)976 4107 y Fr(2)1025 4095 y Fv(ds)1107 3978 y Fp(\023)1192 4095 y Fs(\024)22 b Fv(\027)5 b Fs(j)p Fv(t)1378 4107 y Fr(2)1434 4095 y Fs(\000)18 b Fv(t)1547 4107 y Fr(1)1585 4095 y Fs(j)1618 4038 y Ff(1)p 1618 4047 29 3 v 1618 4081 a(2)1674 4028 y Fp(\000)1726 3982 y(Z)1809 4002 y Fu(t)1834 4010 y Ff(2)1772 4170 y Fu(t)1797 4178 y Ff(1)1884 4095 y Fs(k)p Fv(u)p Fs(k)2016 4061 y Fr(2)2016 4115 y(2)2066 4095 y Fv(ds)2148 4028 y Fp(\001)2196 4022 y Ff(1)p 2196 4031 V 2196 4065 a(2)2261 4095 y Fs(\024)23 b Fv(\027)5 b Fs(j)p Fv(t)2448 4107 y Fr(2)2504 4095 y Fs(\000)18 b Fv(t)2617 4107 y Fr(1)2654 4095 y Fs(j)2687 4038 y Ff(1)p 2687 4047 V 2687 4081 a(2)2744 4028 y Fp(\000)2795 3982 y(Z)2878 4002 y Fu(n)2842 4170 y Fo(\000)p Fu(n)2952 4095 y Fs(k)p Fv(u)p Fs(k)3084 4061 y Fr(2)3084 4115 y(2)3134 4095 y Fv(ds)3216 4028 y Fp(\001)3264 4022 y Ff(1)p 3264 4031 V 3264 4065 a(2)3307 4095 y Fv(:)515 4318 y Fw(Due)28 b(to)f(\(2.12\))o(,)1406 4464 y Fq(E)1469 4397 y Fp(\000)1527 4351 y(Z)1610 4372 y Fu(n)1573 4540 y Fo(\000)p Fu(n)1684 4464 y Fs(k)p Fv(u)p Fs(k)1816 4430 y Fr(2)1816 4485 y(2)1851 4397 y Fp(\001)1899 4392 y Ff(1)p 1899 4401 V 1899 4434 a(2)1965 4464 y Fs(\024)c Fw(\()p Fv(nL)2192 4430 y Fo(\000)p Fr(2)2280 4464 y Fv(B)2343 4476 y Fr(1)2381 4464 y Fw(\))2423 4408 y Ff(1)p 2423 4417 V 2423 4450 a(2)2465 4464 y Fv(:)639 4663 y Fw(Since)j Fs(k)p Fv(B)t Fw(\()p Fv(u;)h(u)p Fw(\))p Fs(k)1215 4675 y Fo(\000)p Fr(1)1326 4663 y Fs(\024)c Fv(C)6 b Fs(k)p Fv(u)p Fs(k)1611 4633 y Fr(2)1611 4684 y(1)1647 4663 y Fw(,)26 b(then)g Fv(H)1959 4633 y Fo(\000)p Fr(1)2047 4663 y Fw(-norm)f(of)g(the)h(second)e(in)n(tegral)g(is)h(b)r (oun-)515 4763 y(ded)j(b)n(y)978 4937 y Fs(j)p Fv(t)1031 4949 y Fr(2)1087 4937 y Fs(\000)18 b Fv(t)1200 4949 y Fr(1)1237 4937 y Fs(j)43 b Fw(sup)1274 5011 y Fr([)p Fu(t)1318 5019 y Ff(1)1350 5011 y Fu(;)11 b(t)1406 5019 y Ff(2)1439 5011 y Fr(])1472 4937 y Fs(k)p Fv(u)p Fw(\()p Fv(t)p Fw(\))p Fs(k)1698 4903 y Fr(2)1698 4958 y(1)1757 4937 y Fs(\024)23 b(j)p Fv(t)1898 4949 y Fr(2)1953 4937 y Fs(\000)18 b Fv(t)2066 4949 y Fr(1)2104 4937 y Fs(j)2179 4833 y Fu(n)p Fo(\000)p