Content-Type: multipart/mixed; boundary="-------------0104251439836" This is a multi-part message in MIME format. ---------------0104251439836 Content-Type: text/plain; name="01-156.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-156.comments" To appear: International J. Math and Mathematical Sciences ---------------0104251439836 Content-Type: text/plain; name="01-156.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-156.keywords" Ergodicity, geophysical fluid dynamics, stochastic PDEs, invariant measure ---------------0104251439836 Content-Type: application/postscript; name="ergodic.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ergodic.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ergodic.dvi %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips ergodic -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.03.19:1110 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 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2397 4157 78 4 v 89 x Fk(QdW)41 b(;)752 b Fm(\(3\))1541 4392 y Fk(!)t Fm(\(0\))54 b(giv)m(en)p Fk(;)1403 b Fm(\(4\))324 4612 y(where)28 b Fk(W)14 b Fm(\()p Fk(x;)j(y)t(;)g(t)p Fm(\))26 b(is)g(a)h(Wiener)g(pro)s(cess)i(de\014ned)f(on)f(a)g (probabilit)m(y)e(space)j(\(\012)p Fk(;)17 b Fh(F)10 b Fk(;)17 b Fe(P)p Fm(\).)324 4732 y(The)28 b(co)m(v)-5 b(ariance)26 b(op)s(erator)g Fk(Q)i Fm(:)g Fk(H)35 b Fh(!)28 b Fk(H)34 b Fm(for)26 b(this)g(Wiener)h(pro)s(cess)h(is)f(a)f (nonnegativ)m(e)324 4852 y(and)38 b(symmetric)e(linear)g(con)m(tin)m (uous)j(op)s(erator)e(to)g(b)s(e)h(sp)s(eci\014ed)g(b)s(elo)m(w.)59 b(The)38 b(term)1918 5251 y(3)p eop %%Page: 4 4 4 3 bop 324 548 a Fm(with)28 b(Ito)g(deriv)-5 b(ativ)m(e,)1163 473 y Fh(p)p 1247 473 78 4 v 1247 548 a Fk(QdW)14 b Fm(,)29 b(is)f(a)g(mo)s(del)f(for)h(the)h(white-in-time)d(noise)i(represen)m (t-)324 668 y(ing)40 b(the)i(random)f(wind)g(forcing.)69 b(This)41 b(equation)h(can)f(b)s(e)h(rewritten)f(in)g(the)h(mild)324 789 y(\(in)m(tegral\))31 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4950 V 977 4991 a(8)1021 5002 y Fm(.)1918 5251 y(4)p eop %%Page: 5 5 5 4 bop 470 548 a Fm(W)-8 b(e)33 b(further)g(assume)g(that)833 761 y Fk(\024)p Fm(\()p Fk(D)s Fm(\))28 b(=)196 b(inf)1180 827 y Fi(0)p Fj(<\032)27 b Fm(0)p Fk(;)324 999 y Fm(where)43 b Fk(diam)p Fm(\()p Fk(D)s Fm(\))f(is)f(the)i(diameter)d(of)i Fk(D)i Fm(\(the)f(least)e(upp)s(er)i(b)s(ound)f(of)f(t)m(w)m(o-p)s(oin)m(t)324 1119 y(distances)29 b(in)f Fk(D)s Fm(\),)h Fk(meas)p Fm(\()p Fh(\001)p Fm(\))f(denotes)i(the)f(Leb)s(esgue)g(measure,)h(and) f Fk(B)5 b Fm(\()p Fk(x;)17 b(y)t Fm(;)g Fk(\032)p Fm(\))27 b(is)h(the)324 1240 y(op)s(en)43 b(disk)h(cen)m(tered)h(at)e(\()p Fk(x;)17 b(y)t Fm(\))42 b(and)h(with)g(radius)g Fk(\032)p Fm(.)76 b(W)-8 b(e)43 b(also)f(assume)i(that)f(the)324 1360 y(eigenfunctions)32 b Fk(e)1006 1375 y Fj(k)1082 1360 y Fm(satisfy)1332 1542 y Fk(e)1377 1557 y Fj(k)1448 1542 y Fh(2)c Fk(C)1612 1557 y Fi(0)1651 1542 y Fm(\()1712 1517 y(\026)1689 1542 y Fk(D)s Fm(\))p Fk(;)114 b Fh(j)p Fk(e)2025 1557 y 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Fm(for)32 