Content-Type: multipart/mixed; boundary="-------------9911301239551" This is a multi-part message in MIME format. ---------------9911301239551 Content-Type: text/plain; name="99-454.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-454.keywords" Quasigeostrophic equation, random dynamical systems, random fixed point theorem, stable stationary solutions ---------------9911301239551 Content-Type: application/postscript; name="stable2d.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="stable2d.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: stable2d.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips stable2d -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.11.24:1514 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S 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Fo(Intr)n(oduction)456 2687 y Fp(The)24 b(quasigeostrophic)d(\(QG\))j (mo)r(del)g(is)g(a)g(simpli\014ed)g(geoph)n(ysical)e(\015uid)i(mo)r (del)h(at)e(asymp-)456 2812 y(totically)38 b(high)i(rotation)e(rate)g (or)h(at)g(small)g(Rossb)n(y)f(n)n(um)n(b)r(er.)h(It)h(is)f(deriv)n(ed) f(as)h(an)g(ap-)456 2936 y(pro)n(ximation)27 b(of)h(the)h(rotating)e (shallo)n(w)h(w)n(ater)f(equations)h(b)n(y)g(a)g(con)n(v)n(en)n(tional) f(asymptotic)456 3061 y(expansion)e(for)g(small)h(Rossb)n(y)f(n)n(um)n (b)r(er)h([13)o(].)g(The)g(lo)n(w)n(est)f(order)g(appro)n(ximation)f (giv)n(es)h(the)456 3185 y(barotropic)30 b(QG)i(equation,)f(whic)n(h)h (is)g(also)f(the)i(conserv)-5 b(ation)30 b(la)n(w)i(for)f(the)i (zero{th)e(order)456 3310 y(p)r(oten)n(tial)d(v)n(orticit)n(y)-7 b(.)28 b(W)-7 b(arn)28 b(et)h(al.)f([19)o(])h(and)f(V)-7 b(allis)28 b([21)o(])h(emphasize)f(that)h(this)g(asymptotic)456 3434 y(expansion)24 b(is)h(generally)f(secular)h(for)g(all)g(but)h(the) 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(del)h([13)o(,)g(14)o(,)f(12)o(])1127 4636 y(\001)p Fn( )1250 4648 y Fm(t)1298 4636 y Fp(+)18 b Fn(J)8 b Fp(\()p Fn( )s(;)14 b Fp(\001)p Fn( )s Fp(\))19 b(+)f Fn(\014)t( )1926 4648 y Fm(x)1991 4636 y Fp(=)23 b Fn(\027)5 b Fp(\001)2194 4601 y Fl(2)2232 4636 y Fn( )21 b Fk(\000)d Fn(r)r Fp(\001)p Fn( )23 b Fp(+)2691 4615 y(_)2658 4636 y Fn(W)2736 4648 y Fl(2)456 4636 y Fp(\(1\))p 456 4813 499 4 v 555 4921 a Fj(Date)5 b Fr(:)23 b(Jan)n(uary)i(1,)e(1999.)555 5025 y(1991)k Fj(Mathematics)f(Subje)l(ct)f(Classi\014c)l(ation.)37 b Fr(Primary)21 b(60H25,)j(47H10;)g(Secondary)i(34D35.)555 5112 y Fj(Key)19 b(wor)l(ds)j(and)e(phr)l(ases.)67 b Fr(Quasigeostrophic)18 b(equation,)h(random)d(dynamical)h(systems,)g (random)f(\014xed)456 5216 y(p)r(oin)n(t)24 b(theorem,)f(stable)h (stationary)h(solutions.)1933 5315 y Fl(1)p eop %%Page: 2 2 2 1 bop 456 238 a Fl(2)378 b(JINQIA)n(O)29 b(DUAN,)g(PETER)f(E.)h (KLOEDEN,)f(AND)h(BJ)2417 223 y(\177)2408 238 y(ORN)g(SCHMALFUSS)456 425 y Fp(on)23 b(a)g(rectangle)f Fn(D)j Fp(=)e(\(0)p Fn(;)14 b Fp(1\))c Fk(\002)g Fp(\(0)p Fn(;)k Fp(1\))22 b Fk(\032)h Fi(R)1784 395 y Fl(2)1851 425 y Fp(,)h(where)f Fn( )s Fp(\()p Fn(x;)14 b(y)s(;)g(t)p Fp(\))24 b(is)f(the)h(stream)f (function,)h Fn(\014)k Fk(\025)456 550 y Fp(0)h(the)h(meridional)f (gradien)n(t)f(of)i(the)g(Coriolis)e(parameter,)g Fn(\027)35 b(>)29 b Fp(0)g(the)h(viscous)f(dissipation)456 674 y(constan)n(t,)c Fn(r)j(>)e Fp(0)f(the)h(Ekman)f(dissipation)h(constan)n(t,)2263 653 y(_)2230 674 y Fn(W)2308 686 y Fl(2)2371 674 y Fp(the)g(noise)g (due)g(to)f(wind)h(forcing,)456 799 y(and)h Fn(J)37 b Fp(the)28 b(Jacobian)e(op)r(erator,)h(whic)n(h)h(is)g(de\014ned)g(b)n (y)g Fn(J)8 b Fp(\()p Fn(f)t(;)14 b(g)s Fp(\))28 b(=)f Fn(f)2677 811 y Fm(x)2719 799 y Fn(g)2759 811 y Fm(y)2817 799 y Fk(\000)18 b Fn(f)2941 811 y Fm(y)2981 799 y Fn(g)3021 811 y Fm(x)3062 799 y Fp(.)28 b(Equation)456 923 y(\(1\))f(can)h(b)r(e) g(rewritten)f(in)h(terms)f(of)h(the)f(v)n(orticit)n(y)g Fn(q)k Fp(=)c(\001)p Fn( 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(large)e(scale)g(quasigeostrophic)f(motion.)i(Boundary)f(condi-)456 3348 y(tions)22 b(for)f(the)i(quasigeostrophic)d(mo)r(del)i(are)g(not)g (quite)h(w)n(ell)f(understo)r(o)r(d,)g(since)g(this)g(mo)r(del)456 3472 y(describ)r(es)30 b(large)g(scale)g(\015o)n(ws)h(while)g(b)r (oundary)f(conditions)h(also)f(in)n(v)n(olv)n(e)g(small)h(scale)f(mo-) 456 3597 y(tions.)k(F)-7 b(or)33 b(this)h(reason)f(w)n(e)g(b)r(eliev)n (e)h(that,)h(under)f(the)g(random)f(media)h(or)f(random)g(wind)456 3721 y(forcing)20 b(conditions,)i(a)f(random)f(slip)i(condition)f(ma)n (y)g(b)r(e)h(more)f(appropriate)f(than)i(the)g(usual)456 3846 y(slip)h(b)r(oundary)g(conditions)h([14)o(])g(for)f(the)h (deterministic)g(quasigeostrophic)d(mo)r(del.)j(W)-7 b(e)25 b(also)456 3970 y(note)39 b(that)g(the)h(Neumann)g(form)f(for)g (the)g(b)r(oundary)g(condition)g(on)g Fn(q)k Fp(is)c(mathematical)456 4095 y(con)n(v)n(enience.)456 4219 y(In)29 b(this)g(article)g(w)n(e)f 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Fp(=)22 b Fn(id)1403 1175 y Fl(\012)1455 1163 y Fn(;)180 b(\022)1697 1175 y Fm(s)p Fl(+)p Fm(t)1831 1163 y Fp(=)23 b Fn(\022)1958 1175 y Fm(s)2012 1163 y Fk(\016)18 b Fn(\022)2111 1175 y Fm(t)2223 1163 y Fp(for)27 b(all)g Fn(s;)14 b(t)23 b Fk(2)h Fi(R)p Fn(;)456 1352 y Fp(suc)n(h)e(that)i(the)f(mapping)g(\() p Fn(t;)14 b(!)s Fp(\))23 b Fk(7!)g Fn(\022)1643 1364 y Fm(t)1672 1352 y Fn(!)j Fp(from)d Fi(R)9 b Fk(\002)g Fp(\012)29 b(in)n(to)23 b(\012)g(is)g(\()p Fk(B)s Fp(\()p Fi(R)p Fp(\))9 b Fk(\002)g(F)f Fn(;)14 b Fk(F)8 b Fp(\){measurable)456 1477 y(and)19 b(the)h(probabilit)n(y)e(measure)h Fi(P)f Fp(is)i(ergo)r(dic)e(\(hence)i(in)n(v)-5 b(arian)n(t\))19 b(with)h(resp)r(ect)f(to)g(the)h(\015o)n(w)f Fn(\022)r Fp(.)456 1726 y(A)35 b Fh(r)l(andom)j(dynamic)l(al)h(system)c Fp(consists)g(of)h(a)f(metric)g(dynamical)g(system)h(\()p Fn(\022)r(;)14 b Fk(P)7 b Fp(\))36 b(and)f(a)456 1850 y(co)r(cycle)27 b(mapping)g Fn(')h Fp(:)g Fi(R)c Fk(\002)18 b Fp(\012)g Fk(\002)g Fn(H)35 b Fk(7!)28 b Fn(H)7 b Fp(,)27 b(i.e.,)h(satisfying)651 2040 y Fn(')p Fp(\(0)p Fn(;)14 b(!)s(;)g Fk(\001)p Fp(\))23 b(=)f(id)1142 2052 y Fm(H)1205 2040 y Fn(;)97 b(')p Fp(\()p Fn(s)19 b Fp(+)f Fn(t;)c(!)s(;)g Fk(\001)p Fp(\))23 b(=)g Fn(')p Fp(\()p Fn(s;)14 b(\022)2078 2052 y Fm(t)2107 2040 y Fn(;)g Fk(\001)p Fp(\))19 b Fk(\016)f Fn(')p Fp(\()p Fn(t;)c(!)s(;)g Fk(\001)p Fp(\))84 b(for)27 b(all)g Fn(s;)14 b(t)23 b Fk(2)g Fi(R)3165 2006 y Fl(+)3226 2040 y Fn(;)456 2230 y Fp(that)g(is)g(\()p Fk(B)s Fp(\()p Fi(R)p Fp(\))9 b Fk(\012)g(F)23 b(\012)9 b(B)s Fp(\()p Fn(H)e Fp(\))p Fn(;)14 b Fk(B)s Fp(\()p Fn(H)7 b Fp(\)\)-measurable.)20 b(The)j(mapping)g Fn(')g Fp(describ)r(es)g(the)g(dynam-)456 2354 y(ics)h(of)h(the)g(system)f(in)h(the)g(state)f(space)g Fn(H)7 b Fp(,)25 b(whic)n(h)g(will)f(b)r(e)h(a)g(separable)e(Hilb)r (ert)i(space)f(with)456 2479 y(inner)j(pro)r(duct)h(\()p Fk(\001)p Fn(;)14 b Fk(\001)p Fp(\))28 b(and)f(norm)g Fk(j)19 b(\001)f(j)28 b Fp(=)1759 2408 y Fe(p)p 1842 2408 148 4 v 71 x Fp(\()p 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y(of)d(these)h(states)f(is)g(indep)r(enden)n(t)i(of)e Fn(t)p Fp(.)456 5216 y(The)g(follo)n(wing)g(theorem)g(is)g(a)h(sp)r (ecial)f(case)g(of)g(a)g(random)g(\014xed)h(p)r(oin)n(t)f(theorem)g (due)h(to)p eop %%Page: 4 4 4 3 bop 456 238 a Fl(4)378 b(JINQIA)n(O)29 b(DUAN,)g(PETER)f(E.)h (KLOEDEN,)f(AND)h(BJ)2417 223 y(\177)2408 238 y(ORN)g(SCHMALFUSS)456 425 