Fr(1)2182 4858 y Fp(X)2141 5033 y Fu(m)p Fr(=)p Fo(\000)p Fu(n)2424 4937 y Fw(sup)2357 5011 y Fr([)p Fu(m;m)p Fr(+1])2630 4937 y Fs(k)p Fv(u)p Fw(\()p Fv(t)p Fw(\))p Fs(k)2856 4903 y Fr(2)2856 4958 y(1)2892 4937 y Fv(:)1905 5255 y Fw(14)p eop %%Page: 15 15 15 14 bop 515 523 a Fw(An)33 b(application)g(of)g(the)h(Ito)f(lemma)g (to)g Fs(k)p Fv(u)p Fw(\()p Fv(t)p Fw(\))p Fs(k)2083 493 y Fr(2)2083 544 y(1)2153 523 y Fw(\(cf.)55 b(\(4.6\))33 b(in)g(the)h(next)f(subsection\))515 623 y(b)r(ounds)d(the)h(exp)r (ectation)f(of)g(oscillation)f(of)i Fs(k)p Fv(u)p Fw(\()p Fv(t)p Fw(\))p Fs(k)2218 593 y Fr(2)2218 643 y(1)2284 623 y Fw(on)f(an)n(y)g(segmen)n(t)g([)p Fv(m;)14 b(m)20 b Fw(+)f(1])30 b(b)n(y)515 722 y Fv(C)580 662 y Fs(p)p 649 662 47 4 v 60 x Fv(\027)5 b Fw(.)37 b(So)28 b(due)g(to)f(\(2.11\))o (,)1389 979 y Fq(E)1511 875 y Fu(n)p Fo(\000)p Fr(1)1514 900 y Fp(X)1472 1074 y Fu(m)p Fr(=)p Fo(\000)p Fu(n)1756 979 y Fw(sup)1689 1052 y Fr([)p Fu(m;m)p Fr(+1])1962 979 y Fs(k)p Fv(u)p Fw(\()p Fv(t)p Fw(\))p Fs(k)2188 944 y Fr(2)2188 999 y(1)2247 979 y Fs(\024)c Fv(C)2394 991 y Fr(1)2432 979 y Fv(n:)639 1233 y Fw(Finally)-7 b(,)27 b(due)h(to)f(basic)f(prop)r(erties)g(of)h(the)h(Bro)n(wnian)d (motion,)i(the)h Fv(H)7 b Fw(-norm)26 b(of)h(the)515 1333 y(third)h(term)f(in)h(the)g(righ)n(t-hand)e(side)i(of)34 b(\(4.3\))27 b(is)h(b)r(ounded)g(b)n(y)1683 1459 y Fs(p)p 1753 1459 V 1753 1523 a Fv(\027)19 b Fs(j)p Fv(t)1866 1535 y Fr(2)1921 1523 y Fs(\000)f Fv(t)2034 1535 y Fr(1)2072 1523 y Fs(j)2105 1467 y Ff(1)p 2105 1476 29 3 v 2105 1509 a(3)2147 1523 y Fv(\030)t(;)515 1701 y Fw(where)27 b Fv(\030)g Fs(\025)c Fw(0)k(and)g Fq(E)c Fv(\030)33 b Fs(\024)22 b Fv(C)1415 1713 y Fr(2)1453 1701 y Fw(\()p Fv(n)p Fw(\).)639 1816 y(Let)h(us)f(consider)f(the)i(H\177)-42 b(older)21 b(space)h Fv(C)1886 1786 y Fr(1)p Fu(=)p Fr(3)2004 1749 y Fp(\000)2042 1816 y Fw([)p Fs(\000)p Fv(n;)28 b(n)p Fw(])p Fv(;)f(H)2430 1786 y Fo(\000)p Fr(1)2519 1749 y Fp(\001)2580 1816 y Fw(=:)c Fs(C)2740 1773 y Fr(1)p Fu(=)p Fr(3)2735 1826 y Fu(n)2844 1816 y Fw(.)35 b(Due)23 b(to)f(\(2.11\))515 1916 y(and)27 b(the)h(estimates)f(for)h(the)g (norms)e(of)i(the)g(three)f(terms)h(in)f(the)h(r.h.s.)37 b(of)d(\(4.3\))o(,)1621 2093 y Fq(E)22 b Fs(j)p Fv(u)1755 2105 y Fu(\027)1802 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2870 y Ft(n)1853 2850 y Fs(\024)h Fv(C)2006 2816 y Fo(0)2029 2850 y Fw(\()p Fv(n)p Fw(\))84 b Fs(8)14 b Fv(\027)q(:)515 3028 y Fw(As)38 b(the)g(space)f Fv(Y)1081 3040 y Fu(n)1165 3028 y Fw(is)g(compactly)h(em)n(b)r(edded)g(in)g(the)h(separable)d (space)h Fv(Z)2995 3040 y Fu(n)3078 3028 y Fw(\([VF88],)515 3127 y(Lemma)30 b(X.3.1\),)g(then)h(the)g(family)f(of)g(measures)f Fv(\026)2191 3097 y Fu(n)2191 3148 y(\027)2263 3127 y Fw(=)e Fs(D)r Fv(u)2469 3139 y Fu(\027)2510 3127 y Fs(j)2533 3142 y Fr([)p Fo(\000)p Fu(n;)11 b(n)p Fr(])2771 3127 y Fw(is)30 b(tigh)n(t)g(in)g Fv(Z)3216 3139 y Fu(n)3291 3127 y Fw(b)n(y)515 3227 y(the)e(Prokhoro)n(v)c(theorem.)36 b(Theorem)27 b(3.1)g(is)g(pro)n(v)n(en.)p 3318 3227 4 57 v 3322 3174 50 4 v 3322 3227 V 3372 3227 4 57 v 515 3506 a Fc(4.2)112 b(Pro)s(of)37 b(of)h(Lemma)f(3.2)515 3660 y Fw(Let)24 b(us)g(apply)g(the)g(Ito)g(form)n(ula)g(to)g(the)g (functional)g Fv(\030)2216 3672 y Fu(\027)2258 3660 y Fw(\()p Fv(t)p Fw(\))g(=)e Fs(k)p Fv(U)2562 3672 y 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4938 y Fr(2)2589 4993 y(1)2632 4972 y Fv(:)1905 5255 y Fw(16)p eop %%Page: 17 17 17 16 bop 515 523 a Fw(That)27 b(is,)1404 657 y Fq(E)22 b Fs(k)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(k)1711 623 y Fr(2)1711 678 y(1)1777 657 y Fw(=)1874 601 y(1)p 1874 638 42 4 v 1874 714 a(2)1926 657 y Fv(B)1989 669 y Fr(0)2109 657 y Fs(8)14 b Fv(t)22 b Fs(2)i Fw([0)p Fv(;)14 b Fw(1])p Fv(:)717 b Fw(\(4.8\))639 845 y(No)n(w)31 b(w)n(e)g(return)f(to)i (estimate)f(\(4.6\))o(.)48 b(As)31 b Fs(j)p Fv(\030)2083 857 y Fu(\027)2125 845 y Fw(\()p Fv(t)p Fw(\))21 b Fs(\000)2326 778 y Fp(R)2381 798 y Fr(1)2365 874 y(0)2432 845 y Fv(\030)2468 857 y Fu(\027)2510 845 y Fw(\()p Fv(\034)9 b Fw(\))p Fv(d\034)g Fs(j)31 b(\024)e Fw(osc)42 b Fv(\030)3045 857 y Fu(\027)3086 845 y Fs(j)3109 860 y Fr([0)p Fu(;)11 b Fr(1])3280 845 y Fw(for)515 944 y(0)22 b Fs(\024)h Fv(t)g Fs(\024)g