b Fk(x)c Fh(2)g Fk(H)40 b Fm(and)33 b(\000)27 b Fh(2)h(B)s Fm(\()p Fk(H)8 b Fm(\),)33 b(the)g Fk(\033)t Fm(-algebra)e(of)h(Borel)g(sets)i(in)d Fk(H)8 b Fm(.)470 668 y(A)33 b(probabilit)m(y)d(measure)j Fk(\026)f Fm(on)h(\()p Fk(H)r(;)17 b Fh(B)s Fm(\()p Fk(H)8 b Fm(\)\))32 b(is)g(called)f(in)m(v)-5 b(arian)m(t)32 b(if)1555 785 y Ff(Z)1671 920 y Fk(g)t(d\026)27 b Fm(=)1962 785 y Ff(Z)2078 920 y Fk(P)2141 935 y Fj(t)2171 920 y Fk(g)t(d\026)324 1173 y Fm(for)32 b(an)m(y)h Fk(t)28 b(>)g Fm(0)k(and)g Fk(g)f Fh(2)e Fk(B)1341 1188 y Fj(b)1375 1173 y Fm(\()p Fk(H)8 b Fm(\),)32 b(or,)g(equiv)-5 b(alen)m(tly)d(,) 1477 1289 y Ff(Z)1533 1515 y Fj(H)1616 1425 y Fk(P)1679 1440 y Fj(t)1709 1425 y Fm(\()p Fk(x;)17 b Fm(\000\))p Fk(d\026)27 b Fm(=)h Fk(\026)p Fm(\(\000\))p Fk(;)324 1681 y Fm(for)k(an)m(y)h Fk(t)28 b(>)g Fm(0,)k Fk(x)c Fh(2)g Fk(H)40 b Fm(and)33 b(\000)27 b Fh(2)i(B)s Fm(\()p Fk(H)8 b Fm(\).)470 1801 y(The)26 b(existence)h(of)e(an)g(in)m(v)-5 b(arian)m(t)24 b(measure)i(for)e(the)i(quasigeostrophic)f(\015o)m(w)h (mo)s(del)324 1922 y(\(3\))35 b(follo)m(ws)g(from)f(a)i(tigh)m(tness)g (or)g(,)h(equiv)-5 b(alen)m(tly)d(,)36 b(a)f(compactness)i(argumen)m(t) f([21)o(].)324 2042 y(If)28 b(the)i(mean-square)f(norm)e(of)i(the)g (solution)e(is)h(b)s(ounded)h(for)g(all)d(time)h Fk(t)h(>)g Fm(0)g(and)h(for)324 2162 y(all)f(initial)e(data,)k(then)g(b)m(y)h(the) f(Cheb)m(yshev)j(inequalit)m(y)-8 b(,)30 b(the)g(solution)e(is)i(b)s (ounded)g(in)324 2283 y(probabilit)m(y)-8 b(,)30 b(whic)m(h)j(further)g (implies)d(that)j(the)g(family)d(of)i(measures)h(on)g(\()p Fk(H)r(;)17 b Fh(B)s Fm(\()p Fk(H)8 b Fm(\)\))1445 2487 y(1)p 1434 2531 71 4 v 1434 2622 a Fk(T)1531 2418 y Ff(Z)1631 2445 y Fj(T)1586 2644 y Fi(0)1702 2554 y Fk(P)1765 2569 y Fj(t)1795 2554 y Fm(\()p Fk(x;)17 b Fh(\001)p Fm(\))p Fk(dt;)71 b(T)42 b Fh(\025)28 b Fm(1)p Fk(;)324 2810 y Fm(is)34 b(tigh)m(t)g(for)h(some)g Fk(x)d Fh(2)g Fk(H)8 b Fm(;)36 b(see)g([4],)g(page)f(89-90.)49 b(Th)m(us)37 b(b)m(y)f(Corollary)d(3.1.2)h(in)h([4)o(],)324 2931 y(there)28 b(exists)h(an)f(in)m(v)-5 b(arian)m(t)26 b(measure)i(for)g(the)g (quasigeostrophic)f(\015o)m(w)i(mo)s(del)d(\(3\).)42 b(So)324 3051 y(in)32 b(the)h(rest)g(of)f(this)g(section,)h(w)m(e)h (estimate)d(the)i(mean-square)g(norm)f Fe(E)12 b Fh(k)p Fk(!)t Fm(\()p Fk(t)p Fm(\))p Fh(k)3310 3015 y Fi(2)3355 3051 y Fm(.)470 3171 y(W)-8 b(e)33 b(assume)g(that)1259 3265 y Ff(Z)1359 3292 y Fd(1)1315 3491 y Fi(0)1450 3401 y Fh(k)p Fk(S)6 b Fm(\()p Fk(r)s Fm(\))1689 3311 y Ff(p)p 1788 3311 78 4 v 1788 3401 a Fk(Q)p Fh(k)1915 3360 y Fi(2)1915 3426 y Fj(H)f(S)2029 3401 y Fk(dr)30 b(<)e Fm(+)p Fh(1)p Fk(;)927 b Fm(\(11\))324 3657 y(where)34 b Fh(k)22 b(\001)f(k)777 3672 y Fj(H)5 b(S)924 3657 y Fm(is)32 b(the)h(Hilb)s(ert-Sc)m(hmidt)d(norm.)42 b(W)-8 b(e)33 b(rewrite)g(\(5\))f(as)1432 3861 y Fk(!)t Fm(\()p Fk(t)p Fm(\))27 b(=)g Fk(Y)22 b Fm(\()p Fk(t)p Fm(\))g(+)g Fk(Z)7 b Fm(\()p Fk(t)p Fm(\))28 b Fk(;)1100 b Fm(\(12\))324 4065 y(where)1130 4223 y Fk(Y)21 b Fm(\()p Fk(t)p Fm(\))28 b(=)f Fk(S)6 b Fm(\()p Fk(t)p Fm(\))p Fk(x)23 b Fm(+)1803 4087 y