y Fs(Theorem)h(2.1.)40 b Fh(L)l(et)21 b Fn(')h Fh(b)l(e)f(a)h(r)l (andom)g(dynamic)l(al)h(system)e(with)h(a)f(sep)l(ar)l(able)i(Banach)g (sp)l(ac)l(e,)456 550 y Fp(\()p Fn(H)r(;)14 b Fk(j)g(\001)g(j)p Fp(\))28 b Fh(as)h(its)e(state)h(sp)l(ac)l(e)g Fp(\()p Fn(H)r(;)14 b Fk(j)h(\001)f(j)p Fp(\))p Fh(.)29 b(Assume)d(that)i(the)g (mapping)h Fn(x)g Fk(7!)e Fh(is)i(c)l(ontinuous)d(for)456 674 y(every)32 b Fn(t)f Fk(\025)f Fp(0)h Fh(and)h Fn(!)i Fk(2)d Fp(\012)p Fh(.)h(In)f(addition,)i(let)e(that)g Fk(X)44 b Fp(=)31 b Fk(fX)12 b Fp(\()p Fn(!)s Fp(\))p Fk(g)2581 686 y Fm(!)r Fg(2)p Fl(\012)2752 674 y Fh(b)l(e)31 b(a)g(close)l(d)i(r)l(andom)456 799 y(set)c(in)h Fn(H)36 b Fh(such)30 b(that)1119 969 y Fi(E)161 b Fp(sup)1182 1043 y Fm(h)1221 1051 y Fd(1)1260 1043 y Fg(6)p Fl(=)p Fm(h)1350 1051 y Fd(2)1382 1043 y Fg(2X)9 b Fl(\()p Fm(!)r Fl(\))1592 969 y Fp(log)1723 913 y Fk(j)p Fn(')p Fp(\(1)p Fn(;)14 b(!)s(;)g(h)2051 925 y Fl(1)2088 913 y Fp(\))19 b Fk(\000)f Fn(')p Fp(\(1)p Fn(;)c(!)s(;)g(h)2527 925 y Fl(2)2563 913 y Fp(\))p Fk(j)p 1723 950 896 4 v 2012 1026 a(j)p Fn(h)2083 1038 y Fl(1)2139 1026 y Fk(\000)k Fn(h)2270 1038 y Fl(2)2307 1026 y Fk(j)2652 969 y Fn(<)k Fp(0)-2325 b(\(7\))456 1180 y Fh(and)28 b(that)g(the)g(r)l(e)l(al)g (value)l(d)h(r)l(andom)g(variable)h(de\014ne)l(d)e(by)g Fn(!)j Fk(!)d Fp(sup)2639 1200 y Fm(h)p Fg(2X)9 b Fl(\()p Fm(!)r Fl(\))2893 1180 y Fk(j)p Fn(h)p Fk(j)27 b Fh(is)i(temp)l(er)l(e) l(d.)456 1304 y(Mor)l(e)l(over,)i(assume)f(that)1232 1599 y Fp(sup)1198 1673 y Fm(t)p Fg(2)p Fl([0)p Fm(;)p Fl(1])1538 1599 y Fp(sup)1405 1673 y Fm(h)1444 1681 y Fd(1)1477 1673 y Fg(6)p Fl(=)p Fm(h)1567 1681 y Fd(2)1599 1673 y Fg(2X)9 b Fl(\()p Fm(!)r Fl(\))1819 1543 y Fk(j)p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(h)2135 1555 y Fl(1)2173 1543 y Fp(\))k Fk(\000)g Fn(')p Fp(\()p Fn(t;)c(!)s(;)g(h)2599 1555 y Fl(2)2636 1543 y Fp(\))p Fk(j)p 1819 1580 873 4 v 2097 1656 a(j)p Fn(h)2168 1668 y Fl(1)2223 1656 y Fk(\000)k Fn(h)2354 1668 y Fl(2)2391 1656 y Fk(j)456 1599 y Fp(\(8\))456 1810 y Fh(is)30 b(temp)l(er)l(e)l(d)g(with)g(r)l (esp)l(e)l(ct)f(to)h Fk(f)p Fn(\022)1523 1822 y Fm(n)1568 1810 y Fk(g)1610 1822 y Fm(n)p Fg(2)p Ff(Z)1738 1810 y Fh(.)456 1934 y(Then)e(ther)l(e)g(exists)g(a)g Fn(\022)r Fh(-invariant)h(set)e(of)i(ful)t(l)g Fi(P)p Fh({me)l(asur)l(e)e Fp(\012)2441 1904 y Fg(0)2492 1934 y Fh(and)h(a)g(r)l(andom)h(variable) h Fn(X)3407 1904 y Fg(\003)456 2059 y Fh(satisfying)h(\(6\))f(on)g Fp(\012)1144 2029 y Fg(0)1167 2059 y Fh(.)g(F)-6 b(uthermor)l(e,)30 b Fn(X)1768 2029 y Fg(\003)1835 2059 y Fh(is)g(exp)l(onential)t(ly)h (attr)l(acting,)f(i.e.,)1182 2227 y Fp(lim)1161 2277 y Fm(t)p Fg(!1)1332 2227 y Fk(j)p Fn(')p Fp(\()p Fn(t;)14 b(!)s(;)g(X)7 b Fp(\()p Fn(!)s Fp(\)\))19 b Fk(\000)f Fn(X)2005 2193 y Fg(\003)2042 2227 y Fp(\()p Fn(\022)2113 2239 y Fm(t)2143 2227 y Fn(!)s Fp(\))p Fk(j)23 b Fp(=)f(0)p Fn(;)184 b Fh(a.s.)456 2395 y(exp)l(onential)t(ly)31 b(fast)f(for)g(any)g(me)l(asur)l(able)h(sele)l(ction)f Fn(X)36 b Fh(of)31 b Fk(X)12 b Fh(.)456 2560 y Fp(2.1.)46 b Fs(Linear)36 b(sto)s(c)m(hastic)f(ev)m(olution)f(equations.)41 b Fp(In)30 b(the)h(follo)n(wing)e(w)n(e)h(will)h(consider)456 2685 y(the)d(motion)g(relativ)n(e)g(to)g(a)g(spatially)g(constan)n(t)f (\015o)n(w.)h(By)g(the)h(particular)e(structure)h(of)g(the)456 2809 y(co)r(e\016cien)n(ts)33 b(this)i(spatially)e(constan)n(t)g(\015o) n(w)h(can)f(b)n(y)h(calculated)f(separately)-7 b(.)33 b(T)-7 b(o)34 b(\014nd)g(this)456 2934 y(spatially)25 b(constan)n(t)g(\015o)n(w)g(w)n(e)h(ha)n(v)n(e)f(to)h(solv)n(e)e(a)i (simpler)g(equation.)f(Henceforth)h(w)n(e)g(tak)n(e)f(for)456 3058 y Fn(H)i Fp(the)21 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4428 y(Let)g(\001)h(b)r(e)g(the)g(Laplacian)f (op)r(erator)e(on)j Fn(D)r Fp(.)g(The)f(b)r(oundary)g(v)-5 b(alue)27 b(problem)1338 4610 y Fk(\000)p Fn(\027)5 b Fp(\001)p Fn(u)23 b Fp(=)g Fn(f)t(;)1959 4554 y(@)p 1934 4591 99 4 v 1934 4667 a(@)5 b(n)2043 4610 y(u)22 b Fp(=)h Fn(g)86 b Fp(on)27 b Fn(@)5 b(D)456 4795 y Fp(with)26 b Fn(f)34 b Fk(2)26 b Fn(H)7 b Fp(,)26 b Fn(g)i Fk(2)e Fn(L)1130 4807 y Fl(2)1167 4795 y Fp(\()p Fn(U)9 b Fp(\))26 b(and)f Fn(\027)k(>)22 b Fp(0,)j(has)h(a)f(unique)h(solution)f Fn(u)g Fp(=)2705 4774 y(~)2686 4795 y Fn(G)p Fp(\()p Fn(f)t(;)14 b(g)s Fp(\).)26 b(The)g(solution)456 4920 y(op)r(erator)817 4899 y(~)798 4920 y Fn(G)36 b Fp(:)f Fn(H)c Fk(\002)23 b Fn(U)44 b Fk(!)35 b Fn(W)1454 4890 y Fl(2)1442 4941 y(2)1491 4920 y Fp(\()p Fn(D)r Fp(\))h(is)g(a)e(b)r (ounded)i(linear)f(op)r(erator,)e(i.e.,)j(there)f(exists)g(a)456 5044 y(constan)n(t)26 b Fn(c)826 5056 y Fm(G)910 5044 y Fp(suc)n(h)h(that)1325 5213 y Fk(k)1386 5192 y Fp(~)1367 5213 y Fn(G)p Fp(\()p 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Fm(G)2231 1111 y Fk(j)p Fn(f)9 b Fk(j)p Fn(:)456 1298 y Fp(In)33 b(the)g(follo)n(wing)f(w)n(e)h(will)g(denote)h(b)n(y)e Fn(C)1812 1310 y Fm(G;x)1926 1298 y Fp(the)h(constan)n(t)g(whic)n(h)g (estimates)g Fn(G)p Fp(\()p Fn(f)9 b Fp(\))3208 1310 y Fm(x)3283 1298 y Fp(with)456 1423 y(resp)r(ect)27 b(to)g Fn(L)898 1435 y Fl(2)935 1423 y Fp(\()p Fn(D)r Fp(\)-norm)g(of)h Fn(f)9 b Fp(.)456 1547 y(In)25 b(order)g(to)g(in)n(tro)r(duce)h(a)f Fh(white)k(noise)d Fp(on)g(the)g(b)r(oundary)f Fn(@)5 b(D)r Fp(,)25 b(w)n(e)h(consider)e(a)i Fn(tempor)r(al)r(l)r(y)456 1672 y Fh(two{side)l(d)f Fp(Wiener)e(pro)r(cess)g Fn(W)1473 1684 y Fl(1)1534 1672 y Fp(with)h(v)-5 b(alues)23 b(in)h Fn(U)32 b Fp(and)23 b(denote)h(b)n(y)f Fk(fF)2778 1684 y Fm(t)2806 1672 y Fk(g)2848 1684 y Fm(t)p Fg(2)p Ff(R)2987 1672 y Fp(the)h(\014ltration)456 1796 y(corresp)r(onding)h(to)i(this)g (Wiener)g(pro)r(cess,)f(where,)h(roughly)e(sp)r(eaking,)i Fk(F)2840 1808 y Fm(t)2896 1796 y Fp(is)g(generated)f(b)n(y)456 1921 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Fp(+)f Fk(\001)p Fp(\))g Fk(\000)g Fn(W)2366 2618 y Fl(1)2404 2606 y Fp(\()p Fn(!)s(;)c(t)p Fp(\))p Fn(:)456 2793 y Fp(The)22 b(measure)g Fn(P)35 b Fp(is)22 b(ergo)r(dic)g(with)h(resp)r(ect)g(to)g(the)g(\015o) n(w)f Fn(\022)j Fp(of)e(the)g(metric)g(dynamical)f(system)456 2918 y(formed)27 b(b)n(y)g(the)h(Wiener)g(shift.)456 3042 y(Since)38 b(the)h(solution)f(op)r(erator)1526 3021 y(~)1507 3042 y Fn(G)p Fp(\(0)p Fn(;)14 b Fk(\001)p Fp(\))39 b(ab)r(o)n(v)n(e)e(is)h(a)g(linear)g(and)g(b)r(ounded)g(op)r(erator,)f (the)456 3167 y(pro)r(cess)29 b Fn(t)i Fk(7!)939 3146 y Fe(f)922 3167 y Fn(W)1000 3179 y Fl(1)1038 3167 y Fp(\()p Fn(!)s(;)14 b(t)p Fp(\))31 b(:=)1392 3146 y(~)1373 3167 y Fn(G)p Fp(\(0)p Fn(;)14 b(W)1627 3179 y Fl(1)1665 3167 y Fp(\()p Fn(!)s(;)g(t)p Fp(\)\))31 b(also)e(de\014nes)i(a)f(Wiener)g (pro)r(cess)g(with)h(tra)5 b(jec-)456 3291 y(tories)31 b(in)h(the)g(Sob)r(olev)g(space)f Fn(W)1565 3261 y Fl(2)1553 3312 y(2)1602 3291 y Fp(\()p Fn(D)r Fp(\))i(for)e(whic)n(h)h(the)g(b)r 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4670 a(@)g(n)2513 4613 y Fp(=)23 b(0)82 b(on)28 b Fn(@)5 b(D)r(:)456 4800 y Fp(with)28 b Fn(\027)33 b(>)27 b Fp(0.)g(This)h(semigroup)e(has)h (the)h(generator)e Fk(\000)p Fn(A)p Fp(.)456 4925 y(W)-7 b(e)28 b(consider)e(an)h(expression)g(of)g(the)h(form)1175 5140 y Fn(z)t Fp(\()p Fn(!)s(;)14 b(t)p Fp(\))22 b(=)h Fn(S)5 b Fp(\()p Fn(t)p Fp(\))p Fn(z)1703 5152 y Fl(0)1759 5140 y Fp(+)1842 5027 y Fe(Z)1925 5047 y Fm(t)1888 5216 y