Fw(1,)k(then)h(due)g(to)f(\(4.6\))h(w)n(e)f(ha)n(v)n (e)1354 1108 y(0)22 b Fs(\024)h Fq(E)f Fv(g)1609 1120 y Fu(N)1678 1108 y Fw(\()p Fv(U)1767 1120 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Fw(,)27 b Fv(n)c Fs(2)h Fq(Z)o Fw(,)e(w)n(e)27 b(get)g(that)h(a.s.)1389 1644 y Fs(k)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(k)1633 1610 y Fr(2)1633 1665 y(1)1693 1644 y Fw(=)22 b(\012)97 b(for)27 b(a)p Fv(:)p Fw(a)p Fv(:)96 b(t)23 b Fs(2)h Fq(R)p Fv(:)709 b Fw(\(4.9\))515 1808 y(Here)27 b(\012)h(is)f(a)g (random)g(constan)n(t)g(whic)n(h)g(can)h(b)r(e)g(de\014ned,)g(sa)n(y)-7 b(,)26 b(as)1453 2027 y(\012)d(=)g Fs(E)7 b Fw(\()p Fv(U)i Fw(\))23 b(=)1916 1914 y Fp(Z)1999 1934 y Fr(1)1962 2102 y(0)2050 2027 y Fs(k)p Fv(U)9 b Fw(\()p Fv(t)p Fw(\))p Fs(k)2294 1992 y Fr(2)2294 2047 y(1)2344 2027 y Fv(dt:)515 2237 y Fw(Using)28 b(\(4.8\))g(w)n(e)f(see)g(that)h(exp)r(ectation)f (of)h(\012)g(satis\014es)e(the)i(\014rst)g(relation)e(in)i(\(3.7\).)639 2337 y(Applying)j(the)g(Ito)f(form)n(ula)f(to)h(the)h(functional)g Fs(j)p Fv(U)2312 2349 y Fu(\027)2353 2337 y Fs(j)2376 2306 y Fr(2)2413 2337 y Fw(,)g(w)n(e)f(get)g(a)g(b)r(ound)h(for)f (oscil-)515 2436 y(lation)25 b(of)i Fs(j)p Fv(U)922 2448 y 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Fv(ds)2204 3573 y Fp(\023)2295 3563 y Ft(pq)p 2275 3577 100 3 v 2275 3610 a(q)q Ff(+1)1595 3946 y Fs(\024)1682 3829 y Fp(\022)1744 3946 y Fv(C)1803 3958 y Fr(1)1840 3946 y Fv(p)1882 3912 y Fo(\000)p Fr(1)p Fu(=)p Fr(2)2038 3875 y Fs(p)p 2107 3875 30 4 v 71 x Fv(t)2151 3879 y Fp(\000)2203 3833 y(Z)2286 3854 y Fu(t)2249 4022 y Fr(0)2329 3946 y Fs(k)p Fv(u)p Fw(\()p Fv(s)p Fw(\))p Fs(k)2564 3912 y Fr(2)2564 3967 y(2)2614 3946 y Fv(ds)2696 3879 y Fp(\001)2734 3896 y Fr(1)p Fu(=)p Fr(2)2839 3829 y Fp(\023)2930 3820 y Ft(pq)p 2910 3834 100 3 v 2910 3866 a(q)q Ff(+1)3024 3946 y Fv(:)639 4179 y Fw(Using)28 b(\(3.14\))f(and)g(passing)f(to)h(the)h(limit)g(as)e Fv(p)d Fs(!)g(1)p Fw(,)28 b(w)n(e)f(see)f(that)i Fv(g)s Fw(\()p Fv(t)p Fw(\))23 b(=)g(0)k(for)f(all)515 4278 y(0)c Fs(\024)h Fv(t)g Fs(\024)g Fv(T)12 b Fw(,)26 b(as)h(stated.)p 515 4332 