Ff(Z)1902 4114 y Fj(t)1858 4313 y Fi(0)1949 4223 y Fk(S)6 b Fm(\()p Fk(t)22 b Fh(\000)g Fk(s)p Fm(\))p Fk(F)14 b Fm(\()p Fk(!)t Fm(\()p Fk(s)p Fm(\)\))p Fk(ds;)324 4443 y Fm(with)40 b(initial)c(data)k Fk(!)t Fm(\(0\))g(=)h Fk(x)p Fm(,)i(and)d Fk(Z)7 b Fm(\()p Fk(t)p Fm(\))41 b(is)e(the)i(Ornstein-Uhlen)m(b)s(ec)m(k)h(pro)s(cess)f(in)324 4563 y(\(6\).)470 4684 y(By)33 b(Corollary)e(4.14)h(in)g([3],)g(for)g (an)m(y)i Fk(x)28 b Fh(2)g Fk(H)8 b Fm(,)816 4950 y(sup)832 5031 y Fj(t)p Fd(\025)p Fi(0)979 4950 y Fe(E)13 b Fh(k)p Fk(Z)7 b Fm(\()p Fk(t)p Fm(\))p Fh(k)1325 4909 y Fi(2)1398 4950 y Fm(=)27 b(sup)1517 5031 y Fj(t)p Fd(\025)p Fi(0)1665 4950 y Fe(E)1742 4815 y Ff(Z)1847 4841 y Fj(t)1803 5040 y Fi(0)1894 4950 y Fh(k)p Fk(S)6 b Fm(\()p Fk(r)s Fm(\))2133 4861 y Ff(p)p 2231 4861 V 2231 4950 a Fk(Q)q Fh(k)2359 4909 y Fi(2)2359 4975 y Fj(H)f(S)2472 4950 y Fk(dr)30 b(<)e Fm(+)p Fh(1)p Fk(:)484 b Fm(\(13\))1918 5251 y(6)p eop %%Page: 7 7 7 6 bop 470 548 a Fm(By)28 b([2])f(or)g(follo)m(w)e(a)i(Y)-8 b(osida)26 b(appro)m(ximation)f(com)m(bined)i(with)g Fk(L)2882 512 y Fi(2)2922 548 y Fm(-norm)e(estimate)324 668 y(as)33 b(in)e(Prop)s(osition)g(6.1.6)h(in)g([4],)h(w)m(e)g(ha)m(v) m(e,)h(for)e(an)m(y)i Fk(x)28 b Fh(2)g Fk(H)8 b Fm(,)1414 867 y(sup)1429 947 y Fj(t)p Fd(\025)p Fi(0)1577 867 y Fe(E)13 b Fh(k)p Fk(Y)27 b Fm(\()p Fk(t)p Fm(\))p Fh(k)1933 826 y Fi(2)2000 867 y Fk(<)g Fm(+)p Fh(1)p Fk(:)1082 b Fm(\(14\))324 1124 y(Note)32 b(that)658 1323 y Fh(k)p Fk(!)t Fm(\()p Fk(t)p Fm(\))p Fh(k)934 1282 y Fi(2)1000 1323 y Fm(=)p Fk(<)c(Y)43 b Fm(+)22 b Fk(Z)r(;)17 b(Y)43 b Fm(+)22 b Fk(Z)35 b(>)83 b Fm(=)h Fh(k)p Fk(Y)21 b Fh(k)2288 1282 y Fi(2)2349 1323 y Fm(+)h(2)28 b Fk(<)f(Y)5 b(;)17 b(Z)35 b(>)27 b Fm(+)p Fh(k)p Fk(Z)7 b Fh(k)3188 1282 y Fi(2)1949 1469 y Fh(\024)84 b(k)p Fk(Y)21 b Fh(k)2288 1427 y Fi(2)2349 1469 y Fm(+)h(2)p Fh(k)p Fk(Y)f Fh(k)28 b(k)p Fk(Z)7 b Fh(k)21 b Fm(+)h Fh(k)p Fk(Z)7 b Fh(k)3169 1427 y Fi(2)1949 1614 y Fh(\024)84 b Fm(2\()p Fh(k)p Fk(Y)21 b Fh(k)2375 1573 y Fi(2)2436 1614 y Fm(+)h Fh(k)p Fk(Z)7 b Fh(k)2708 1573 y Fi(2)2747 1614 y Fm(\))p Fk(:)324 1813 y Fm(Th)m(us,)34 b(b)m(y)f(\(13\))f(and)h(\(14\),)1421 2011 y(sup)1436 2092 y Fj(t)p Fd(\025)p Fi(0)1584 2011 y Fe(E)12 b Fh(k)p Fk(!)t Fm(\()p Fk(t)p Fm(\))p Fh(k)1920 1970 y Fi(2)1993 2011 y Fk(<)27 b Fm(+)p Fh(1)p Fk(:)1089 b Fm(\(15\))324 2269 y(By)39 b(the)h(argumen)m(t)e(in)h(the)g(b)s (eginning)f(of)g(this)h(section,)i(there)e(exists)h(at)f(least)f(one) 324 2389 y(in)m(v)-5 b(arian)m(t)32 b(measure)i(for)f(the)g(the)h (quasigeostrophic)g(\015o)m(w)g(mo)s(del)e(\(3\).)45 b(W)-8 b(e)34 b(ha)m(v)m(e)h(the)324 2509 y(main)c(result)h(in)g(this)g (section.)324 2713 y Fa(Theorem)37 b(1)49 b Fb(Assume)34 b(that)1465 2633 y Ff(R)1532 2660 y Fi(+)p Fd(1)1512 2748 y Fi(0)1678 2713 y Fh(k)p Fk(S)6 b Fm(\()p Fk(r)s Fm(\))1917 2639 y Fh(p)p 1999 2639 78 4 v 1999 2713 a Fk(Q)p Fh(k)2126 2677 y Fi(2)2126 2739 y Fj(H)f(S)2240 2713 y Fk(dr)30 b(<)d Fm(+)p Fh(1)p Fb(.)44 b(Then)34 b(ther)-5 b(e)34 b(exists)f(at)324 2834 y(le)-5 b(ast)43 b(one)g(invariant)f(pr)-5 b(ob)g(ability)43 