Content-Type: multipart/mixed; boundary="-------------0012221124945" This is a multi-part message in MIME format. ---------------0012221124945 Content-Type: text/plain; name="00-513.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-513.comments" Submitted to Physica D ---------------0012221124945 Content-Type: text/plain; name="00-513.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-513.keywords" 3D Quasigeostrophic flows, wind forcing on boundary, stable stationary flows, random dynamical systems, random attractors ---------------0012221124945 Content-Type: application/postscript; name="duan.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="duan.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: duan.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips duan -o 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y(tion)32 b(of)g(the)h(geoph)n(ysical)d (\015o)n(w)h(system.)51 b(Sto)r(c)n(hastically)31 b(forced)h(QGE)f(has) h(b)r(een)g(used)h(to)456 1621 y(in)n(v)n(estigate)26 b(v)-5 b(arious)26 b(mec)n(hanisims)h(in)h(geoph)n(ysical)e(\015o)n(ws) g([13)o(,)i(17)o(,)g(25)o(,)g(4)o(].)456 1720 y(The)36 b(QGE)g(can)h(b)r(e)g(transformed)e(in)n(to)i(an)f(ev)n(olution)g (equation)g(with)i(standard)d(b)r(ound-)456 1820 y(ary)g(conditions.)63 b(F)-7 b(or)36 b(this)h(transformation)e(w)n(e)h(need)h(an)f (Ornstein-Uhlen)n(b)r(ec)n(k)g(pro)r(cess)456 1920 y(ful\014lling)27 b(our)g(dynamical)g(random)f(or)g(time-p)r(erio)r(dic)i(b)r(oundary)e (conditions.)36 b(This)28 b(trans-)456 2019 y(formation)e(will)i(b)r(e) g(in)n(tro)r(duced)g(in)f(Section)h(2.)456 2119 y(In)i(Section)g(3)g(w) n(e)g(in)n(v)n(estigate)f(the)i(co)r(e\016cien)n(ts)f(of)g(the)h (transformed)e(ev)n(olution)h(equation,)456 2218 y(and)35 b(further)h(obtain)g(a)f(global)g(existence)g(and)h(uniqueness)g (result)f(and)h(some)f(regularit)n(y)456 2318 y(result.)456 2418 y(In)d(Section)h(4,)g(w)n(e)g(study)f(the)h(random)f(dynamics)g (of)h(the)g(transformed)e(QGE.)h(Based)g(on)456 2517 y(the)22 b(uniqueness)g(result)g(ab)r(o)n(v)n(e,)g(the)h(transformed)e (ev)n(olution)h(equation)g(generates)e(a)i(nonau-)456 2617 y(tonomous)h(dynamical)g(system.)35 b(In)24 b(addition,)h(if)f(w)n (e)g(restrict)f(this)h(system)g(to)g(discrete)f(time)456 2717 y(step)g(of)g(the)h(p)r(erio)r(d)f(of)g(the)h(p)r(erio)r(dic)f (rotation)f(of)i(the)f(earth,)h(w)n(e)f(obtain)g(a)g(random)f(dynam-) 456 2816 y(ical)31 b(system.)49 b(This)32 b(random)e(dynamical)i (system)f(has)g(a)h(random)e(attractor.)48 b(This)32 b(result)456 2916 y(can)27 b(b)r(e)h(extended)g(to)f(the)h(dynamical)f (system)g(on)h(the)g(real)e(time)i(axis.)456 3015 y(The)f(Section)h(5)f (con)n(tains)g(the)h(pro)r(ofs.)36 b(Finally)-7 b(,)27 b(w)n(e)h(summarize)e(our)h(results)g(in)h(Section)g(6.)1158 3262 