Fl(0)1968 5140 y Fn(AS)g Fp(\()p Fn(t)18 b Fk(\000)g Fn(\034)9 b Fp(\))p Fn(d)2387 5119 y Fe(f)2369 5140 y Fn(W)2447 5152 y Fl(1)2486 5140 y Fp(\()p Fn(!)s(;)14 b(\034)9 b Fp(\))p Fn(;)-2254 b Fp(\(10\))p eop %%Page: 6 6 6 5 bop 456 238 a Fl(6)378 b(JINQIA)n(O)29 b(DUAN,)g(PETER)f(E.)h (KLOEDEN,)f(AND)h(BJ)2417 223 y(\177)2408 238 y(ORN)g(SCHMALFUSS)456 425 y Fp(as)e(the)h(solution)f(of)g(the)h(linear)f(sto)r(c)n(hastic)g (ev)n(olution)g(equation)870 564 y Fn(dz)p 870 601 86 4 v 876 677 a(dt)984 620 y Fk(\000)18 b Fn(\027)5 b Fp(\001)14 b Fn(z)27 b Fp(=)22 b(0)p Fn(;)180 b(z)t Fp(\(0\))22 b(=)h Fn(z)1892 632 y Fl(0)1952 620 y Fk(2)g Fn(H)r(;)2339 564 y(@)p 2314 601 99 4 v 2314 677 a(@)5 b(n)2423 620 y(z)t Fp(\()p Fn(t)p Fp(\))p Fk(j)2583 632 y Fm(@)t(D)2705 620 y Fp(=)2826 599 y(_)2793 620 y Fn(W)2871 632 y Fl(1)2908 620 y Fp(\()p Fn(t)p Fp(\);)-2569 b(\(11\))456 805 y(see)27 b(\([15)o(])h(Section)g(13.2\).)e(The)i(expression)e(\(10\))h(is)h (meaningful)f(if,)h(for)g(example,)838 906 y Fe(Z)921 926 y Fm(t)884 1094 y Fl(0)964 1019 y Fk(k)p Fn(AS)5 b Fp(\()p Fn(t)18 b Fk(\000)g Fn(\034)9 b Fp(\))p Fn(GQ)1495 984 y Fm(W)1557 992 y Fd(1)1612 962 y(1)p 1612 971 29 3 v 1612 1004 a(2)1655 1019 y Fk(k)1697 984 y Fl(2)1697 1041 y Fg(L)1743 1025 y Fd(2)1775 1041 y Fl(\()p Fm(U)o(;H)t Fl(\))1954 1019 y Fn(d\034)32 b(<)23 b Fk(1)p Fn(;)97 b Fp(for)27 b(all)g(0)c Fk(\024)g Fn(t)g(<)f Fp(+)p Fk(1)p Fn(:)456 1217 y Fp(Since)27 b(b)n(y)h(the)g(in)n(v)-5 b(ariance)26 b(of)i(the)f(incremen)n(ts)h(of)f(the)h(Wiener)g(pro)r (cess,)e(w)n(e)h(then)h(ha)n(v)n(e)847 1445 y Fi(E)911 1325 y Fe(\014)911 1374 y(\014)911 1424 y(\014)911 1474 y(\014)944 1445 y Fn(A)1020 1332 y Fe(Z)1103 1353 y Fl(0)1066 1521 y Fg(\0001)1202 1445 y Fn(S)5 b Fp(\()p Fk(\000)p Fn(\034)k Fp(\))p Fn(d)1492 1424 y Fe(f)1475 1445 y Fn(W)1553 1457 y Fl(1)1592 1325 y Fe(\014)1592 1374 y(\014)1592 1424 y(\014)1592 1474 y(\014)1619 1336 y Fl(2)1680 1445 y Fk(\024)1794 1341 y Fg(1)1767 1366 y Fe(X)1774 1543 y Fm(i)p Fl(=0)1901 1445 y Fn(e)1940 1411 y Fg(\000)p Fl(2)p Fm(\025)2064 1419 y Fd(1)2097 1411 y Fm(i)2124 1445 y Fi(E)2188 1325 y Fe(\014)2188 1374 y(\014)2188 1424 y(\014)2188 1474 y(\014)2221 1445 y Fn(A)2297 1332 y Fe(Z)2380 1353 y Fl(0)2343 1521 y Fg(\000)p Fl(1)2446 1445 y Fn(S)c Fp(\()p Fn(\034)k Fp(\))p Fn(d)2672 1424 y Fe(f)2654 1445 y Fn(W)2732 1457 y Fl(1)2771 1325 y Fe(\014)2771 1374 y(\014)2771 1424 y(\014)2771 1474 y(\014)2799 1336 y Fl(2)2859 1445 y Fn(<)23 b Fk(1)p Fn(:)456 1674 y Fp(The)k(random)g(v)-5 b(ariable)26 b Fn(z)1281 1686 y Fm(W)1343 1694 y Fd(1)1408 1674 y Fp(de\014ned)i(b)n(y)1330 1890 y Fn(z)1369 1902 y Fm(W)1431 1910 y Fd(1)1468 1890 y Fp(\()p Fn(!)s Fp(\))23 b(:=)g Fn(A)1797 1777 y Fe(Z)1880 1797 y Fl(0)1843 1965 y Fg(\0001)1979 1890 y Fn(S)5 b Fp(\()p Fk(\000)p Fn(\034)k Fp(\))p Fn(d)2269 1869 y Fe(f)2252 1890 y Fn(W)2330 1902 y Fl(1)2368 1890 y Fp(\()p Fn(!)s(;)14 b(\034)9 b Fp(\))456 2095 y(is)29 b(th)n(us)h(w)n(ell)f (de\014ned)h(and)f(has)g(\014nite)i(second)e(momen)n(t)g(with)h(resp)r (ect)g(to)f(the)h(norm)f(of)h Fn(H)7 b Fp(.)456 2220 y(Moreo)n(v)n(er,)25 b(w)n(e)i(ha)n(v)n(e)f(formally)650 2429 y Fn(S)5 b Fp(\()p Fn(t)p Fp(\))p Fn(z)839 2441 y Fm(W)901 2449 y Fd(1)938 2429 y Fp(\()p Fn(!)s Fp(\))83 b(+)g Fn(A)1364 2316 y Fe(Z)1447 2337 y Fm(t)1410 2505 y Fl(0)1490 2429 y Fn(S)5 b Fp(\()p Fn(t)19 b Fk(\000)f Fn(\034)9 b Fp(\))p Fn(d)1847 2408 y Fe(f)1830 2429 y Fn(W)1908 2441 y Fl(1)1947 2429 y Fp(\()p Fn(!)s(;)14 b(\034)9 b Fp(\))1140 2674 y(=)83 b Fn(S)5 b Fp(\()p Fn(t)p Fp(\))p Fn(A)1514 2561 y Fe(Z)1597 2582 y Fl(0)1561 2750 y Fg(\0001)1697 2674 y Fn(S)g Fp(\()p Fk(\000)p Fn(\034)k Fp(\))p Fn(d)1987 2653 y Fe(f)1970 2674 y Fn(W)2048 2686 y Fl(1)2086 2674 y Fp(\()p Fn(!)s(;)14 b(\034)9 b Fp(\))19 b(+)f Fn(A)2465 2561 y Fe(Z)2548 2582 y Fm(t)2512 2750 y Fl(0)2592 2674 y Fn(S)5 b Fp(\()p Fn(t)18 b Fk(\000)g Fn(\034)9 b Fp(\))p Fn(d)2948 2653 y Fe(f)2931 2674 y Fn(W)3009 2686 y Fl(1)3048 2674 y Fp(\()p Fn(!)s(;)14 b(\034)9 b Fp(\))1140 2926 y(=)83 b Fn(A)1364 2813 y Fe(Z)1447 2834 y Fl(0)1410 3002 y Fg(\0001)1546 2926 y Fn(S)5 b Fp(\()p Fn(t)19 b Fk(\000)f Fn(\034)9 b Fp(\))p Fn(d)1903 2905 y Fe(f)1886 2926 y Fn(W)1964 2938 y Fl(1)2003 2926 y Fp(\()p Fn(!)s(;)14 b(\034)9 b Fp(\))19 b(+)f Fn(A)2382 2813 y Fe(Z)2465 2834 y Fm(t)2428 3002 y Fl(0)2508 2926 y Fn(S)5 b Fp(\()p Fn(t)19 b Fk(\000)f Fn(\034)9 b Fp(\))p Fn(d)2865 2905 y Fe(f)2848 2926 y Fn(W)2926 2938 y Fl(1)2964 2926 y Fp(\()p Fn(!)s(;)14 b(\034)9 b Fp(\))1140 3176 y(=)83 b Fn(A)1364 3063 y Fe(Z)1447 3084 y Fm(t)1410 3252 y Fg(\0001)1546 3176 y Fn(S)5 b Fp(\()p Fn(t)19 b Fk(\000)f Fn(\034)9 b Fp(\))p Fn(d)1903 3155 y Fe(f)1886 3176 y Fn(W)1964 3188 y Fl(1)2003 3176 y Fp(\()p Fn(!)s(;)14 b(\034)9 b Fp(\))1140 3428 y(=)83 b Fn(A)1364 3315 y Fe(Z)1447 3336 y Fl(0)1410 3504 y Fg(\0001)1546 3428 y Fn(S)5 b Fp(\()p Fk(\000)p Fn(\034)k Fp(\))p Fn(d)1836 3407 y Fe(f)1819 3428 y Fn(W)1897 3440 y Fl(1)1936 3428 y Fp(\()p Fn(\022)2007 3440 y Fm(t)2036 3428 y Fn(!)s(;)14 b(\034)9 b Fp(\))24 b(=)e Fn(z)2355 3440 y Fm(W)2417 3448 y Fd(1)2454 3428 y Fp(\()p Fn(\022)2525 3440 y Fm(t)2555 3428 y Fn(!)s Fp(\))p Fn(;)456 3633 y Fp(so)27 b(the)h(stationary)e(pro)r(cess)1086 3845 y Fn(t)d Fk(7!)g Fn(z)t Fp(\()p Fn(\022)1359 3857 y Fm(t)1388 3845 y Fn(!)s Fp(\))g(=)f Fn(A)1661 3732 y Fe(Z)1744 3752 y Fl(0)1707 3920 y Fg(\0001)1844 3845 y Fn(S)5 b Fp(\()p Fk(\000)p Fn(\034)k Fp(\))p Fn(d)2134 3824 y Fe(f)2117 3845 y Fn(W)2195 3857 y Fl(1)2233 3845 y Fp(\()p Fn(\022)2304 3857 y Fm(t)2333 3845 y Fn(!)s(;)14 b(\034)9 b Fp(\))p Fn(;)98 b(t)23 b Fk(2)g Fi(R)456 4054 y Fp(solv)n(es)j(the)i (b)r(oundary)f(v)-5 b(alue)27 b(problem)g(\(11\).)456 4179 y(Since)g Fi(E)9 b Fk(j)p Fn(z)784 4191 y Fm(W)846 4199 y Fd(1)888 4179 y Fk(j)911 4149 y Fl(2)976 4179 y Fp(is)28 b(\014nite,)g(w)n(e)f(can)h(apply)f(the)h(Burkholder)e (inequalit)n(y)h(to)h(obtain)1499 4360 y Fi(E)63 b Fp(sup)1563 4434 y Fm(t)p Fg(2)p Fl([0)p Fm(;)p Fl(1])1776 4360 y Fk(j)p Fn(z)1838 4372 y Fm(W)1900 4380 y Fd(1)1936 4360 y Fp(\()p Fn(\022)2007 4372 y Fm(t)2037 4360 y Fn(!)s Fp(\))p Fk(j)2147 4326 y Fl(2)2207 4360 y Fn(<)23 b Fk(1)p Fn(;)456 4584 y Fp(and)k(it)h(then)g(follo)n(ws)f(from)g(the)h (Birkho\013)f(Ergo)r(dic)f(Theorem)h([15)o(])h(that)1378 4805 y(lim)1331 4857 y Fm(i)p Fg(!\0061)1563 4737 y Fp(sup)1688 4758 y Fm(\034)7 b Fg(2)p Fl([0)p Fm(;)p Fl(1])1911 4737 y Fk(j)p Fn(z)1973 4749 y Fm(W)2035 4757 y Fd(1)2072 4737 y Fp(\()p Fn(\022)2143 4749 y Fm(i)p Fl(+)p Fm(\034)2259 4737 y Fn(!)s Fp(\))p Fk(j)2369 4707 y Fl(2)p 1563 4786 844 4 v 1970 4862 a Fn(i)2440 4805 y Fp(=)22 b(0)456 4995 y(on)27 b(a)g Fn(\022)r Fp(-in)n(v)-5 b(arian)n(t)27 b(subset)g(of)h(\012)f(of)h(full)g Fn(P)12 b Fp({measure.)26 b(Hence)1595 5184 y(lim)1548 5234 y Fm(t)p Fg(!