b(me)-5 b(asur)g(e)43 b(for)g(the)g(quasige)-5 b(ostr)g(ophic)42 b(\015ow)i(mo)-5 b(del)324 2954 y(\(3\))34 b(in)h(the)g(sp)-5 b(ac)g(e)34 b Fk(L)1090 2918 y Fi(2)1129 2954 y Fm(\()p Fk(D)s Fm(\))h Fb(of)f(squar)-5 b(e-inte)g(gr)g(able)34 b(vorticities.)324 3284 y Fl(4)161 b(Uniqueness)52 b(of)i(an)g(In)l(v)-9 b(arian)l(t)54 b(Measure)324 3503 y Fm(No)m(w)24 b(w)m(e)h(consider)f (the)g(uniqueness)i(of)e(in)m(v)-5 b(arian)m(t)22 b(measure)i(for)f (the)i(quasigeostrophic)324 3623 y(\015o)m(w)40 b(mo)s(del)e(\(3\).)64 b(As)41 b(w)m(e)f(kno)m(w)h(in)e(Chapter)h(4)g(in)e([4],)k(the)e (uniqueness)i(of)d(in)m(v)-5 b(ari-)324 3744 y(an)m(t)41 b(measure)g(is)f(a)h(consequence)j(of)c(regularit)m(y)g(of)h(the)g (transition)e(semigroup)h Fk(P)3505 3759 y Fj(t)3535 3744 y Fm(,)324 3864 y(b)m(y)d(the)g(Do)s(ob's)g(Theorem.)56 b(Due)36 b(to)h(Khasminskii's)e(Theorem,)j(strong)f(F)-8 b(eller)34 b(and)324 3984 y(irreducibilit)m(y)39 b(prop)s(erties)j (imply)e(the)j(regularit)m(y)-8 b(.)71 b(So)42 b(w)m(e)h(no)m(w)g(try)f (to)g(pro)m(v)m(e)h(the)324 4105 y(strong)32 b(F)-8 b(eller)31 b(and)i(irreducibilit)m(y)d(prop)s(erties)i(for)g(the)h(transition)e (semigroup)h Fk(P)3409 4120 y Fj(t)3438 4105 y Fm(.)470 4225 y(Strong)j(F)-8 b(eller)33 b(prop)s(ert)m(y)i(means)g(that)g(for)f (ev)m(ery)j Fk(g)t Fm(\()p Fk(x)p Fm(\))d(in)h Fk(B)2789 4240 y Fj(b)2823 4225 y Fm(\()p Fk(H)8 b Fm(\),)35 b(the)g(space)h(of) 324 4346 y(b)s(ounded)f(Borel)f Fb(me)-5 b(asur)g(able)34 b Fm(functions)h(on)g Fk(H)8 b Fm(,)35 b Fk(P)2257 4361 y Fj(t)2286 4346 y Fk(g)t Fm(\()p Fk(x)p Fm(\))g(is)f(in)g Fk(C)2789 4361 y Fj(b)2823 4346 y Fm(\()p Fk(H)8 b Fm(\),)35 b(the)g(space)h(of)324 4466 y(b)s(ounded)d Fb(c)-5 b(ontinuous)32 b Fm(functions)h(on)f Fk(H)8 b Fm(.)470 4586 y(Irreducibilit)m(y)27 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2148 31 3 v 2307 2189 a(2)2297 2246 y Fj(t)2352 2224 y Fk(:)1009 b Fm(\(17\))324 2444 y(Then)27 b(follo)m(w)c(a)j(similar)c(argumen)m(t)j(as)h(in)f(the)h(pro)s(ofs)f (of)g(Theorem)h(7.2.4)f(in)g([4])g(and)h(of)324 2564 y(Theorem)j(3.1)g(in)f([10)o(],)i(w)m(e)g(conclude)f(that)g Fk(P)2012 2579 y Fj(t)2042 2564 y Fm(,)g Fk(t)f(>)g Fm(0,)h(is)g(a)f (strong)h(F)-8 b(eller)27 b(semigroup.)324 2884 y Fa(Irreducibilit)m(y) 33 b(Prop)s(ert)m(y)470 3004 y Fm(No)m(w)48 b(w)m(e)g(consider)f (irreducibilit)m(y)d(prop)s(ert)m(y)-8 b(.)87 b(W)-8 b(e)48 b(further)f(assume)g(that)g(the)324 3124 y(co)m(v)-5 b(ariance)23 b(op)s(erator)f Fk(Q)h Fm(is)g(one-to-one)f(\(or)h (injectiv)m(e\),)i(i.e.,)g(the)e(k)m(ernel)h(k)m(er)18 b Fk(Q)28 b Fm(=)f Fh(f)p Fm(0)p Fh(g)p Fm(.)324 3245 y(Then,)33 b(as)e(in)g(the)g(pro)s(of)g(of)f(Theorem)i(7.4.2)f(in)f ([4])i(and)f(of)g(Theorem)g(3.1)g(in)g([10)o(],)h Fk(P)3505 3260 y Fj(t)3535 3245 y Fm(,)324 3365 y Fk(t)c(>)f Fm(0,)33 b(is)f(irreducible.)470 3685 y(Th)m(us,)40 b(with)e(the)g(strong)f(F)-8 b(eller)36 b(and)h(irreducibilit)m(y)e(prop)s(erties)j(pro)m(v)m(ed)h (ab)s(o)m(v)m(e,)324 3805 y(using)29 b(Do)s(ob's)h(Theorem)g(4.2.1)g (in)f([4],)h(there)h(exists)g(a)f(unique)g(in)m(v)-5 b(arian)m(t)29 b(measure)h