y(2.)45 b Fn(The)32 b(3D)f(quasigeostr)n(ophic)h(equa)-6 b(tion)456 3411 y Fo(Let)40 b Fm(O)45 b Fo(=)e(\(0)p Fm(;)14 b Fo(2)p Fm(\031)s Fo(\))1068 3381 y Fp(3)1145 3411 y Fo(b)r(e)40 b(the)h(cub)r(e)f(whic)n(h)f(is)h(a)f(mo)r(del)h (for)g(a)f(piece)h(of)f(the)h(o)r(cean.)73 b(F)-7 b(or)456 3511 y Fm(x;)28 b(y)s(;)f(z)45 b Fj(2)e Fo(\(0)p Fm(;)14 b Fo(2)p Fm(\031)s Fo(\))38 b(and)h(smo)r(oth)g(functions)g Fm(u)p Fo(\()p Fm(x;)14 b(y)s(;)g(z)t Fo(\))p Fm(;)27 b(v)s Fo(\()p Fm(x;)14 b(y)s(;)g(z)t Fo(\))39 b(w)n(e)f(de\014ne)h(the) h(Jacobi)456 3610 y(op)r(erator)1540 3772 y Fm(J)8 b Fo(\()p Fm(u;)14 b(v)s Fo(\))23 b(=)g Fm(u)1945 3784 y Fl(x)1986 3772 y Fm(v)2026 3784 y Fl(y)2085 3772 y Fj(\000)18 b Fm(u)2216 3784 y Fl(y)2255 3772 y Fm(v)2295 3784 y Fl(x)2337 3772 y Fm(:)456 3952 y Fo(In)27 b(addition,)h(the)g (di\013eren)n(tial)f(op)r(erator)1817 3931 y(~)1803 3952 y(\001)h(is)f(de\014ned)h(b)n(y)1454 4094 y(~)1440 4115 y(\001)23 b(:=)g Fm(@)1687 4127 y Fl(xx)1785 4115 y Fo(+)18 b Fm(@)1912 4127 y Fl(y)r(y)2006 4115 y Fo(+)g(\()p Fm(F)12 b Fo(\()p Fm(z)t Fo(\))p Fm(@)2337 4127 y Fl(z)2376 4115 y Fo(\))2408 4139 y Fl(z)456 4313 y Fo(for)36 b Fm(f)633 4325 y Fp(0)707 4313 y Fj(2)j Fh(R)30 b Fj(n)24 b Fo(0)p Fm(;)j(F)12 b Fo(\()p Fm(z)t Fo(\))38 b(:=)1441 4270 y Fl(f)1480 4245 y Fg(2)1473 4287 y(0)p 1388 4294 178 4 v 1388 4341 a Fl(N)1447 4325 y Fg(2)1480 4341 y Fp(\()p Fl(z)r Fp(\))1612 4313 y Fo(and)e Fm(N)9 b Fo(\()p Fm(z)t Fo(\))37 b(is)f(de\014ned)h(to)f(b)r(e)h(a)f(p)r(ositiv)n(e)g Fm(C)3080 4283 y Ff(1)3151 4313 y Fo(-smo)r(oth)456 4426 y(function)c Fm(N)9 b Fo(\()p Fm(z)t Fo(\))29 b Fm(>)g Fo(0)j(on)f([0)p Fm(;)14 b Fo(2)p Fm(\031)s Fo(].)48 b(Let)32 b Fm(\027)q(;)27 b(\014)36 b Fo(b)r(e)c(p)r(ositiv)n(e)g (constan)n(ts.)48 b(In)31 b(the)h(follo)n(wing)f(w)n(e)456 4526 y(in)n(v)n(estigate)26 b(the)i(3D)f(QGE)g(\015o)n(w)g(mo)r(del)h ([19)o(,)g(24)o(,)g(31)o(]:)1355 4672 y(~)1342 4693 y(\001)p Fm( )1465 4705 y Fl(t)1512 4693 y Fo(+)18 b Fm(J)8 b Fo(\()p Fm( )s(;)1790 4672 y Fo(~)1775 4693 y(\001)q Fm( )s Fo(\))19 b(+)f Fm(\014)t( )2141 4705 y Fl(x)2206 4693 y Fo(=)23 b Fm(\027)2354 4672 y Fo(~)2340 4693 y(\001)2423 4672 y(~)2409 4693 y(\001)p Fm( )s(:)-2102 b Fo(\(1\))456 4855 y(W)-7 b(e)18 b(imp)r(ose)h(the)f(follo)n(wing)g(b)r(oundary)f (conditions)h(for)g(this)h(equation.)33 b(Let)18 b Fm(O)2928 4867 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(0)3045 4855 y Fm(;)28 b(O)3159 4867 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(2)p Fl(\031)3316 4855 y Fm(;)14 b Fj(\001)g(\001)g(\001)456 4955 y Fo(b)r(e)33 b(the)h(faces)e(of)i(the)f(cub)r(e)h Fm(O)i Fo(then)d(w)n(e)g(assume)g(p)r(erio)r(dic)g(b)r(oundary)f (conditions)h(in)h Fm(x;)28 b(y)456 5054 y Fo(directions)1195 5216 y Fm( )s Fj(j)1275 5228 y