\0061)1781 5128 y Fk(j)p Fn(z)1843 5140 y Fm(W)1905 5148 y Fd(1)1942 5128 y Fp(\()p Fn(\022)2013 5140 y Fm(t)2042 5128 y Fn(!)s Fp(\))p Fk(j)2152 5097 y Fl(2)p 1781 5165 409 4 v 1970 5241 a Fn(t)2223 5184 y Fp(=)c(0)p eop %%Page: 7 7 7 6 bop 831 226 a Fl(ST)-5 b(ABILITY)29 b(OF)g(THE)g(STOCHASTIC)h (QUASIGEOSTR)n(OPHIC)g(EQUA)-5 b(TION)341 b(7)456 425 y Fp(on)31 b(a)h Fn(\022)r Fp(-in)n(v)-5 b(arian)n(t)31 b(subset)h(of)g(\012)g(of)g(full)h Fn(P)12 b Fp({measure,)30 b(i.e.,)j Fk(j)p Fn(z)2451 437 y Fm(W)2513 445 y Fd(1)2549 425 y Fk(j)g Fp(is)f(temp)r(ered.)g(Note)g(that)456 550 y(similar)c(tec)n(hniques)h(can)f(b)r(e)i(used)e(to)h(sho)n(w)f(that)i Fn(z)2127 562 y Fm(W)2189 570 y Fd(1)2254 550 y Fp(is)f(de\014ned)h(on) e(a)h Fn(\022)r Fp({in)n(v)-5 b(arian)n(t)28 b(set)h(of)456 674 y(full)f(measure.)456 923 y(Finally)-7 b(,)26 b(equations)f(for)g (the)h(generalized)f(spatial)g(deriv)-5 b(ativ)n(es)25 b(of)g Fn(z)2626 935 y Fm(W)2688 943 y Fd(1)2751 923 y Fp(can)g(b)r(e)i(in)n(v)n(estigated)456 1048 y(if)34 b(w)n(e)f(supp)r(ose)g(that)h(the)g(co)n(v)-5 b(ariance)31 b Fn(Q)1792 1018 y Fm(W)1854 1026 y Fd(1)1925 1048 y Fp(is)i(su\016cien)n(tly)h(regular.)e(Conditions)h(are)f(for-)456 1172 y(m)n(ulated)27 b(in)i(DaPrato)d(and)i(Zab)r(czyk)f([15)o(])h (Theorem)f(13.3.1.)f(In)i(particular,)f Fk(r)p Fn(z)3093 1184 y Fm(W)3155 1192 y Fd(1)3220 1172 y Fp(is)h(w)n(ell)456 1297 y(de\014ned)g(and)f(temp)r(ered.)816 1545 y(3.)46 b Fo(Transf)n(orma)-6 b(tion)32 b(of)f(the)h(quasigeostr)n(ophic)g (equa)-6 b(tion)456 1719 y Fp(W)f(e)36 b(return)f(to)h(the)g(QG)g(v)n (orticit)n(y)e(equation)h(\(2\),)i(in)f(whic)n(h)f(w)n(e)h(no)n(w)f (write)h(\001)p Fn( )j Fp(for)c(the)456 1844 y(v)n(orticit)n(y)-7 b(.)26 b(That)i(is,)f(w)n(e)g(consider)1007 1993 y Fn(d)p Fp(\001)p Fn( )p 1007 2030 170 4 v 1055 2106 a(dt)1205 2049 y Fp(+)18 b Fn(J)8 b Fp(\()p Fn( )s(;)14 b Fp(\001)p Fn( )s Fp(\))19 b(+)f Fn(\014)t( )1833 2061 y Fm(x)1898 2049 y Fp(=)23 b Fn(\027)5 b Fp(\001)2101 2015 y Fl(2)2139 2049 y Fn( )21 b Fk(\000)d Fn(r)r Fp(\001)p Fn( )23 b Fp(+)2598 2028 y(_)2565 2049 y Fn(W)2643 2061 y Fl(2)2680 2049 y Fp(\()p Fn(!)s(;)14 b(t)p Fp(\))p Fn(;)-2433 b Fp(\(12\))456 2241 y(with)28 b(a)f(non)g(zero)g(b)r(oundary)g (condition)1267 2390 y Fn(@)p 1242 2427 99 4 v 1242 2503 a(@)5 b(n)1351 2446 y Fp(\001)p Fn( )s Fp(\()p Fn(t)p Fp(\))23 b(=)1715 2425 y(_)1682 2446 y Fn(W)1760 2458 y Fl(1)1798 2446 y Fp(\()p Fn(!)s(;)14 b(t)p Fp(\))p Fn(;)97 b( )26 b Fp(=)c(0)83 b(on)27 b Fn(@)5 b(D)r(;)-2198 b Fp(\(13\))456 2655 y(that)32 b(in)n(v)n(olv)n(es)e(a)h(white)h(noise) 1505 2634 y(_)1472 2655 y Fn(W)1550 2667 y Fl(1)1619 2655 y Fp(on)g(the)g(b)r(oundary)f Fn(@)5 b(D)34 b Fp(as)d(describ)r (ed)g(in)h(the)h(previous)456 2779 y(section.)23 b(In)h(addition,)g (the)g(wind)g(forcing)f(white)h(noise)2268 2758 y(_)2234 2779 y Fn(W)2312 2791 y Fl(2)2374 2779 y Fp(is)f(based)h(on)f(a)h(temp) r(orally)f(t)n(w)n(o{)456 2904 y(sided)j(noise)g(adapted)g(Wiener)h (pro)r(cess)e Fn(W)1845 2916 y Fl(2)1909 2904 y Fp(with)i(v)-5 b(alues)27 b(in)f Fn(V)46 b Fp(and)26 b(co)n(v)-5 b(ariance)25 b Fn(Q)3160 2874 y Fm(W)3222 2882 y Fd(2)3285 2904 y Fp(suc)n(h)456 3028 y(that)i(tr)700 3040 y Fm(V)758 3028 y Fn(Q)824 2998 y Fm(W)886 3006 y Fd(2)946 3028 y Fn(<)22 b Fk(1)p Fp(.)28 b(In)g(particular,)e Fn(W)1755 3040 y Fl(1)1820 3028 y Fp(and)i Fn(W)2060 3040 y Fl(2)2125 3028 y Fp(are)f(assumed)g(to)g(b)r(e)h(indep)r(enden)n(t.)456 3153 y(W)-7 b(e)36 b(can)h(no)n(w)e(de\014ne)i(a)f(metric)h(dynamical)e (system)i(with)g(the)f(prop)r(erties)g(of)g(our)g(white)456 3277 y(noise)25 b(terms.)h(F)-7 b(or)25 b(\012)h(w)n(e)f(c)n(ho)r(ose)g (an)h(appropriate)e(subset)i(of)g(the)g(function)g(space)g Fn(C)3180 3289 y Fl(0)3217 3277 y Fp(\()p Fi(R)p Fn(;)15 b(U)9 b Fp(\))456 3402 y Fk(\002)37 b Fn(C)617 3414 y Fl(0)655 3402 y Fp(\()p Fi(R)p Fn(;)14 b(V)25 b Fp(\))39 b(with)f(the)h(usual)f(Borel)f Fn(\033)s Fp({algebra)f(of)i(a)g(F)-7 b(r)n(\023)-39 b(ec)n(het)37 b(space,)g(i.e.,)h(an)g(elemen)n(t)456 3526 y Fn(!)k Fp(is)e(a)f(con)n(tin)n(uous)g(path)h(from)g Fi(R)46 b Fp(in)n(to)39 b(\()p Fn(U;)14 b(V)19 b Fp(\))40 b(with)h Fn(!)s Fp(\(0\))i(=)g(0.)c(Then)h(w)n(e)g(tak)n(e)f Fi(P)g Fp(=)456 3651 y Fn(P)521 3621 y Fm(W)583 3629 y Fd(1)641 3651 y Fk(\012)20 b Fn(P)791 3621 y Fm(W)853 3629 y Fd(2)922 3651 y Fp(to)31 b(b)r(e)h(the)g(pro)r(duct)g(measure)e (of)i(the)g(Wiener)g(measures)e(corresp)r(onding)g(to)456 3775 y Fn(W)534 3787 y Fl(1)600 3775 y Fp(and)e Fn(W)840 3787 y Fl(2)878 3775 y Fp(,)g(whic)n(h)h(is)f(ergo)r(dic)f(since)i(b)r (oth)g Fn(P)2008 3745 y Fm(W)2070 3753 y Fd(1)2135 3775 y Fp(and)f Fn(P)2362 3745 y Fm(W)2424 3753 y Fd(2)2490 3775 y Fp(are)f(ergo)r(dic.)h(The)g(\015o)n(w)g Fn(\022)j Fp(on)456 3900 y(\012)22 b(is)g(de\014ned)h(in)f(terms)g(of)g(shift)h (op)r(erators)d Fn(\022)25 b Fp(applied)d(to)g(the)h(sample)e(paths)i (of)f Fn(W)3090 3912 y Fl(1)3150 3900 y Fp(and)g Fn(W)3384 3912 y Fl(2)3421 3900 y Fp(.)456 4149 y(The)33 b(ab)r(o)n(v)n(e)f(QG)h (equation)f(has)h(structural)f(similarities)h(to)g(equations)f(of)h(Na) n(vier{Stok)n(es)456 4274 y(t)n(yp)r(e.)27 b(T)-7 b(o)26 b(b)r(e)h(able)g(to)g(adapt)f(w)n(ell)h(kno)n(wn)f(results)g(of)h(suc)n (h)f(equations,)g(w)n(e)h(need)g(to)f(replace)456 4398 y(these)d(b)r(oundary)f(conditions)h(b)n(y)f(zero)g(b)r(oundary)h (conditions,)f(whic)n(h)h(is)g(p)r(ossible)g(with)h(par-)456 4523 y(ticular)e(t)n(yp)r(es)i(of)f(stationary)f(transformations;)f (see)i(Crauel)f(and)h(Flandoli)g([5])g(or)g(Brannan,)456 4647 y(Duan)30 b(and)g(W)-7 b(anner)30 b([4)o(],)h(or)e(in)h(a)g(more)f (general)g(con)n(text)h(Keller)f(and)h(Sc)n(hmalfuss)g([11)o(])g(or)456 4772 y(Imk)n(eller)c(and)i(Sc)n(hmalfuss)f([10)o(].)h(In)g(particular,) e(w)n(e)h(transform)g(\(12\))g(in)n(to)1121 4925 y Fn(d)p 1106 4963 74 4 v 1106 5039 a(dt)1189 4982 y(u)18 b Fp(+)g Fn(J)8 b Fp(\()p Fn(G)p Fp(\()p Fn(u)p Fp(\))p Fn(;)14 b(u)p Fp(\))19 b(+)f Fn(\014)t(G)p Fp(\()p Fn(u)p Fp(\))2048 4994 y Fm(x)2114 4982 y Fp(=)k Fn(\027)5 b Fp(\001)p Fn(u)19 b Fk(\000)f Fn(r)e(u)i Fp(+)2702 4961 y(_)2668 4982 y Fn(W)2746 4994 y Fl(2)2784 4982 y Fp(\()p Fn(!)s(;)c(t)p Fp(\))-2514 b(\(14\))1297 5136 y Fn(@)p 1272 5173 99 4 v 1272 5249 a(@)5 b(n)1380 5192 y(u)p Fp(\()p Fn(t)p Fp(\))p Fk(j)24 b Fp(=)1690 5171 y(_)1656 5192 y Fn(W)1734 5204 y Fl(1)1772 5192 y Fp(\()p Fn(!)s(;)14 b(t)p Fp(\))83 b(on)27 b Fn(@)5 b(D)p eop %%Page: 8 8 8 7 bop 456 238 a Fl(8)378 b(JINQIA)n(O)29 b(DUAN,)g(PETER)f(E.)h (KLOEDEN,)f(AND)h(BJ)2417 223 y(\177)2408 238 y(ORN)g(SCHMALFUSS)456 425 y Fp(where)e Fn(G)h Fp(is)f(the)h(solution)f(op)r(erator)f(of)i (the)g(b)r(oundary)e(v)-5 b(alue)28 b(problem)f(\(cf.)h(\(9\)\))1529 593 y(\001)p Fn( )e Fp(=)d Fn(u;)179 b( )s Fk(j)2096 605 y Fm(@)t(D)2219 593 y Fp(=)22 b(0)p Fn(;)456 761 y Fp(i.e.,)f(with)h(the)g(solution)e Fn( )25 b Fp(=)c Fn(G)p Fp(\()p Fn(u)p Fp(\).)g(W)-7 b(e)22 b(consider)e(equation)h (\(14\))g(as)g(an)g(ev)n(olution)f(equation)456 886 y(on)27 b(the)h(triple)g Fn(V)41 b Fk(\032)23 b Fn(H)30 b Fk(\032)23 b Fn(V)1366 856 y Fg(0)1417 886 y Fp(where)k Fn(V)1724 856 y Fg(0)1775 886 y Fp(is)h(the)g(dual)f(space)g(of)h Fn(V)19 b Fp(.)456 1135 y(The)39 b(prop)r(erties)g(of)g(the)h (nonlinear)f(term)g(of)h(equation)e(\(14\))i(follo)n(w)e(from)h(those)h (of)f(the)456 1260 y(bilinear)27 b(op)r(erator)e Fn(B)32 b Fp(:)c Fn(L)1293 1272 y Fl(2)1330 1260 y Fp(\()p Fn(D)r Fp(\))19 b Fk(\002)f Fn(W)1657 1229 y Fl(1)1645 1280 y(2)1694 1260 y Fp(\()p Fn(D)r Fp(\))28 b Fk(!)g Fn(V)2035 1229 y Fg(0)2086 1260 y Fp(de\014ned)g(b)n(y)1501 1428 y Fn(B)t Fp(\()p Fn(v)1640 1440 y Fl(1)1678 1428 y Fn(;)14 b(v)1755 1440 y Fl(2)1792 1428 y Fp(\))24 b(=)e Fn(J)8 b Fp(\()p Fn(G)p Fp(\()p Fn(v)2158 1440 y Fl(1)2197 1428 y Fp(\))p Fn(;)14 b(v)2306 1440 y Fl(2)2343 1428 