Fk(\026)324 3926 y Fm(on)k(\()p Fk(H)r(;)17 b Fh(B)s Fm(\()p Fk(H)8 b Fm(\)\),)34 b(and)g(all)d(other)k (transition)d(probabilit)m(y)g(measures)j Fk(P)2971 3941 y Fj(t)3000 3926 y Fm(\()p Fk(x;)17 b Fh(\001)p Fm(\),)34 b Fk(x)d Fh(2)f Fk(H)8 b Fm(,)324 4046 y(approac)m(h)33 b(this)f(unique)h(in)m(v)-5 b(arian)m(t)31 b(measure)i Fk(\026)f Fm(as)h(time)e(go)s(es)i(to)f(in\014nit)m(y)-8 b(.)470 4166 y(Therefore,)34 b(w)m(e)g(ha)m(v)m(e)f(the)g(follo)m(wing) d(main)h(theorem)i(in)e(this)i(section.)324 4395 y Fa(Theorem)k(2)49 b Fb(Assume)34 b(that)470 4515 y(\(i\))615 4435 y Ff(R)681 4461 y Fi(+)p Fd(1)662 4550 y Fi(0)827 4515 y Fh(k)p Fk(S)6 b Fm(\()p Fk(r)s Fm(\))1066 4440 y Fh(p)p 1148 4440 78 4 v 1148 4515 a Fk(Q)q Fh(k)1276 4479 y Fi(2)1276 4540 y Fj(H)f(S)1389 4515 y Fk(dr)30 b(<)e Fm(+)p Fh(1)p Fb(,)470 4678 y(\(ii\))34 b(Image)27 b Fk(S)6 b Fm(\()p Fk(t)p Fm(\))27 b Fh(\032)i Fb(Image)d Fk(Q)1610 4590 y Fc(1)p 1610 4602 31 3 v 1610 4643 a(2)1600 4700 y Fj(t)1655 4678 y Fb(,)35 b(wher)-5 b(e)34 b Fk(Q)2072 4693 y Fj(t)2137 4678 y Fb(is)g(de\014ne)-5 b(d)34 b(in)g(\(16\),)h(and)470 4798 y(\(iii\))f(The)h(c)-5 b(ovarianc)g(e)33 b(op)-5 b(er)g(ator)35 b Fk(Q)28 b Fm(:)55 b Fk(L)1990 4762 y Fi(2)2030 4798 y Fm(\()p Fk(D)s Fm(\))27 b Fh(!)g Fk(L)2410 4762 y Fi(2)2450 4798 y Fm(\()p Fk(D)s Fm(\))35 b Fb(is)f(one-to-one.) 470 4918 y(Then)1918 5251 y Fm(8)p eop %%Page: 9 9 9 8 bop 470 548 a Fb(\(A\))38 b(Ther)-5 b(e)36 b(exists)h(a)h(unique)f (invariant)g(pr)-5 b(ob)g(ability)37 b(me)-5 b(asur)g(e)36 b Fk(\026)h Fb(for)h(the)f(quasi-)324 668 y(ge)-5 b(ostr)g(ophic)24 b(\015ow)i(system)f(\(3\))g(in)g(the)h(sp)-5 b(ac)g(e)25 b Fk(L)2052 632 y Fi(2)2091 668 y Fm(\()p Fk(D)s Fm(\))g Fb(of)h(squar)-5 b(e-inte)g(gr)g(able)24 b(vorticities;)470 789 y(\(B\))45 b(Mor)-5 b(e)g(over,)48 b(for)e(any)f Fk(!)52 b Fh(2)c Fk(L)1793 753 y Fi(2)1833 789 y Fm(\()p Fk(D)s Fm(\))p Fb(,)g(the)d(tr)-5 b(ansition)46 b(pr)-5 b(ob)g(ability)45 b(me)-5 b(asur)g(es)324 909 y Fk(P)387 924 y Fj(t)416 909 y Fm(\()p Fk(!)t(;)17 b Fh(\001)p Fm(\))43 b Fb(appr)-5 b(o)g(ach)43 b(the)i(unique)f(invariant)g(pr)-5 b(ob)g(ability)44 b(me)-5 b(asur)g(e)43 b Fk(\026)p Fb(.)73 b(Namely,)47 b(for)324 1029 y(any)35 b Fm(\000)27 b Fh(2)h(B)s Fm(\()p Fk(H)8 b Fm(\))p Fb(,)1521 1150 y Fm(lim)1478 1210 y Fj(t)p Fd(!)p Fi(+)p Fd(1)1716 1150 y Fk(P)1779 1165 y Fj(t)1809 1150 y Fm(\()p Fk(!)t(;)17 b Fm(\000\))26 b(=)i Fk(\026)p Fm(\(\000\);)324 1358 y Fb(and)470 1478 y(\(C\))34 b(Quasige)-5 b(ostr)g(ophic)34 b(\015ow)h(system)f(\(3\))h (is)f(er)-5 b(go)g(dic,)34 b(namely,)1146 1765 y Fm(lim)1090 1827 y Fj(T)10 b Fd(!)p Fi(+)p Fd(1)1354 1630 y Ff(Z)1453 1656 y Fj(T)1409 1855 y Fi(0)1525 1765 y Fk(g)t Fm(\()p Fk(!)t Fm(\()p Fk(t)p Fm(\)\))p Fk(dt)26 b Fm(=)2044 1630 y Ff(Z)2099 1855 y Fj(L)2147 1836 y Fc(2)2202 1765 y Fk(g)t(d\026;)71 b Fe(P)24 b Fh(\000)f Fk(a:s:)324 2042 y Fb(for)29 b(al)5 b(l)30 b(solution)f Fk(!)t Fm(\()p Fk(t)p Fm(\))g Fb(with)h(initial)f(date)g(in)h Fk(L)2052 2006 y Fi(2)2092 2042 y Fm(\()p Fk(D)s Fm(\))f Fb(and)g(al)5 b(l)29 b(Bor)-5 b(el)29 b(me)-5 