Fl(O)1325 5236 y Fg(0)p Fe(;)p Fd(\001)p Fe(;)p Fd(\001)1461 5216 y Fo(=)23 b Fm( )s Fj(j)1629 5228 y Fl(O)1679 5236 y Fg(2)p Fe(\031)q(;)p Fd(\001)p Fe(;)p Fd(\001)1994 5216 y Fm( )2048 5228 y Fl(x)2090 5216 y Fj(j)2113 5228 y Fl(O)2163 5236 y Fg(0)p Fe(;)p Fd(\001)p Fe(;)p Fd(\001)2299 5216 y Fo(=)g Fm( )2441 5228 y Fl(x)2483 5216 y Fj(j)2506 5228 y Fl(O)2556 5236 y Fg(2)p Fe(\031)q(;)p Fd(\001)p Fe(;)p Fd(\001)p eop %%Page: 4 4 4 3 bop 456 238 a Fp(4)738 b(JINQIA)n(O)29 b(DUAN)g(AND)g(BJ)2028 223 y(\177)2019 238 y(ORN)g(SCHMALFUSS)456 425 y Fo(and)c(similar)f (for)h(the)h(faces)f Fm(O)1413 437 y Ff(\001)p Fl(;)p Fp(0)p Fl(;)p Ff(\001)1529 425 y Fm(;)j(O)1643 437 y Ff(\001)p Fl(;)p Fp(2)p Fl(\031)r(;)p Ff(\001)1826 425 y Fo(together)c(with)i(deriv)-5 b(ativ)n(e)24 b(in)i Fm(y)i Fo(direction.)36 b(With)456 525 y(resp)r(ect)j(to)g(the)h Fm(z)j Fo(direction,)f Fm( )g Fo(ful\014lls)f(Neumann)e(b)r(oundary)g (conditions)g(on)g(top)h(and)456 624 y(b)r(ottom)27 b(b)r(oundaries;)g (see)g([19],)g(p.367.)36 b(Namely)-7 b(,)28 b(w)n(e)f(imp)r(ose)g(that) 1200 773 y Fm(@)5 b( )p 1200 810 106 4 v 1204 886 a(@)g(n)1338 829 y Fo(=)23 b Fm( )1480 841 y Fl(z)1541 829 y Fo(=)g(0)83 b(on)27 b(b)r(ottom)h(b)r(oundary)22 b Fm(O)2594 841 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(0)456 1014 y Fo(and)1193 1142 y Fm(@)5 b( )p 1193 1180 V 1197 1256 a(@)g(n)1332 1199 y Fo(=)22 b Fm( )1473 1211 y Fl(z)1535 1199 y Fo(=)1656 1178 y(_)1622 1199 y Fm(W)95 b Fo(on)28 b(top)f(b)r(oundary)22 b Fm(O)2490 1211 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(2)p Fl(\031)2694 1199 y Fm(;)456 1402 y Fo(where)729 1381 y(_)696 1402 y Fm(W)39 b Fo(is)28 b(a)f(temp)r(oral)g(white)h (noise)f(and)g Fm(n)h Fo(denotes)f(the)h(unit)h(outer)e(normal)f(v)n (ector.)456 1503 y(Moreo)n(v)n(er,)857 1482 y(~)844 1503 y(\001)p Fm( )k Fo(is)e(supp)r(osed)f(to)h(b)r(e)g(p)r(erio)r(dic)f(in) h Fm(x;)14 b(y)31 b Fo(directions:)1070 1645 y(~)1057 1666 y(\001)p Fm( )s Fj(j)1206 1678 y Fl(O)1256 1686 y Fg(0)p Fe(;)p Fd(\001)p Fe(;)p Fd(\001)1392 1666 y Fo(=)1494 1645 y(~)1480 1666 y(\001)p Fm( )s Fj(j)1629 1678 y Fl(O)1679 1686 y Fg(2)p Fe(\031)q(;)p Fd(\001)p Fe(;)p Fd(\001)2008 1645 y Fo(~)1994 1666 y(\001)p Fm( )2117 1678 y Fl(x)2159 1666 y Fj(j)2182 1678 y Fl(O)2232 1686 y Fg(0)p Fe(;)p Fd(\001)p Fe(;)p Fd(\001)2369 1666 y Fo(=)2470 1645 y(~)2456 1666 y(\001)p Fm( )2579 1678 y Fl(x)2621 1666 y Fj(j)2644 1678 y Fl(O)2694 1686 y Fg(2)p Fe(\031)q(;)p Fd(\001)p Fe(;)p Fd(\001)456 1828 y Fo(and)d(similar)f(for)g(the)i(faces)e Fm(O)1426 1840 y Ff(\001)p Fl(;)p Fp(0)p Fl(;)p Ff(\001)1543 1828 y Fm(;)h(O)1657 1840 y Ff(\001)p Fl(;)p Fp(2)p Fl(\031)r(;)p Ff(\001)1842 1828 y Fo(with)h(deriv)-5 b(ativ)n(e)27 b(in)h Fm(y)j Fo(direction.)38 b(F)-7 b(or)28 b(the)g(other)456 