1146 4 v 607 4385 a Fn(5)642 4409 y Fm(In)21 b(the)g(statemen)n(t)h(of)e(the)h(theorem)f(in)g([LL97)q(])g(the)h(v)l (alue)g(of)f(the)i(constan)n(t)g(that)g(b)r(ounds)f(the)g(norm)515 4488 y(of)29 b(the)h(Sob)r(olev)g(em)n(b)r(edding)g(is)f(giv)n(en)h (incorrectly)-6 b(,)31 b(but)f(righ)n(t)f(v)l(alue)h(of)f(the)h (constan)n(t)i(immediately)515 4566 y(follo)n(ws)22 b(from)g(the)j(pro) r(of)e(\(whic)n(h)h(is)f(correct\).)1905 5255 y Fw(18)p eop %%Page: 19 19 19 18 bop 515 523 a Fy(References)515 705 y Fw([BKL02])64 b(J.)30 b(Bricmon)n(t,)g(A.)g(Kupiainen,)g(and)g(R.)h(Lefev)n(ere,)e FB(Exp)l(onential)k(mixing)f(for)885 805 y(the)f(2D)f(sto)l(chastic)i (Navier{Stokes)g(dynamics)p Fw(,)e(Comm.)f(Math.)f(Ph)n(ys.)g Fx(230)885 904 y Fw(\(2002\),)e(no.)i(1,)f(87{132.)515 1069 y([BKM84])40 b(J.)j(Beale,)48 b(T.)43 b(Kato,)k(and)d(A.)g(Ma)5 b(jda,)47 b FB(R)l(emarks)e(on)g(the)f(br)l(e)l(akdown)i(of)885 1169 y(smo)l(othness)g(for)g(the)g(3-D)f(Euler)h(e)l(quations)p Fw(,)k(Comm.)45 b(Math.)g(Ph)n(ys.)f Fx(94)885 1268 y Fw(\(1984\),)26 b(61{66.)515 1433 y([CF88])126 b(P)-7 b(.)34 b(Constan)n(tin)f(and)h(C.)h(F)-7 b(oias,)35 b FB(Navier-Stokes)h(Equations)p Fw(,)h(Univ)n(ersit)n(y)c(of)885 1533 y(Chicago)26 b(Press,)g(Chicago,)h(1988.)515 1698 y([Che98])97 b(J.-Y.)19 b(Chemin,)j FB(Perfe)l(ct)h(Inc)l(ompr)l (essible)g(Fluids)p Fw(,)f(Oxford)d(Univ)n(ersit)n(y)g(Press,)885 1797 y(Oxford,)27 b(1998.)515 1962 y([Dud89])85 b(R.)29 b(M.)g(Dudley)-7 b(,)31 b FB(R)l(e)l(al)g(Analysis)g(and)h(Pr)l(ob)l (ability)p Fw(,)g(W)-7 b(adsw)n(orth)28 b(&)g(Bro)r(oks/)885 2062 y(Cole,)f(P)n(aci\014c)f(Gro)n(v)n(e,)g(California,)h(1989.)515 2227 y([EM01])107 b(W.)28 b(E)f(and)h(J.)f(C.)h(Mattingly)-7 b(,)27 b FB(Er)l(go)l(dicity)32 b(for)f(the)f(Navier{Stokes)h(e)l (quation)885 2326 y(with)41 b(de)l(gener)l(ate)h(r)l(andom)f(for)l (cing:)62 b(\014nite)40 b(dimensional)j(appr)l(oximation)p Fw(,)885 2426 y(Comm.)27 b(Pure)g(Appl.)h(Math)g Fx(54)f Fw(\(2001\),)g(1386{1402.)515 2591 y([EMS01])61 b(W.)41 b(E,)f(J.)g(C.)g(Mattingly)-7 b(,)44 b(and)c(Y)-7 b(a.)40 b(G.)h(Sinai,)j FB(Gibbsian)f(dynamics)f(and)885 