y Fp(\))p Fn(:)-1942 b Fp(\(15\))456 1614 y Fs(Lemma)29 b(3.1.)125 b Fn(B)34 b Fh(is)c(a)g(wel)t(l)h(de\014ne)l(d,)f(c)l(ontinuous)f(op)l (er)l(ator)i(and)1157 1782 y Fn(i)p Fp(\))336 b Fk(h)p Fn(B)t Fp(\()p Fn(v)1725 1794 y Fl(1)1763 1782 y Fn(;)14 b(v)1840 1794 y Fl(2)1878 1782 y Fp(\))p Fn(;)g(v)1987 1794 y Fl(3)2024 1782 y Fk(i)24 b Fp(=)e Fk(\000h)p Fn(B)t Fp(\()p Fn(v)2403 1794 y Fl(1)2441 1782 y Fn(;)14 b(v)2518 1794 y Fl(3)2555 1782 y Fp(\))p Fn(;)g(v)2664 1794 y Fl(2)2702 1782 y Fk(i)p Fn(;)1129 1938 y(ii)p Fp(\))335 b Fk(h)p Fn(B)t Fp(\()p Fn(v)1725 1950 y Fl(1)1763 1938 y Fn(;)14 b(v)1840 1950 y Fl(2)1878 1938 y Fp(\))p Fn(;)g(v)1987 1950 y Fl(2)2024 1938 y Fk(i)24 b Fp(=)e(0)p Fn(:)456 2106 y Fh(for)30 b Fn(v)628 2118 y Fl(1)689 2106 y Fk(2)23 b Fn(L)824 2118 y Fl(2)861 2106 y Fp(\()p Fn(D)r Fp(\))p Fh(,)31 b Fn(v)1092 2118 y Fl(2)1152 2106 y Fk(2)24 b Fn(W)1321 2076 y Fl(1)1309 2126 y(2)1358 2106 y Fp(\()p Fn(D)r Fp(\))p Fn(;)k(v)1584 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y(Sc)n(hmalfuss)g([16)o(].)i(The)f(linear)f(terms)h Fn(r)r(u)g Fp(and)g Fn(\014)t(G)p Fp(\()p Fn(u)p Fp(\))2201 1754 y Fm(x)2272 1742 y Fp(app)r(earing)f(in)h(\(14\))g(but)h(not)f(in) g(the)456 1866 y(Na)n(vier{Stok)n(es)i(equation)i(are)g(not)h(essen)n (tial)f(for)h(a)f(pro)r(of)g(of)h(existence)g(and)g(uniqueness.)456 1991 y(See)c(Brannan,)f(Duan)h(and)g(W)-7 b(anner)29 b([4)o(])h(for)e(another)g(pro)r(of)h(of)g(existence)g(and)g (uniqueness)456 2115 y(based)e(on)g(mild)h(solutions.)1376 2345 y(4.)45 b Fo(The)32 b(st)-6 b(a)g(tionar)g(y)32 b(solution)456 2519 y Fp(W)-7 b(e)29 b(no)n(w)g(transform)e(the)j(sto)r (c)n(hastic)e(ev)n(olution)g(equation)h(\(14\))g(in)n(to)g(a)f(random)g (ev)n(olution)456 2644 y(equation)38 b(in)h Fn(V)58 b Fk(\032)39 b Fn(H)46 b Fk(\032)38 b Fn(V)1409 2613 y Fg(0)1432 2644 y Fp(,)i(i.e.,)f(with)g(stationary)f(random)g(co)r (e\016cien)n(ts)g(rather)g(than)456 2768 y(white)31 b(noise)g(driving)f 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4300 y Fm(t)2624 4288 y Fn(!)s Fp(\))g(+)g Fn(z)2851 4300 y Fm(W)2913 4308 y Fd(2)2950 4288 y Fp(\()p Fn(\022)3021 4300 y Fm(t)3050 4288 y Fn(!)s Fp(\))p Fn(;)c(z)t Fp(\))1139 4444 y Fk(\000)k Fn(B)t Fp(\()p Fn(z)1360 4456 y Fm(W)1422 4464 y Fd(1)1459 4444 y Fp(\()p Fn(\022)1530 4456 y Fm(t)1560 4444 y Fn(!)s Fp(\))g(+)g Fn(z)1787 4456 y Fm(W)1849 4464 y Fd(2)1886 4444 y Fp(\()p Fn(\022)1957 4456 y Fm(t)1986 4444 y Fn(!)s Fp(\))p Fn(;)c(z)2149 4456 y Fm(W)2211 4464 y Fd(1)2248 4444 y Fp(\()p Fn(\022)2319 4456 y Fm(t)2348 4444 y Fn(!)s Fp(\))k(+)g Fn(z)2575 4456 y Fm(W)2637 4464 y Fd(2)2674 4444 y Fp(\()p Fn(\022)2745 4456 y Fm(t)2775 4444 y Fn(!)s Fp(\)\))1139 4600 y Fk(\000)g Fn(\014)t(G)p Fp(\()p Fn(z)1409 4612 y Fm(W)1471 4620 y Fd(1)1509 4600 y Fp(\()p Fn(\022)1580 4612 y Fm(t)1609 4600 y Fn(!)s Fp(\))h(+)f Fn(z)1837 4612 y Fm(W)1899 4620 y Fd(2)1935 4600 y Fp(\()p Fn(\022)2006 4612 y Fm(t)2036 4600 y Fn(!)s Fp(\)\))2155 4612 y Fm(x)2215 4600 y Fk(\000)g Fn(r)r Fp(\()p Fn(z)2408 4612 y Fm(W)2470 4620 y Fd(1)2508 4600 y Fp(\()p Fn(\022)2579 4612 y Fm(t)2608 4600 y Fn(!)s Fp(\))h(+)f Fn(z)2836 4612 y Fm(W)2898 4620 y Fd(2)2934 4600 y Fp(\()p Fn(\022)3005 4612 y Fm(t)3035 4600 y Fn(!)s Fp(\)\))456 4343 y(\(16\))456 4771 y(with)28 b Fn(z)t Fp(\(0\))22 b(=)h Fn(z)943 4783 y Fl(0)1003 4771 y Fk(2)g Fn(H)7 b Fp(.)456 5091 y Fs(Lemma)29 b(4.1.)40 b Fh(The)c(r)l(andom)g(evolution)g(e)l(quation)f(\(16\))h(has)g(a)g (unique)f(solution)g(for)h(any)456 5216 y(initial)30 b(c)l(ondition)f Fn(z)1099 5228 y Fl(0)1165 5216 y Fk(2)g Fn(H)35 b Fh(and)30 b(this)f(solution)g(de\014nes)f(a)h(r)l(andom)h (dynamic)l(al)g(system)f(with)p eop %%Page: 10 10 10 9 bop 456 238 a Fl(10)345 b(JINQIA)n(O)29 b(DUAN,)g(PETER)f(E.)h (KLOEDEN,)f(AND)h(BJ)2417 223 y(\177)2408 238 y(ORN)g(SCHMALFUSS)456 425 y Fh(r)l(esp)l(e)l(ct)41 b(to)g(the)g(metric)h(dynamic)l(al)h (system)e Fn(\022)i Fh(intr)l(o)l(duc)l(e)l(d)e(in)h(Se)l(ction)f(3)g (for)h(which)h(the)456 550 y(asso)l(ciate)l(d)31 b(c)l(o)l(cycle)g (mapping)g(is)f(de\014ne)l(d)g(by)g Fp(\()p Fn(t;)14 b(!)s(;)g(z)2164 562 y Fl(0)2201 550 y Fp(\))30 b Fk(7!)g Fn(z)t Fp(\()p Fn(t;)14 b(!)s(;)g(z)2649 562 y Fl(0)2685 550 y Fp(\))p Fh(.)456 737 y Fp(F)-7 b(or)21 b(the)g(pro)r(of)g(of)h (this)g(lemma)f(w)n(e)g(can)h(use)f(the)h(fact)g(that)f(equation)g (\(16\))h(is)f(quite)h(similar)f(to)456 861 y(the)26 b(Na)n(vier{Stok)n(es)d(equation.)i(Although)g(some)g(linear)g(terms)g (are)g(also)f(presen)n(t,)h(similar)g(a)456 986 y(priori)e(estimates)i (can)f(b)r(e)h(obtained)g(to)f(those)h(in)g(T)-7 b(emam)25 b(\([20)o(],)g(Chapter)f(I)r(I)r(I\))i(or)e(Benssou-)456 1110 y(san)j(and)h(T)-7 b(emam)27 b([2].)h(Because)f(of)g(the)i(prop)r (erties)d(of)i(the)g(op)r(erator)e Fn(B)32 b Fp(in)n(tro)r(duced)c(in)g (the)456 1235 y(previous)i(section.)h(Moreo)n(v)n(er,)e(the)j(random)f (terms)g(app)r(earing)g(inside)g(the)h(co)r(e\016cien)n(ts)g(of)456 1360 y(equation)27 b(\(16\))g(are)f(giv)n(en)h(b)n(y)g(stationary)f (pro)r(cesses,)g(so)h(w)n(e)g(obtain)g(a)h(random)e(dynamical)456 1484 y(system,)h(see)g(Arnold)h(\([1)o(],)g(page)f(58\).)456 1796 y Fs(Remark)j(4.2.)41 b Fp(T)-7 b(o)27 b(see)g(that)h Fn(t)g Fk(7!)g Fn(z)t Fp(\()p Fn(t;)14 b(!)s(;)g(z)1887 1808 y Fl(0)1923 1796 y Fp(\))28 b(is)g(con)n(tin)n(uous)e(for)i(an)n (y)f Fn(z)2806 1808 y Fl(0)2870 1796 y Fk(2)i Fn(H)34 b Fp(and)28 b Fn(!)i Fk(2)e Fp(\012)456 1920 y(w)n(e)f(can)g(use)g (Lemma)h(I)r(I)r(I.1.2)f(in)h([20)o(])g(since)f(the)h(solution)f(of)h (equation)f(\(16\))g(satis\014es)1509 2009 y Fe(Z)1592 2029 y Fm(t)1556 2197 y Fl(0)1635 2122 y Fk(k)p Fn(z)t Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(z)1961 2134 y Fl(0)1997 2122 y Fp(\))p Fk(k)2071 2087 y Fl(2)2108 2122 y Fn(d\034)33 b(<)23 b Fk(1)456 2307 y Fp(for)k(an)n(y)f Fn(z)778 2319 y Fl(0)843 2307 y Fk(2)i Fn(H)7 b Fp(.)28 b(Indeed,)f(b)n(y)h(the)g(c)n (hain)f(rule,)468 2504 y Fk(j)p Fn(z)t Fp(\()p Fn(t)p Fp(\))p Fk(j)651 2470 y Fl(2)706 2504 y Fp(+)18 b(2)p Fn(\027)891 2391 y Fe(Z)974 2412 y Fm(t)937 2580 y Fl(0)1017 2504 y Fk(k)p Fn(z)t Fp(\()p Fn(\034)9 b Fp(\))p Fk(k)1253 2470 y Fl(2)1290 2504 y Fn(d\034)93 b Fk(\024)82 b(j)p Fn(z)1671 2516 y Fl(0)1708 2504 y Fk(j)1731 2470 y Fl(2)1787 2504 y Fp(+)18 b(2\()p Fn(\014)t(c)2031 2516 y Fm(G;x)2163 2504 y Fk(\000)g Fn(r)r Fp(\))2331 2391 y Fe(Z)2415 2412 y Fm(t)2378 2580 y Fl(0)2458 2504 y Fk(j)p Fn(z)t Fp(\()p Fn(\034)9 b Fp(\))p Fk(j)2656 2470 y Fl(2)2694 2504 y Fn(d\034)1609 2748 y Fp(+2)p Fn(c)1752 2760 y Fm(B)1822 2635 y Fe(Z)1906 2655 y Fm(t)1869 2823 y Fl(0)1949 2748 y Fk(j)p Fn(z)t Fp(\()p Fn(\034)g Fp(\))p Fk(jk)p Fn(z)t Fp(\()p Fn(\034)g Fp(\))p Fk(kjr)p Fn(z)2514 2760 y Fm(W)2576 2768 y Fd(1)2613 2748 y Fp(\()p Fn(\022)2684 2760 y Fm(\034)2725 2748 y Fn(!)s Fp(\))19 b(+)f Fk(r)p Fn(z)3022 2760 y Fm(W)3084 2768 y Fd(2)3121 2748 y Fp(\()p Fn(\022)3192 2760 y Fm(\034)3233 2748 y Fn(!)s Fp(\))p Fk(j)p Fn(d\034)1609 2991 y Fp(+2)1730 2878 y Fe(Z)1812 2898 y Fm(t)1776 3067 y Fl(0)1855 2991 y Fk(k)p Fn(f)9 b Fp(\()p Fn(\034)g Fp(\))p Fk(k)2098 3003 y Fg(\000)p Fl(1)2187 2991 y Fk(k)p Fn(z)t Fp(\()p Fn(\034)g Fp(\))p Fk(k)p Fn(d\034)c(;)456 3181 y Fp(where)27 b Fn(f)36 b Fp(consists)27 b(of)g(all)h(the)g(terms) f(in)h(\(16\))f(that)h(do)f(not)h(con)n(tain)f Fn(z)t