b(asur)g(able)29 b(func-)324 2162 y(tion)35 b Fk(g)30 b Fm(:)e Fk(L)724 2126 y Fi(2)764 2162 y Fm(\()p Fk(D)s Fm(\))f Fh(!)h Fk(R)36 b Fb(such)e(that)1609 2082 y Ff(R)1657 2197 y Fj(L)1705 2178 y Fc(2)1739 2197 y Fi(\()p Fj(D)r Fi(\))1875 2162 y Fh(k)p Fk(g)t Fh(k)p Fk(d\026)26 b(<)h Fh(1)p Fb(.)470 2404 y Fm(The)39 b(ergo)s(dicit)m(y)f (in)f(P)m(art)i(\(C\))g(ab)s(o)m(v)m(e)g(is)f(a)g(consequence)j(of)d (the)h(uniqueness)h(of)324 2524 y(the)33 b(in)m(v)-5 b(arian)m(t)31 b(measure)i Fk(\026)p Fm(;)f(see)i(Theorem)f(3.2.6)f(in) g([4)o(].)324 2857 y Fl(5)161 b(Summary)324 3076 y Fm(In)27 b(this)f(pap)s(er,)i(w)m(e)f(ha)m(v)m(e)h(studied)f(ergo)s(dicit)m(y)e (of)h(large)g(scale)g(quasigeostrophic)g(\015o)m(ws)324 3196 y(under)32 b(random)f(wind)h(forcing.)42 b(W)-8 b(e)32 b(ha)m(v)m(e)h(sho)m(wn)g(that)f(the)g(quasigeostrophic)f(\015o) m(ws)324 3317 y(are)e(ergo)s(dic)g(under)h(suitable)f(conditions)f(on)i (the)g(random)f(forcing)f(and)i(on)f(the)h(\015uid)324 3437 y(domain,)35 b(and)h(under)h(no)f(restrictions)f(on)h(viscosit)m (y)-8 b(,)37 b(Ekman)f(constan)m(t)h(or)f(Coriolis)324 3557 y(parameter.)88 b(When)49 b(these)g(conditions)e(are)g (satis\014ed,)52 b(then)d(for)e(an)m(y)h(observ)-5 b(able)324 3678 y(of)39 b(the)h(quasigeostrophic)f(\015o)m(ws,)j(its)d(time)f(a)m (v)m(erage)j(appro)m(ximates)e(the)h(statistical)324 3798 y(ensem)m(ble)33 b(a)m(v)m(erage,)h(as)e(long)g(as)g(the)h(time)f (in)m(terv)-5 b(al)31 b(is)h(su\016cien)m(tly)h(long.)470 3919 y(There)54 b(is)e(recen)m(t)i(w)m(ork)g(on)f(random)e(dynamical)g (attractors)i(for)f(the)h(quasi-)324 4039 y(geostrophic)45 b(\015o)m(w)i(mo)s(del)d(b)m(y)j(Duan)e(et)h(al.)82 b([6].)h(A)46 b(consequence)j(of)c(that)h(w)m(ork)324 4159 y(implies)37 b(that,)42 b(when)f(viscosit)m(y)g(is)f(su\016cien)m(tly)g(large)f(and) h(when)i(the)e(trace)h(of)e(the)324 4280 y(co)m(v)-5 b(ariance)23 b(op)s(erator)f(for)h(the)h(Wiener)f(pro)s(cess)i(is)d (su\016cien)m(tly)i(small,)f(then)h(all)d(quasi-)324 4400 y(geostrophic)27 b(motions)e(approac)m(h)j(a)f(p)s(oin)m(t)f (random)h(attractor)f(exp)s(onen)m(tially)g(fast)h(as)324 4521 y(time)h(go)s(es)i(in\014nit)m(y)-8 b(.)41 b(This)30 b(is)f(a)g(v)m(ery)i(rare)f(case.)43 b(This)30 b(p)s(oin)m(t)f(random)f (attractor)h(cor-)324 4641 y(resp)s(onds)39 b(to)e(a)g(unique)i(in)m(v) -5 b(arian)m(t)36 b(Dirac)g(measure,)j(i.e.,)g(the)f(supp)s(orting)f(p) s(oin)m(t)g(of)324 4761 y(the)c(Dirac)e(measure)i(is)f(a)g(global)e (\(p)s(oin)m(t\))h(attractor,)h(and)h(th)m(us)g(under)h(these)g(condi-) 324 4882 y(tions,)d(quasigeostrophic)h(\015o)m(ws)i(are)e(also)f(ergo)s (dic.)42 b(These)34 b(conditions)d(are)h(di\013eren)m(t)324 5002 y(from)25 b(the)i(ergo)s(dic)e(conditions)g(in)h(the)h(curren)m(t) g(pap)s(er.)42 b(F)-8 b(or)25 b(example,)i(in)f(the)g(curren)m(t)1918 5251 y(9)p eop %%Page: 10 10 10 9 bop 324 548 a Fm(pap)s(er,)27 b(w)m(e)g(do)e(not)g(imp)s(ose)g(an) m(y)h(condition)e(on)i(viscosit)m(y)-8 b(,)27 b(or)e(on)h(the)f(size)h (of)f(the)h(trace)324 668 y(of)32 b(the)h(co)m(v)-5 b(ariance)32 b(op)s(erator)g(for)g(the)h(Wiener)g(pro)s(cess.)470 1059 