2691 y(er)l(go)l(dicity)24 b(for)g(the)f(sto)l(chastic)l(al)t(ly)i(for)l(c)l (e)l(d)f(Navier{Stokes)g(e)l(quation)p Fw(,)e(Comm.)885 2790 y(Math.)28 b(Ph)n(ys.)e Fx(224)h Fw(\(2001\),)f(83{106.)515 2955 y([Gal01])110 b(G.)57 b(Galla)n(v)n(otti,)63 b FB(F)-6 b(oundations)57 b(of)g(Fluid)h(Dynamics)p Fw(,)65 b(Springer-V)-7 b(erlag,)885 3055 y(Berlin,)27 b(2001.)515 3220 y([Har64])103 b(Ph.)31 b(Hartman,)i FB(Or)l(dinary)h(Di\013er)l(ential)g(Equations)p Fw(,)g(John)d(Wiley)h(&)f(Sons,)885 3319 y(New)d(Y)-7 b(ork,)27 b(1964.)515 3484 y([IW89])125 b(N.)23 b(Ik)n(eda)f(and)h(S.)g (W)-7 b(atanab)r(e,)24 b FB(Sto)l(chastic)i(Di\013er)l(ential)g (Equations)g(and)g(Dif-)885 3584 y(fusion)k(Pr)l(o)l(c)l(esses)p Fw(,)e(North-Holland,)f(Amsterdam,)h(1989.)515 3749 y([KM80])99 b(R.)34 b(H.)h(Kraic)n(hnan)d(and)i(D.)h(Mon)n(tgomery)-7 b(,)34 b FB(Two-dimensional)k(turbulenc)l(e)p Fw(,)885 3848 y(Rep.)28 b(Prog.)e(Ph)n(ys.)g Fx(43)h Fw(\(1980\),)g(547{619.)515 4013 y([Kra67])100 b(R.)20 b(H.)h(Kraic)n(hnan,)f FB(Inertial)j(r)l (anges)g(in)h(two-dimensional)g(turbulenc)l(e)p Fw(,)d(Ph)n(ys.)885 4113 y(Fluids)28 b Fx(10)f Fw(\(1967\),)f(1417{1423.)515 4278 y([KS00])129 b(S.)19 b(B.)g(Kuksin)f(and)h(A.)g(Shiriky)n(an,)h FB(Sto)l(chastic)i(dissip)l(ative)i(PDE's)f(and)f(Gibbs)885 4378 y(me)l(asur)l(es)p Fw(,)27 b(Comm.)h(Math.)g(Ph)n(ys.)e Fx(213)h Fw(\(2000\),)f(291{330.)515 4542 y([KS02])129 b(S.)40 b(B.)f(Kuksin)g(and)h(A.)g(Shiriky)n(an,)i FB(Coupling)g(appr)l (o)l(ach)h(to)d(white-for)l(c)l(e)l(d)885 4642 y(nonline)l(ar)30 b(PDE's)p Fw(,)f(J.)e(Math.)h(Pures)e(Appl.)i Fx(81)g Fw(\(2002\),)e(567{602.)515 4807 y([KS03])129 b(S.)27 b(B.)f(Kuksin)g(and)g(A.)h(Shiriky)n(an,)e FB(Some)k(limiting)h(pr)l (op)l(erties)f(of)h(r)l(andomly)885 4907 y(for)l(c)l(e)l(d)g(2D)f (Navier{Stokes)i(e)l(quations)p Fw(,)c(Pro)r(ceedings)e(A)j(of)f(the)g (Ro)n(y)n(al)e(So)r(ci-)885 5006 y(et)n(y)i(of)h(Edin)n(burgh)e (\(2003\),)h(to)g(app)r(ear.)1905 5255 y(19)p eop %%Page: 20 20 20 19 bop 515 523 a Fw([Kuk02])85 b(S.)21 b(B.)h(Kuksin,)g FB(Er)l(go)l(dic)j(the)l(or)l(ems)g(for)f(2D)g(statistic)l(al)h(hydr)l (o)l(dynamics)p Fw(,)g(Rev.)885 623 y(Math.)j(Ph)n(ys.)e Fx(14)h Fw(\(2002\),)g(585{600.)515 789 y([LL97])136 b(E.)26 b(H.)g(Lieb)h(and)f(M.)g(Loss,)g FB(A)n(nalysis)p Fw(,)h(Graduate)e(Studies)i(in)f(Mathematics,)885 888 y(v)n(ol.)h(14,)f(AMS,)j(Pro)n(vidence,)c(1997.)515 1054 y([Ons49])96 b(L.)52 b(Onsager,)k FB(Statistic)l(al)c(hydr)l(o)l (dynamics)p Fw(,)61 b(Nuo)n(v)n(o)50 b(Cimen)n(to)i(Suppl.)g Fx(6)885 1154 y Fw(\(1949\),)20 b(279{287,)e(also)h(in)h(\\The)g (collected)g(w)n(orks)e(of)i(Lars)e(Onsager",)h(W)-7 b(orld)885 1254 y(Scien)n(ti\014c,)28 b(1996.)515 1420 y([RS94])133 b(R.)39 b(Rob)r(ert)g(and)g(J.)g(Sommeria,)i FB(Statistic)l(al)f(e)l(quilibrium)h(states)f(for)h(two-)885 1519 y(dimensional)31 b(\015ows)p Fw(,)d(J.)g(Fluid)g(Mec)n(h.)g Fx(229)e Fw(\(1994\),)h(291{310.)515 1685 y([Shi02])125 b(A.)57 b(Shiriky)n(an,)63 b FB(A)n(nalyticity)57 b(of)h(solutions)f (for)g(r)l(andomly)h(for)l(c)l(e)l(d)f(two-)885 1785 y(dimensional)c(Navier{Stokes)g(e)l(quations)p Fw(,)58 b(Russian)51 b(Math.)g(Surv)n(eys)g Fx(57)885 1885 y Fw(\(2002\),)26 b(785{799.)515 2051 y([VF88])124 b(M.)23 b(I.)g(Vishik)f(and)h(A.)g(V.)g(F)-7 b(ursik)n(o)n(v,)22 b FB(Mathematic)l(al)27 b(Pr)l(oblems)f(in)g(Statistic)l(al)885 2150 y(Hydr)l(ome)l(chanics)p Fw(,)j(Klu)n(w)n(er,)e(Dordrec)n(h)n(t,)f (1988.)515 2316 y([Y)-7 b(ud63])93 b(V.)32 b(I.)h(Y)-7 b(udo)n(vic)n(h,)32 b FB(Non)i(stationary)h(\015ow)f(of)h(an)f(ide)l (al)h(inc)l(ompr)l(essible)h(\015uid)885 2416 y(\(in)30 b(Russ.\))p Fw(,)d(Zh.)g(Vyc)n(h.)h(Mat.)g Fx(3)f Fw(\(1963\),)f (1032{1066.)639 2685 y Fx(Sergei)31 b(B.)g(Kuksin)639 2785 y FB(Dep)l(artment)e(of)i(Mathematics)639 2885 y(Heriot-Watt)f (University)639 2984 y(Edinbur)l(gh)h(EH14)g(4AS)639 3084 y(Sc)l(otland,)g(UK)724 3183 y Fw(and)639 3283 y FB(Steklov)f(Institute)f(of)h(Mathematics)639 3383 y(8)g(Gubkina)h (St.,)f(117966)i(Mosc)l(ow)639 3482 y(R)n(ussia)639 3607 y(E-mail:)40 b Fa(kuksin@ma.hw.ac.)o(uk)639 3731 y FB(web-addr)l(ess)p Fw(:)f Fa(www.ma.hw.ac.uk)o(/)p Fs(\030)p Fa(k)o(uk)o(sin)o(/)1905 5255 y Fw(20)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0306030517132--