Fp(.)g(Then,)h(using)803 3364 y(2)p Fk(j)p Fn(z)t Fk(jk)p Fn(z)t Fk(kjr)p Fn(z)1192 3376 y Fm(W)1254 3384 y Fd(1)1307 3364 y Fp(+)18 b Fk(r)p Fn(z)1498 3376 y Fm(W)1560 3384 y Fd(2)1597 3364 y Fk(j)p Fn(d\034)93 b Fk(\024)1949 3308 y Fn(\027)p 1949 3345 47 4 v 1951 3421 a Fp(2)2005 3364 y Fk(k)p Fn(z)t Fk(k)2132 3330 y Fl(2)2186 3364 y Fp(+)2282 3308 y(2)p 2279 3345 V 2279 3421 a Fn(\027)2336 3364 y Fk(j)p Fn(z)t Fk(j)2425 3330 y Fl(2)2461 3364 y Fk(jr)p Fn(z)2592 3376 y Fm(W)2654 3384 y Fd(1)2710 3364 y Fp(+)18 b Fk(r)p Fn(z)2901 3376 y Fm(W)2963 3384 y Fd(2)2999 3364 y Fk(j)3022 3330 y Fl(2)3060 3364 y Fn(;)1320 3570 y Fp(2)p Fk(k)p Fn(f)9 b Fk(k)1496 3582 y Fg(\000)p Fl(1)1583 3570 y Fk(k)p Fn(z)t Fk(k)82 b(\024)1949 3514 y Fn(\027)p 1949 3551 V 1951 3627 a Fp(2)2005 3570 y Fk(k)p Fn(z)t Fk(k)2132 3536 y Fl(2)2186 3570 y Fp(+)2282 3514 y(2)p 2279 3551 V 2279 3627 a Fn(\027)2336 3570 y Fk(k)p Fn(f)9 b Fk(k)2470 3536 y Fl(2)2470 3591 y Fg(\000)p Fl(1)2557 3570 y Fn(;)456 3739 y Fp(the)19 b(asserted)f(estimate)h (follo)n(ws)f(b)n(y)h(an)g(application)f(of)h(the)h(Gron)n(w)n(all)d (inequalit)n(y)-7 b(.)19 b(Moreo)n(v)n(er,)456 3864 y(b)n(y)27 b(the)h(prop)r(erties)f(of)g(the)h(op)r(erators)e Fn(A)i Fp(and)f Fn(B)t Fp(,)h(w)n(e)f(also)g(ha)n(v)n(e)1292 3947 y Fe(Z)1375 3968 y Fm(t)1338 4136 y Fl(0)1418 4060 y Fk(k)p Fn(z)t Fp(\()p Fn(\034)5 b(;)14 b(!)s(;)g(z)1744 4072 y Fl(0)1780 4060 y Fp(\))p Fk(k)1854 4026 y Fl(2)1854 4081 y Fm(V)1907 4064 y Fc(0)1934 4060 y Fn(d\034)33 b(<)22 b Fk(1)p Fn(;)97 b(z)2375 4072 y Fl(0)2435 4060 y Fk(2)24 b Fn(H)r(:)456 4541 y Fp(W)-7 b(e)28 b(no)n(w)f(de\014ne)g (random)g(isomorphism)f Fn(i)p Fp(\()p Fn(!)s Fp(\))i(:)g Fn(H)34 b Fk(!)28 b Fn(H)34 b Fp(b)n(y)1342 4709 y Fn(i)p Fp(\()p Fn(!)s Fp(\))p Fn(a)23 b Fp(=)g Fk(\000)p Fp(\()p Fn(z)1781 4721 y Fm(W)1843 4729 y Fd(1)1879 4709 y Fp(\()p Fn(!)s Fp(\))c(+)f Fn(z)2139 4721 y Fm(W)2201 4729 y Fd(2)2238 4709 y Fp(\()p Fn(!)s Fp(\)\))h(+)f Fn(a;)456 4878 y Fp(for)27 b(whic)n(h)g(the)h(in)n(v)n(erse)e(isomorphism)h Fn(i)1753 4848 y Fg(\000)p Fl(1)1842 4878 y Fp(\()p Fn(!)s Fp(\))g(is)h(giv)n(en)f(b)n(y)1330 5047 y Fn(i)1359 5013 y Fg(\000)p Fl(1)1448 5047 y Fp(\()p Fn(!)s Fp(\))p Fn(a)c Fp(=)g(\()p Fn(z)1793 5059 y Fm(W)1855 5067 y Fd(1)1892 5047 y Fp(\()p Fn(!)s Fp(\))18 b(+)g Fn(z)2151 5059 y Fm(W)2213 5067 y Fd(2)2250 5047 y Fp(\()p Fn(!)s Fp(\)\))h(+)f Fn(a:)456 5216 y Fp(Note)27 b(that)h(the)g(random)f(v)-5 b(ariable)26 b Fn(i)p Fp(\()p Fn(!)s Fp(\))p Fn(a)p Fp(\()p Fn(!)s Fp(\))i(is)f(temp)r(ered)h(for)f(an)n(y)g(temp)r(ered)h Fn(a)p Fp(\()p Fn(!)s Fp(\).)p eop %%Page: 11 11 11 10 bop 831 226 a Fl(ST)-5 b(ABILITY)29 b(OF)g(THE)g(STOCHASTIC)h (QUASIGEOSTR)n(OPHIC)g(EQUA)-5 b(TION)308 b(11)456 425 y Fs(Lemma)29 b(4.3.)40 b Fh(L)l(et)29 b Fn(z)t Fp(\()p Fk(\001)p Fn(;)14 b(!)s(;)g(z)1393 437 y Fl(0)1429 425 y Fp(\))30 b Fh(b)l(e)g(the)g(solution)g(of)h(\(16\).)g(Then)f(the)g (pr)l(o)l(c)l(ess)1237 739 y Fn(u)p Fp(\()p Fn(t;)14 b(!)s(;)g(u)1524 751 y Fl(0)1560 739 y Fp(\))23 b(=)g Fn(i)1732 705 y Fg(\000)p Fl(1)1821 739 y Fp(\()p Fn(\022)1892 751 y Fm(t)1921 739 y Fn(!)s Fp(\))c Fk(\016)f Fn(z)t Fp(\()p Fn(t;)c(!)s(;)g(i)p Fp(\()p Fn(!)s Fp(\))j Fk(\016)h Fn(u)2594 751 y Fl(0)2631 739 y Fp(\))456 1052 y Fh(solves)30 b(\(14\).)h(In)f(p)l(articular,)h Fn(u)e Fh(satis\014es)h(the)g(b)l (oundary)g(c)l(onditions)h(\(13\).)456 1322 y(Pr)l(o)l(of.)43 b Fp(The)27 b(assertion)f(follo)n(ws)h(b)n(y)g(replacing)g Fn(z)k Fp(b)n(y)c Fn(u)g Fk(\000)g Fn(z)2368 1334 y Fm(W)2430 1342 y Fd(1)2467 1322 y Fp(\()p Fn(\022)2538 1334 y Fm(t)2568 1322 y Fn(!)s Fp(\))g Fk(\000)g Fn(z)2813 1334 y Fm(W)2875 1342 y Fd(2)2912 1322 y Fp(\()p Fn(\022)2983 1334 y Fm(t)3012 1322 y Fn(!)s Fp(\).)p 3384 1322 4 57 v 3388 1270 50 4 v 3388 1322 V 3437 1322 4 57 v 456 1663 a(W)-7 b(e)29 b(will)h(no)n(w)f(c)n(hec)n(k)f(in)i(the)f(follo)n(wing)g(Lemmata)f (that)i(the)g(assumptions)e(of)i(the)f(random)456 1787 y(\014xed)i(p)r(oin)n(t)h(theorem)f(2.1)f(are)h(satis\014ed.)g(First,)g (w)n(e)g(sho)n(w)g(that)h(there)f(exists)g(a)g(temp)r(ered)456 1912 y(random)19 b(set)h Fk(X)12 b Fp(\()p Fn(!)s Fp(\))21 b(of)g(\(single)f(v)-5 b(alued\))21 b(random)e(v)-5 b(ariables)19 b(that)i(will)f(b)r(e)h(mapp)r(ed)g(in)n(to)f(itself.)456 2182 y Fs(Lemma)29 b(4.4.)40 b Fh(L)l(et)45 b Fk(X)12 b Fp(\()p Fn(!)s Fp(\))46 b Fh(b)l(e)g(the)g(b)l(al)t(l)g Fn(B)t Fp(\(0)p Fn(;)14 b(\032)p Fp(\()p Fn(!)s Fp(\)\))46 b Fh(in)g Fn(H)52 b Fh(with)46 b(c)l(enter)f(zer)l(o)h(and)g Fk(F)3365 2194 y Fl(0)3402 2182 y Fh({)456 2306 y(me)l(asur)l(able)30 b(r)l(adius)673 2634 y Fn(\032)p Fp(\()p Fn(!)s Fp(\))83 b(=)1065 2517 y Fe(\022)1140 2521 y(Z)1223 2541 y Fl(0)1186 2709 y Fg(\0001)1323 2634 y Fp(exp)1463 2517 y Fe(\022)1524 2634 y Fp(\()p Fn(\025)1604 2646 y Fl(1)1642 2634 y Fn(\027)24 b Fk(\000)18 b Fp(2)p Fn(\014)t(C)1942 2646 y Fm(G;x)2074 2634 y Fp(+)g(2)p Fn(r)r Fp(\))p Fn(\034)1150 2905 y Fp(+)1225 2848 y(3)p Fn(c)1303 2818 y Fl(2)1303 2871 y Fm(B)p 1225 2886 135 4 v 1269 2962 a Fn(\027)1383 2792 y Fe(Z)1466 2812 y Fl(0)1429 2980 y Fm(\034)1517 2905 y Fk(jr)p Fn(z)1648 2917 y Fm(W)1710 2925 y Fd(1)1747 2905 y Fp(\()p Fn(\022)1818 2917 y Fm(s)1854 2905 y Fn(!)s Fp(\))g(+)g Fk(r)p Fn(z)2150 2917 y Fm(W)2212 2925 y Fd(2)2249 2905 y Fp(\()p Fn(\022)2320 2917 y Fm(s)2356 2905 y Fn(!)s Fp(\)\))p Fk(j)2498 2870 y Fl(2)2535 2905 y Fn(ds)2617 2788 y Fe(\023)2697 2905 y Fk(\001)h Fn(R)q Fp(\()p Fn(\022)2874 2917 y Fm(\034)2915 2905 y Fn(!)s Fp(\))p Fn(d\034)3090 2788 y Fe(\023)3162 2782 y Fd(1)p 3162 2791 29 3 v 3162 2825 a(2)3204 2905 y Fn(;)456 3239 y Fh(wher)l(e)750 3560 y Fn(R)q Fp(\()p Fn(!)s Fp(\))83 b(=)g(3)1216 3503 y(\()p Fn(C)1307 3515 y Fm(G;x)1420 3503 y Fn(\014)23 b Fp(+)18 b Fn(r)r Fp(\))1644 3473 y Fl(2)p 1215 3541 467 4 v 1383 3617 a Fn(\027)5 b(\025)1477 3629 y Fl(1)1692 3560 y Fk(j)p Fn(z)1754 3572 y Fm(W)1816 3580 y Fd(1)1853 3560 y Fp(\()p Fn(\022)1924 3572 y Fm(t)1953 3560 y Fn(!)s Fp(\))19 b(+)f Fn(z)2181 3572 y Fm(W)2243 3580 y Fd(2)2279 3560 y Fp(\()p Fn(\022)2350 3572 y Fm(t)2380 3560 y Fn(!)s Fp(\))p Fk(j)2490 3525 y Fl(2)1016 3792 y Fp(+)1174 3736 y(3)p Fn(c)1252 3706 y Fl(2)1252 3759 y Fm(B)p 1174 3773 135 4 v 1218 3849 a Fn(\027)1318 3792 y Fk(j)p Fn(z)1380 3804 y Fm(W)1442 3812 y Fd(1)1479 3792 y Fp(\()p Fn(\022)1550 3804 y Fm(t)1580 3792 y Fn(!)s Fp(\))g(+)g Fn(z)1807 3804 y Fm(W)1869 3812 y Fd(2)1906 3792 y Fp(\()p Fn(\022)1977 3804 y Fm(t)2006 3792 y Fn(!)s Fp(\))p Fk(j)2116 3758 y Fl(2)2153 3792 y Fk(jr)p Fn(z)2284 3804 y Fm(W)2346 3812 y Fd(1)2383 3792 y Fp(\()p Fn(\022)2454 3804 y Fm(t)2484 3792 y Fn(!)s Fp(\))g(+)g Fk(r)p Fn(z)2780 3804 y Fm(W)2842 3812 y Fd(2)2879 3792 y Fp(\()p Fn(\022)2950 3804 y Fm(t)2979 3792 y Fn(!)s Fp(\))p Fk(j)3089 3758 y Fl(2)3127 3792 y Fn(;)456 4106 y Fh(and)30 b(supp)l(ose)g(that)1138 4443 y Fn(\025)1186 4455 y Fl(1)1224 4443 y Fn(\027)24 b Fp(+)18 b(2)p Fn(r)25 b(>)e Fp(2)p Fn(C)1665 4455 y Fm(G;x)1778 4443 y Fn(\014)f Fp(+)1940 4387 y(3)p Fn(c)2018 4357 y Fl(2)2018 4410 y Fm(B)p 1940 4424 V 1985 4500 a Fn(\027)2085 4443 y Fi(E)8 b Fk(jr)q Fn(z)2266 4455 y Fm(W)2328 4463 y Fd(1)2389 4443 y Fp(+)18 b Fk(r)p Fn(z)2580 4455 y Fm(W)2642 4463 y Fd(2)2678 4443 y Fk(j)2701 4409 y Fl(2)2739 4443 y Fn(;)456 4757 y Fh(wher)l(e)33 b Fn(\025)741 4769 y