y Fa(Ac)m(kno)m(wledgemen)m(t.)58 b Fm(A)39 b(part)e(of)h(this)f (w)m(ork)i(w)m(as)g(done)g(while)e(J.)h(Duan)g(w)m(as)324 1179 y(visiting)27 b(the)i(Univ)m(ersit)m(y)h(of)f(New)h(South)f(W)-8 b(ales,)29 b(Australia.)41 b(This)29 b(w)m(ork)h(w)m(as)g(partly)324 1299 y(supp)s(orted)36 b(b)m(y)g(the)g(Australia)d(Researc)m(h)k (Council)d(and)h(b)m(y)i(the)e(NSF)g(Gran)m(t)g(DMS-)324 1420 y(9973204.)324 1750 y Fl(References)373 1969 y Fm([1])48 b(L.)32 b(Arnold.)40 b Fb(Random)33 b(Dynamic)-5 b(al)32 b(Systems)p Fm(.)41 b(Springer{V)-8 b(erlag,)30 b(Berlin,)g(1998.)373 2168 y([2])48 b(J.)37 b(R.)g(Brannan,)g(J.)g(Duan,)h(and)e(T.)h(W)-8 b(anner.)56 b(Dissipativ)m(e)36 b(quasigeostrophic)524 2288 y(dynamics)45 b(under)g(random)f(forcing.)77 b Fb(J.)46 b(Math.)g(A)n(nal.)f(Appl.)p Fm(,)j(228:221{233,)524 2408 y(1998.)373 2607 y([3])g(G.)39 b(Da)f(Prato)h(and)g(J.)h(Zab)s (czyk,)i Fb(Sto)-5 b(chastic)40 b(Equations)g(in)h(In\014nite)f(Dimen-) 524 2727 y(sions)p Fm(,)32 b(Cam)m(bridge)g(Univ)m(ersit)m(y)h(Press,)i (1992.)373 2926 y([4])48 b(G.)38 b(Da)f(Prato)h(and)g(J.)h(Zab)s(czyk,) h Fb(Er)-5 b(go)g(dicity)40 b(for)g(In\014nite)f(Dimensional)f(Sys-)524 3046 y(tems)p Fm(,)33 b(Cam)m(bridge)f(Univ)m(ersit)m(y)h(Press,)h (1996.)373 3245 y([5])48 b(T.)38 b(DelSole)d(and)j(B.)f(F.)g(F)-8 b(arrell,)36 b(A)h(sto)s(c)m(hastically)f(excited)i(linear)d(system)j (as)524 3365 y(a)e(mo)s(del)f(for)h(quasigeostrophic)g(turbulence:)52 b(Analytic)36 b(results)g(for)g(one-)g(and)524 3486 y(t)m(w)m(o-la)m(y) m(er)d(\015uids,)g Fb(J.)i(A)n(tmos.)f(Sci.)e Fa(52)h Fm(\(1995\))e(2531-2547.)373 3684 y([6])58 b(J.)32 b(Duan,)f(P)-8 b(.)31 b(E.)h(Klo)s(eden)f(and)g(B.)g(Sc)m(hmalfu\031.)40 b(Exp)s(onen)m(tial)31 b(Stabilit)m(y)e(of)i(the)524 3804 y(Quasigeostropic)g(Flo)m(ws)h(under)h(Random)e(P)m(erturbation,)h Fb(Pr)-5 b(o)g(gr)g(ess)34 b(in)g(Pr)-5 b(ob)g(a-)524 3925 y(bility)p Fm(,)33 b(in)f(press,)i(2000.)373 4123 y([7])48 b(J.)38 b(Duan)f(and)h(B.)f(Sc)m(hmalfu\031.)57 b(The)39 b(3D)e(Quasigeostrophic)f(Equation)i(under)524 4244 y(Random)32 b(P)m(erturbation.)43 b(,)33 b(submitted,)f(2000.)373 4442 y([8])48 b(V.)40 b(Dymnik)m(o)m(v)g(and)g(E.)g(Kazan)m(tsev,)k(On) c(the)g(genetic)g(\\memory")e(of)i(c)m(haotic)524 4563 y(attractor)22 b(of)g(the)g(barotropic)f(o)s(cean)i(mo)s(del,)f Fb(F)-7 b(r)i(ench-R)n(ussian)23 b(A.)j(M.)f(Liapunov)524 4683 y(Institute)36 b(MSU,)f(INRIA)p Fm(,)d(25-36.)373 4882 y([9])48 b(L.)40 b(C.)g(Ev)-5 b(ans,)42 b Fb(Partial)f(Di\013er)-5 b(ential)40 b(Equations)p Fm(,)i(Amer.)d(Math.)h(So)s(c.,)h(Pro)m(vi-) 524 5002 y(dence,)34 b(1998.)1894 5251 y(10)p eop %%Page: 11 11 11 10 bop 324 548 a Fm([10])48 b(B.)25 b(F)-8 b(errario,)25 b(Ergo)s(dic)f(results)i(for)e(sto)s(c)m(hastic)i(Na)m(vier-Stok)m(es)g (equation,)g Fb(Sto)-5 b(ch.)524 668 y(Sto)g(chastic)35 b(r)-5 b(ep)g(orts)32 b Fa(60)h Fm(\(1997\),)e(271-288.)324 862 y([11])48 b(B.)30 b(F)-8 b(errario,)29 b(Sto)s(c)m(hastic)h(Na)m (vier-Stok)m(es)h(equations:)42 b(analysis)30 b(of)f(the)h(noise)g(to) 524 982 y(ha)m(v)m(e)j(a)e(unique)h(in)m(v)-5 b(arian)m(t)30 b(measure.)i