Fl(1)808 4757 y Fn(>)28 b Fp(0)33 b Fh(is)g(the)g(\014rst)f(eigenvalue)i(of)g(the)f(op)l(er)l(ator)h Fn(A)p Fh(.)g(Then)f(the)h(r)l(andom)f(set)g Fk(X)45 b Fh(is)456 4881 y(forwar)l(d)31 b(invariant,)g(i.e.,)1331 5195 y Fn(z)t Fp(\()p Fn(t;)14 b(!)s(;)g Fk(X)e Fp(\()p Fn(!)s Fp(\)\))23 b Fk(\032)g(X)12 b Fp(\()p Fn(\022)2040 5207 y Fm(t)2070 5195 y Fn(!)s Fp(\))p Fn(;)183 b(t)23 b Fk(\025)g Fp(0)p Fn(:)p eop %%Page: 12 12 12 11 bop 456 238 a Fl(12)345 b(JINQIA)n(O)29 b(DUAN,)g(PETER)f(E.)h (KLOEDEN,)f(AND)h(BJ)2417 223 y(\177)2408 238 y(ORN)g(SCHMALFUSS)456 425 y Fh(Pr)l(o)l(of.)43 b Fp(W)-7 b(e)28 b(ha)n(v)n(e)f(to)h(estimate) g Fk(j)p Fn(z)t Fk(j)1575 395 y Fl(2)1639 425 y Fp(for)g(whic)n(h)g(w)n (e)f(need)h(the)h(follo)n(wing)d(relations)h(that)h(are)456 550 y(a)f(consequence)f(of)i(Lemma)f(3.1:)622 727 y(2)p Fk(h)p Fn(B)t Fp(\()p Fn(z)t(;)14 b(z)t Fp(\))p Fn(;)g(z)t Fk(i)21 b Fp(=)i(0)p Fn(;)179 b Fp(2)p Fk(h)p Fn(Az)t(;)14 b(z)t Fk(i)22 b Fp(=)h(2)p Fn(\027)5 b Fk(k)p Fn(z)t Fk(k)2031 692 y Fl(2)2089 727 y Fk(\025)23 b Fn(\027)5 b Fk(k)p Fn(z)t Fk(k)2350 692 y Fl(2)2404 727 y Fp(+)18 b Fn(\027)5 b(\025)2581 739 y Fl(1)2619 727 y Fk(j)p Fn(z)t Fk(j)2708 692 y Fl(2)2745 727 y Fn(;)622 954 y Fp(2)p Fn(\014)t Fk(h)p Fn(G)p Fp(\()p Fn(z)t Fp(\))919 966 y Fm(x)961 954 y Fn(;)14 b(z)t Fk(i)22 b(\024)h Fp(2)p Fn(\014)t(C)1335 966 y Fm(G;x)1448 954 y Fk(j)p Fn(z)t Fk(j)1537 920 y Fl(2)1574 954 y Fn(;)180 b Fp(2)p Fn(r)r Fk(h)p Fn(z)t(;)14 b(z)t Fk(i)22 b Fp(=)h(2)p Fn(r)r Fk(j)p Fn(z)t Fk(j)2325 920 y Fl(2)2362 954 y Fn(;)622 1181 y Fp(2)p Fk(jh)p Fn(B)t Fp(\()p Fn(z)t(;)14 b(z)937 1193 y Fm(W)999 1201 y Fd(1)1035 1181 y Fp(\()p Fn(\022)1106 1193 y Fm(t)1135 1181 y Fn(!)s Fp(\))19 b(+)f Fn(z)1363 1193 y Fm(W)1425 1201 y Fd(2)1461 1181 y Fp(\()p Fn(\022)1532 1193 y Fm(t)1562 1181 y Fn(!)s Fp(\)\))p Fn(;)c(z)t Fk(ij)22 b(\024)h Fp(2)p Fn(c)2004 1193 y Fm(B)2061 1181 y Fk(j)p Fn(z)t Fk(jjr)p Fn(z)2281 1193 y Fm(W)2343 1201 y Fd(1)2379 1181 y Fp(\()p Fn(\022)2450 1193 y Fm(t)2479 1181 y Fn(!)s Fp(\))c(+)f Fk(r)p Fn(z)2776 1193 y Fm(W)2838 1201 y Fd(2)2875 1181 y Fp(\()p Fn(\022)2946 1193 y Fm(t)2975 1181 y Fn(!)s Fp(\)\))p Fk(jk)p Fn(z)t Fk(k)1286 1365 y(\024)1383 1309 y Fp(3)p Fn(c)1461 1279 y Fl(2)1461 1332 y Fm(B)p 1383 1346 135 4 v 1428 1422 a Fn(\027)1528 1365 y Fk(jr)p Fn(z)1659 1377 y Fm(W)1721 1385 y Fd(1)1758 1365 y Fp(\()p Fn(\022)1829 1377 y Fm(t)1858 1365 y Fn(!)s Fp(\))h(+)f Fk(r)p Fn(z)2155 1377 y Fm(W)2217 1385 y Fd(2)2254 1365 y Fp(\()p Fn(\022)2325 1377 y Fm(t)2354 1365 y Fn(!)s Fp(\)\))p Fk(j)2496 1331 y Fl(2)2534 1365 y Fk(j)p Fn(z)t Fk(j)2623 1331 y Fl(2)2678 1365 y Fp(+)2771 1309 y Fn(\027)p 2771 1346 47 4 v 2773 1422 a Fp(3)2827 1365 y Fk(k)p Fn(z)t Fk(k)2954 1331 y Fl(2)2989 1365 y Fn(;)622 1595 y Fp(2)p Fk(h)p Fn(B)t Fp(\()p Fn(z)834 1607 y Fm(W)896 1615 y Fd(1)932 1595 y Fp(\()p Fn(\022)1003 1607 y Fm(t)1033 1595 y Fn(!)s Fp(\))g(+)g Fn(z)1260 1607 y Fm(W)1322 1615 y Fd(2)1359 1595 y Fp(\()p Fn(\022)1430 1607 y Fm(t)1459 1595 y Fn(!)s Fp(\)\))p Fn(;)c(z)t Fp(\))p Fn(;)g(z)t Fk(i)23 b Fp(=)f(0)p Fn(;)622 1822 y Fp(2)p Fk(jh)p Fn(B)t Fp(\()p Fn(z)857 1834 y Fm(W)919 1842 y Fd(1)956 1822 y Fp(\()p Fn(\022)1027 1834 y Fm(t)1056 1822 y Fn(!)s Fp(\))c(+)g Fn(z)1283 1834 y Fm(W)1345 1842 y Fd(2)1382 1822 y Fp(\()p Fn(\022)1453 1834 y Fm(t)1483 1822 y Fn(!)s Fp(\))p Fn(;)c(z)1646 1834 y Fm(W)1708 1842 y Fd(1)1744 1822 y Fp(\()p Fn(\022)1815 1834 y Fm(t)1844 1822 y Fn(!)s Fp(\))19 b(+)f Fn(z)2072 1834 y Fm(W)2134 1842 y Fd(2)2170 1822 y Fp(\()p Fn(\022)2241 1834 y Fm(t)2271 1822 y Fn(!)s Fp(\)\))p Fn(;)c(z)t Fk(ij)1037 2006 y(\024)1134 1950 y Fp(3)p Fn(c)1212 1920 y Fl(2)1212 1973 y Fm(B)p 1134 1987 135 4 v 1178 2063 a Fn(\027)1279 2006 y Fk(j)p Fn(z)1341 2018 y Fm(W)1403 2026 y Fd(1)1440 2006 y Fp(\()p Fn(\022)1511 2018 y Fm(t)1540 2006 y Fn(!)s Fp(\))k(+)g Fn(z)1767 2018 y Fm(W)1829 2026 y Fd(2)1866 2006 y Fp(\()p Fn(\022)1937 2018 y Fm(t)1967 2006 y Fn(!)s Fp(\))p Fk(j)2077 1972 y Fl(2)2114 2006 y Fk(jr)p Fn(z)2245 2018 y Fm(W)2307 2026 y Fd(1)2344 2006 y Fp(\()p Fn(\022)2415 2018 y Fm(t)2444 2006 y Fn(!)s Fp(\))h(+)f Fk(r)p Fn(z)2741 2018 y Fm(W)2803 2026 y Fd(2)2839 2006 y Fp(\()p Fn(\022)2910 2018 y Fm(t)2940 2006 y Fn(!)s Fp(\))p Fk(j)3050 1972 y Fl(2)3106 2006 y Fp(+)3199 1950 y Fn(\027)p 3199 1987 47 4 v 3201 2063 a Fp(3)3255 2006 y Fk(k)p Fn(z)t Fk(k)3382 1972 y Fl(2)3417 2006 y Fn(;)622 2236 y Fp(2)p Fk(jh)p Fn(\014)t(G)p Fp(\()p Fn(z)906 2248 y Fm(W)968 2256 y Fd(1)1005 2236 y Fp(\()p Fn(\022)1076 2248 y Fm(t)1106 2236 y Fn(!)s Fp(\))g(+)g Fn(z)1333 2248 y Fm(W)1395 2256 y Fd(2)1432 2236 y Fp(\()p Fn(\022)1503 2248 y Fm(t)1532 2236 y Fn(!)s Fp(\)\))1651 2248 y Fm(x)1711 2236 y Fk(\000)h Fn(r)r Fk(h)p Fn(z)1905 2248 y Fm(W)1967 2256 y Fd(1)2004 2236 y Fp(\()p Fn(\022)2075 2248 y Fm(t)2105 2236 y Fn(!)s Fp(\))f(+)g Fn(z)2332 2248 y Fm(W)2394 2256 y Fd(2)2431 2236 y Fp(\()p Fn(\022)2502 2248 y Fm(t)2531 2236 y Fn(!)s Fp(\))p Fn(;)c(z)t Fk(ij)1037 2420 y(\024)22 b Fp(3)1176 2364 y(\()p Fn(C)1267 2376 y Fm(G;x)1381 2364 y Fn(\014)g Fp(+)d Fn(r)r Fp(\))1605 2334 y Fl(2)p 1176 2401 467 4 v 1343 2477 a Fn(\027)5 b(\025)1437 2489 y Fl(1)1653 2420 y Fk(j)p Fn(z)1715 2432 y Fm(W)1777 2440 y Fd(1)1813 2420 y Fp(\()p Fn(\022)1884 2432 y Fm(t)1914 2420 y Fn(!)s Fp(\))18 b(+)g Fn(z)2141 2432 y Fm(W)2203 2440 y Fd(2)2240 2420 y Fp(\()p Fn(\022)2311 2432 y Fm(t)2340 2420 y Fn(!)s Fp(\))p Fk(j)2450 2386 y Fl(2)2506 2420 y Fp(+)2599 2364 y Fn(\027)5 b(\025)2693 2376 y Fl(1)p 2599 2401 132 4 v 2644 2477 a Fp(3)2741 2420 y Fk(j)p Fn(z)t Fk(j)2830 2386 y Fl(2)2867 2420 y Fn(:)456 2618 y Fp(It)41 b(can)g(b)r(e)g(sho)n(wn)g(b)n(y)f(a)h (comparison)e(argumen)n(t)i(that)g Fk(j)p Fn(z)t Fp(\()p Fn(t;)14 b(!)s(;)g(z)2655 2630 y Fl(0)2691 2618 y Fp(\))p Fk(j)2746 2588 y Fl(2)2825 2618 y Fp(is)41 b(b)r(ounded)g(b)n(y)g(a)456 2743 y(solution)27 b(of)g(the)h(a\016ne)g(random)e(di\013eren)n(tial)i (equation:)692 2887 y Fn(d\020)p 692 2924 86 4 v 698 3000 a(dt)806 2943 y Fp(+)18 b(\()p Fn(\025)969 2955 y Fl(1)1007 2943 y Fn(\027)24 b Fk(\000)18 b Fp(2)p Fn(\014)t(C)1307 2955 y Fm(G;x)1439 2943 y Fp(+)g(2)p Fn(r)i Fk(\000)1714 2887 y Fp(3)p Fn(c)1792 2857 y Fl(2)1792 2910 y Fm(B)p 1714 2924 135 4 v 1759 3000 a Fn(\027)1859 2943 y Fp(\()p Fk(jr)p Fn(z)2022 2955 y Fm(W)2084 2963 y Fd(1)2121 2943 y Fp(\()p Fn(\022)2192 2955 y Fm(t)2222 2943 y Fn(!)s Fp(\))e(+)g Fk(r)p Fn(z)2518 2955 y Fm(W)2580 2963 y Fd(2)2617 2943 y Fp(\()p Fn(\022)2688 2955 y Fm(t)2717 2943 y Fn(!)s Fp(\)\))p Fk(j)2859 2909 y Fl(2)2897 2943 y Fp(\)\))p Fn(\020)30 b Fp(=)22 b Fn(R)q Fp(\()p Fn(\022)3249 2955 y Fm(t)3279 2943 y Fn(!)s Fp(\))1286 3186 y Fn(\020)6 b Fp(\(0\))23 b(=)g Fk(j)p Fn(z)1607 3198 y Fl(0)1644 3186 y Fk(j)1667 3152 y Fl(2)1704 3186 y Fn(;)456 3044 y Fp(\(17\))456 3346 y(for)i(whic)n(h)h(the)g(solution)f(is)h(giv)n(en) f(b)n(y)g(a)h(v)-5 b(ariation)24 b(of)i(constan)n(t)f(form)n(ula.)g(A)h (direct)g(calcula-)456 3470 y(tion)j(v)n(eri\014es)g(that)h Fn(t)f Fk(!)h Fn(\032)1305 3440 y Fl(2)1342 3470 y Fp(\()p Fn(\022)1413 3482 y Fm(t)1442 3470 y Fn(!)s Fp(\))g(is)g(a)f(solution)g (of)g(equation)g(\(17\))h(with)g(initial)f(v)-5 b(alue)30 b Fn(\020)6 b Fp(\(0\))456 3595 y(=)27 b Fn(\032)591 3565 y Fl(2)628 3595 y Fp(\()p Fn(!)s