Fb(A)n(nn.)h(Mat.)h(Pur)-5 b(a)34 b(Appl.)d Fm(\(4\))g(177)g(\(1999\),)524 1102 y(331{347.)324 1296 y([12])48 b(F.)33 b(Flandoli,)d(Dissipativit)m(y)h(and)i(in)m(v)-5 b(arian)m(t)32 b(measures)h(for)g(sto)s(c)m(hastic)g(Na)m(vier-)524 1416 y(Stok)m(es)h(equations,)f Fb(NoDEA.)g Fa(1)70 b Fm(\(1994\),)31 b(403-423.)324 1610 y([13])48 b(F.)28 b(Flandoli)e(and)i(B.)h(Maslo)m(wski,)g(Ergo)s(dicit)m(y)f(of)f(the)i (2D)f(Na)m(vier-Stok)m(es)h(equa-)524 1730 y(tion)40 b(under)i(random)d(p)s(erturbations,)k Fb(Commun.)e(Math.)i(Phys.)e Fa(171)g Fm(\(1995\),)524 1850 y(119-141.)324 2044 y([14])48 b(A.)43 b(Gri\013a)e(and)h(S.)h(Castellari,)g(Nonlinear)e(general)h (circulation)e(of)i(an)g(o)s(cean)524 2164 y(mo)s(del)28 b(driv)m(en)h(b)m(y)h(wind)f(with)g(a)f(sto)s(c)m(hastic)i(comp)s(onen) m(t,)f Fb(J.)j(Marine)f(R)-5 b(ese)g(ar)g(ch)524 2284 y Fa(49)33 b Fm(\(1991\),)e(53-73.)324 2478 y([15])48 b(G.)c(Hollo)m(w)m(a)m(y)-8 b(,)45 b(Ocean)f(circulation:)64 b(Flo)m(w)43 b(in)g(probabilit)m(y)e(under)k(statistical)524 2598 y(dynamical)c(forcing,)i(in)e Fb(Sto)-5 b(chastic)43 b(Mo)-5 b(dels)43 b(in)h(Ge)-5 b(osystems)p Fm(,)44 b(S.)e(Molc)m(hano) m(v)524 2719 y(and)33 b(W.)g(W)-8 b(o)m(yczynski)34 b(\(eds.\),)g (Springer,)e(1996.)324 2912 y([16])48 b(P)-8 b(.)40 b(M)s(\177)-51 b(uller,)38 b(Sto)s(c)m(hastic)i(forcing)e(of)g(quasi-geostrophic)h (eddies,)i(in)d Fb(Sto)-5 b(chastic)524 3032 y(Mo)g(del)5 b(ling)35 b(in)h(Physic)-5 b(al)35 b(Oc)-5 b(e)g(ano)g(gr)g(aphy)p Fm(,)32 b(R.)i(J.)f(Adler,)h(P)-8 b(.)34 b(M)s(\177)-51 b(uller)31 b(and)j(B.)g(Ro-)524 3153 y(zo)m(vskii)f(\(eds.\),)g (Birkh\177)-49 b(auser,)34 b(1996.)324 3346 y([17])48 b(A.)35 b(P)m(azy)-8 b(,)37 b Fb(Semigr)-5 b(oups)36 b(of)h(line)-5 b(ar)36 b(op)-5 b(er)g(ators)36 b(and)h(applic)-5 b(ations)35 b(to)i(p)-5 b(artial)37 b(dif-)524 3467 y(fer)-5 b(ential)35 b(e)-5 b(quations)p Fm(,)32 b(Springer-V)-8 b(erlag,)31 b(1983.)324 3660 y([18])48 b(J.)33 b(P)m(edlosky)-8 b(,)34 b Fb(Ge)-5 b(ophysic)g(al)35 b(Fluid)f(Dynamics)p Fm(,)e(Springer-V)-8 b(erlag,)31 b(2nd)i(edition,)524 3780 y(1987.)324 3974 y([19])48 b(J.)33 b(P)m(edlosky)-8 b(,)34 b Fb(Oc)-5 b(e)g(an)34 b(Cir)-5 b(culation)34 b(The)-5 b(ory)p Fm(.)33 b(Springer{V)-8 b(erlag,)31 b(Berlin,)g(1996.)324 4167 y([20])48 b(R.)c(M.)g(Samelson,)i(Sto)s(c)m (hastically)c(forced)i(curren)m(t)h(\015uctuations)e(in)g(v)m(ertical) 524 4287 y(shear)34 b(and)e(o)m(v)m(er)i(top)s(ograph)m(y)-8 b(,)32 b Fb(J.)j(Ge)-5 b(ophys.)35 b(R)-5 b(es.)32 b Fa(94)h Fm(\(1989\))e(8207-8215.)324 4481 y([21])48 b(D.)32 b(W.)h(Stro)s(o)s(c)m(k,)f Fb(Pr)-5 b(ob)g(ability)35 b(The)-5 b(ory:)44 b(A)n(n)34 b(A)n(nalytic)h(View)p Fm(,)d(Cam)m(bridge)g(Uni-)524 4601 y(v)m(ersit)m(y)i(Press,)g(Cam)m (bridge,)e(1993.)324 4795 y([22])48 b(M.)39 b(I.)g(Vishik)f(and)g(A.)h (V.)f(F)-8 b(ursik)m(o)m(v,)41 b Fb(Mathematic)-5 b(al)39 b(Pr)-5 b(oblems)40 b(of)g(Statistic)-5 b(al)524 4915 y(Hydr)g(o)g(dynamics)p Fm(,)32 b(Klu)m(w)m(er)h(Academic)f (Publishers,)h(Boston,)g(1988.)1894 5251 y(11)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0104251439836--