Fp(\),)h(whic)n(h)g(means)f(that) h Fn(\032)1513 3565 y Fl(2)1578 3595 y Fp(is)g(a)f(random)g(\014xed)h (p)r(oin)n(t)f(of)h(\(17\).)g(It)g(th)n(us)f(follo)n(ws)g(that)456 3719 y Fn(z)t Fp(\()p Fn(t;)14 b(!)s(;)g(z)729 3731 y Fl(0)764 3719 y Fp(\))28 b Fk(2)g(X)12 b Fp(\()p Fn(\022)1049 3731 y Fm(t)1079 3719 y Fn(!)s Fp(\))28 b(whenev)n(er)f Fn(z)1598 3731 y Fl(0)1662 3719 y Fk(2)h(X)12 b Fp(\()p Fn(!)s Fp(\).)p 3384 3719 4 57 v 3388 3667 50 4 v 3388 3719 V 3437 3719 4 57 v 456 3904 a(W)-7 b(e)21 b(note)g(that)g(the)h (random)e(v)-5 b(ariable)20 b Fn(\032)h Fp(is)f(temp)r(ered)i(\(see)f ([17)o(])g(,)g(page)f(110\),)g(so)g(an)n(y)h(selector)456 4029 y(con)n(tained)27 b(in)g Fk(X)40 b Fp(is)28 b(also)e(temp)r(ered.) 456 4278 y(It)33 b(remains)g(to)g(c)n(hec)n(k)f(that)i(the)g(con)n (traction)e(condition)h(\(7\))g(of)h(the)f(random)g(\014xed)g(p)r(oin)n (t)456 4402 y(theorem)27 b(holds.)456 4594 y Fs(Lemma)i(4.5.)40 b Fh(Supp)l(ose)30 b(that)773 4795 y Fk(\000)18 b Fn(\027)5 b(\025)950 4807 y Fl(1)1006 4795 y Fp(+)18 b(2)p Fn(\014)t(C)1241 4807 y Fm(G;x)1373 4795 y Fk(\000)g Fp(2)p Fn(r)j Fp(+)1649 4739 y(3)p Fn(c)1727 4708 y Fl(2)1727 4761 y Fm(B)p 1649 4776 135 4 v 1693 4852 a Fn(\027)1793 4795 y Fi(E)9 b Fk(jr)p 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(the)g(phase)i(sp)l(ac)l(e)456 2795 y(exp)l(onential)t(ly)h(fast.)456 2993 y Fp(Indeed,)36 b(the)g(random)f(v)-5 b(ariable)34 b Fn(u)i Fp(that)g(generates)e(an)h(exp)r(onen)n(tially)h(stable)f (stationary)456 3118 y(solution)27 b(is)g(giv)n(en)g(b)n(y)1323 3306 y Fn(u)1371 3272 y Fg(\003)1408 3306 y Fp(\()p Fn(!)s Fp(\))d(=)e Fn(z)1681 3272 y Fg(\003)1719 3306 y Fp(\()p Fn(!)s Fp(\))c(+)g Fn(z)1978 3318 y Fm(W)2040 3326 y Fd(2)2077 3306 y Fp(\()p Fn(!)s Fp(\))h(+)f Fn(z)2337 3318 y Fm(W)2399 3326 y Fd(1)2435 3306 y Fp(\()p Fn(!)s Fp(\))p Fn(:)1657 3547 y Fp(5.)46 b Fo(Discussions)456 3721 y Fp(W)-7 b(e)38 b(ha)n(v)n(e)g(sho)n(wn)g(that,)g(under)h (suitable)f(conditions)g(on)g(the)h(random)f(forcing,)f(random)456 3846 y(b)r(oundary)27 b(conditions,)h(viscosit)n(y)-7 b(,)27 b(Ekman)h(constan)n(t)f(and)i(Coriolis)d(parameter,)h(all)h (quasi-)456 3970 y(geostrophic)h(motion)j(approac)n(h)d(a)i(unique)h (stationary)e(state)h(exp)r(onen)n(tially)g(fast)g(as)g(time)456 4095 y(go)r(es)d(to)g(in\014nit)n(y)-7 b(.)30 b(In)f(deterministic)g (systems)g(a)f(high)h(lev)n(el)f(of)h(stabilit)n(y)g(is)g(obtained)f (when)456 4219 y(there)i(is)g(an)g(exp)r(onen)n(tial)g(attractor)e (whic)n(h)i(attracts)g(tra)5 b(jectories)28 b(exp)r(onen)n(tially)i (fast.)g(In)456 4344 y(some)35 b(situations)g(this)i(attractor)d(is)i (a)g(single)f(p)r(oin)n(t)h(\(p)r(oin)n(t)h(attractor\))d(whic)n(h)i (describ)r(es)456 4468 y(the)d(laminar)e(b)r(eha)n(vior)h(of)g(the)h (\015o)n(w.)f(W)-7 b(e)34 b(are)d(lo)r(oking)h(for)g(suc)n(h)g (stabilit)n(y)h(in)g(the)g(case)f(of)456 4593 y(quasigeostrophic)d (\015uid)j(motion)g(under)g(random)f(p)r(erturbations.)g(In)h (particular,)f(w)n(e)g(\014nd)456 4717 y(a)38 b(random)g(attractor)f (whic)n(h)i(is)g(de\014ned)g(b)n(y)g(a)f(single)h(random)f(v)-5 b(ariable.)38 b(This)g(random)456 4842 y(v)-5 b(ariable)22 b(attracts)h(all)h(other)f(quasigeostrophic)f(motion)h(exp)r(onen)n (tially)h(fast.)g(This)f(random)456 4967 y(v)-5 b(ariable)35 b(corresp)r(onds)g(to)i(a)f(unique)h(in)n(v)-5 b(arian)n(t)36 b(measure,)g(whic)n(h)g(is)h(the)g(Dirac)g(measure)456 5091 y(with)25 b(the)h(random)f(v)-5 b(ariable)24 b(as)h(the)g(random)g (mass)g(p)r(oin)n(t.)g(The)h(corresp)r(onding)d(stationary)456 5216 y(Mark)n(o)n(v)i(measure)h(is)i(the)g(exp)r(ectation)f(of)h(this)g (random)e(Dirac)i(measure.)p eop %%Page: 15 15 15 14 bop 831 226 a Fl(ST)-5 b(ABILITY)29 b(OF)g(THE)g(STOCHASTIC)h (QUASIGEOSTR)n(OPHIC)g(EQUA)-5 b(TION)308 b(15)1708 425 y Fo(References)491 579 y Fr([1])35 b(L.)23 b(Arnold.)g Fj(Random)k(Dynamic)l(al)f(Systems)p Fr(.)d(Springer{V)-6 b(erlag,)23 b(Berlin,)g(1998.)491 683 y([2])35 b(A.)30 b(Bensoussan)i(and)g(R.)f(T)-6 b(emam.)29 b(Equations)j(sto)r(c)n (hastiques)h(du)f(t)n(yp)r(e)g(Na)n(vier{Stok)n(es.)h Fj(J.)g(F)-5 b(unct.)601 786 y(A)n(nal.)p Fr(,)23 b(13:195{222,)j (1973.)491 890 y([3])35 b(A.)25 b(J.)i(Bourgeois)f(and)h(J.)f(T.)g (Beale.)h(V)-6 b(alidit)n(y)26 b(of)g(the)i(quasigeostrophic)f(mo)r (del)f(for)g(large-scale)g(\015o)n(w)601 994 y(in)d(the)i(atmosphere)e (and)i(o)r(cean.)f Fj(SIAM)i(J.)g(Math.)f(A)n(nal.)p Fr(,)f(25:1023{1068,)i(1994.)491 1098 y([4])35 b(J.)16 b(R.)g(Brannan,)h(J.)g(Duan,)f(and)i(T.)e(W)-6 b(anner.)17 b(Dissipativ)n(e)g(quasigeostrophic)h(dynamics)e(under)h(random)601 1201 y(forcing.)23 b Fj(J.)i(Math.)h(A)n(nal.)g(Appl.)p Fr(,)e(228:221{233,)i(1998.)491 1305 y([5])35 b(H.)22 b(Crauel)h(and)g(F.)g(Flandoli.)f(A)n(ttractors)i(for)e(random)g (dynamical)g(systems.)g Fj(Pr)l(ob.)k(Th.)f(R)l(el.)g(Fields)p Fr(,)601 1409 y(100:)f(365{393,)h(1994.)491 1513 y([6])35 b(B.)19 b(Desjardins)h(and)h(E.)f(Grenier.)f(Deriv)l(ation)i(of)f (quasigeostrophic)i(p)r(oten)n(tial)f(v)n(orticit)n(y)g(equations,)h (to)601 1617 y(app)r(ear)i(in)f Fj(A)l(dv.)j(Di\013.)f(Eqns.)p Fr(,)e(1998.)491 1720 y([7])35 b(P)-6 b(.)25 b(F.)g(Em)n(bid)g(and)i (A.)e(J.)g(Ma)t(jda.)g(Av)n(eraging)i(o)n(v)n(er)f(fast)g(gra)n(vit)n 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b(Da)g(Prato)g(and)h(J.)f(Zab)r(czyk.)h Fj(Er)l(go)l(dicity)i(for)f(In\014nite)g(Dimensional)h(Systems)p Fr(.)c(Cam)n(bridge)g(Uni-)601 3485 y(v)n(ersit)n(y)g(Press,)e(Cam)n (bridge,)h(1996.)456 3588 y([16])35 b(B.)c(Sc)n(hmalfuss.)f(Qualitativ) n(e)j(prop)r(erties)f(of)g(the)h(sto)r(c)n(hastic)g(Na)n(vier)f(Stok)n (es)h(equation.)g Fj(Nonline)l(ar)601 3692 y(A)n(nalysis)26 b(TMA)p Fr(,)d(28:1545{1563,)j(1997.)456 3796 y([17])35 b(B.)17 b(Sc)n(hmalfuss.)e(A)j(random)e(\014xed)j(p)r(oin)n(t)f (theorem)f(and)h(the)g(random)f(graph)h(transformation.)e Fj(J.)k(Math.)601 3900 y(A)n(nal.)26 b(Applns.)p Fr(,)e (225\(1\):91{113,)j(1998.)456 4003 y([18])35 b(S.)d(H.)g(Sc)n(ho)r(c)n (het.)i(Singular)e(limits)f(in)h(b)r(ounded)i(domains)e(for)f (quasilinear)h(symmetric)f(h)n(yp)r(erb)r(olic)601 4107 y(systems)23 b(ha)n(ving)h(a)g(v)n(orticit)n(y)g(equation.)h Fj(J.)h(Di\013.)e(Eqns.)p Fr(,)g(68:400{428,)h(1987.)456 4211 y([19])35 b(T.)23 b(G.)h(Shepherd,)g(T.)f(W)-6 b(arn,)24 b(O.)f(Bokho)n(v)n(e)j(and)e(G.)g(K.)f(V)-6 b(allis.)22 b(Rossb)n(y)i(n)n(um)n(b)r(er)f(expansions,)h(sla)n(ving)601 4315 y(principle)f(and)h(balance)h(dynamics.)e Fj(Quart.)i(J.)h(R)l (oy.)g(Met.)e(So)l(c.)p Fr(,)g(121:723{739,)h(1995.)456 4419 y([20])35 b(R.)23 b(T)-6 b(emam.)22 b Fj(Navier{Stokes)27 b(Equation{The)l(ory)h(and)f(Numeric)l(al)g(A)n(nalysis)p Fr(.)d(North{Holland,)g(Ams-)601 4522 y(terdam,)e(1979.)456 4626 y([21])35 b(G.)23 b(K.)f(V)-6 b(allis.)22 b(P)n(oten)n(tial)i(v)n (orticit)n(y)g(in)n(v)n(ersion)f(and)h(balanced)h(equations)f(of)f (motion)g(for)g(rotating)h(and)601 4730 y(strati\014ed)g(\015o)n(ws.)g Fj(Quart.)h(J.)h(R)l(oy.)f(Met.)g(So)l(c.)p Fr(,)f(122:291{322,)h (1996.)p eop %%Page: 16 16 16 15 bop 456 238 a Fl(16)345 b(JINQIA)n(O)29 b(DUAN,)g(PETER)f(E.)h (KLOEDEN,)f(AND)h(BJ)2417 223 y(\177)2408 238 y(ORN)g(SCHMALFUSS)456 425 y Fr(\(Jinqiao)37 b(Duan\))g Fq(Dep)l(ar)l(tment)f(of)h(Ma)l(thema) l(tical)f(Sciences,)i(Clemson)g(University,)g(Clemson,)456 529 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