1928 y(faces)h(on)h(the)h(top)f(and)g(the)g(b)r(ottom)g(of)g(the)h(o)r (cean,)f(w)n(e)g(imp)r(ose)g(homogeneous)e(Neumann)456 2027 y(b)r(oundary)e(conditions)i(as)e(in)i([31)o(,)g(8,)f(7])1344 2174 y(~)1331 2195 y(\001)p Fm( )1454 2207 y Fl(z)1492 2195 y Fj(j)1515 2207 y Fl(O)1565 2215 y Fd(\001)p Fe(;)p Fd(\001)p Fe(;)p Fg(0)1701 2195 y Fo(=)c(0)p Fm(;)2047 2174 y Fo(~)2033 2195 y(\001)q Fm( )2157 2207 y Fl(z)2195 2195 y Fj(j)2218 2207 y Fl(O)2268 2215 y Fd(\001)p Fe(;)p Fd(\001)p Fe(;)p Fg(2)p Fe(\031)2440 2195 y Fo(=)g(0)456 2357 y(In)k(addition,)h(w)n(e)f(imp)r(ose)h(that)1721 2443 y Fk(Z)1767 2632 y Fl(O)1838 2556 y Fm( )s(dO)e Fo(=)c(0)p Fm(;)456 2760 y Fo(and)1687 2830 y Fk(Z)1733 3019 y Fl(O)1817 2922 y Fo(~)1803 2943 y(\001)p Fm( )s(dO)k Fo(=)d(0)p Fm(:)456 3147 y Fo(W)-7 b(e)28 b(also)e(assume)h(an)g (appropriate)f(initial)i(condition)1498 3309 y Fm( )s Fo(\()p Fm(x;)14 b(y)s(;)g(z)t(;)g Fo(0\))22 b(=)h Fm( )2070 3321 y Fp(0)2107 3309 y Fo(\()p Fm(x;)14 b(y)s(;)g(z)t Fo(\))p Fm(:)456 3470 y Fo(Later)30 b(on)i(w)n(e)f(will)h(see)f(that)h (w)n(e)f(can)h(\014nd)g(an)f(Ornstein-Uhlen)n(b)r(ec)n(k)g(pro)r(cess)f Fm(\021)35 b Fo(satisfying)456 3570 y(the)28 b(linear)e(di\013eren)n (tial)i(equation)1761 3733 y Fm(\021)1802 3745 y Fl(t)1855 3733 y Fo(=)22 b Fm(\027)2002 3712 y Fo(~)1988 3733 y(\001)q Fm(\021)s(;)-1669 b Fo(\(2\))456 3895 y(where)27 b Fm(\021)k Fo(ful\014lls)d(the)g(same)f(b)r(oundary)g(conditions)g(as)g Fm( )j Fo(in)e(some)f(sense.)456 3995 y(W)-7 b(e)37 b(no)n(w)f (transform)f(formally)h(the)h(ab)r(o)n(v)n(e)e(equation)h(to)h(another) e(parab)r(olic)h(di\013eren)n(tial)456 4094 y(equation)30 b(with)h(random)e(co)r(e\016cien)n(ts)h(but)h(with)g(standard)f(b)r (oundary)g(conditions.)45 b(T)-7 b(o)30 b(do)456 4194 y(this)c(w)n(e)f(ha)n(v)n(e)g(to)h(in)n(tro)r(duce)g(the)g(solution)g (op)r(erator)e Fm(G)i Fo(of)g(the)g(follo)n(wing)f(elliptic)i(problem:) 1829 4336 y(~)1815 4357 y(\001)p Fm(\030)h Fo(=)22 b Fm(f)-1620 b Fo(\(3\))456 4519 y(on)30 b Fm(O)k Fo(with)e(p)r(erio)r (dic)f(b)r(oundary)f(conditions)h(in)h Fm(x;)14 b(y)34 b Fo(directions)c(and)h Fi(homo)l(gene)l(ous)h Fo(Neu-)456 4619 y(mann)e(b)r(oundary)g(conditions)g(in)h Fm(z)j Fo(direction.)46 b(F)-7 b(or)30 b(existence)g(and)h(regularit)n(y)d (prop)r(erties)456 4718 y(of)33 b(this)g(op)r(erator)f(see)g(b)r(elo)n (w.)53 b Fm(G)34 b Fo(can)f(b)r(e)g(considered)f(as)h(a)g(linear)f(con) n(tin)n(uous)g(op)r(erator.)456 4818 y(Here)27 b Fm(f)36 b Fo(is)28 b(an)f(elemen)n(t)h(in)f(a)h(particular)e(Hilb)r(ert)i (space.)456 4918 y(F)-7 b(or)27 b(the)h(di\013erence)f(of)h Fm( )i Fo(and)e Fm(\021)j Fo(w)n(e)c(obtain)g(for)g(\(1\))456 5085 y(\()502 5064 y(~)488 5085 y(\001)p Fm( )22 b Fj(\000)729 5064 y Fo(~)716 5085 y(\001)p Fm(\021)s Fo(\))861 5097 y Fl(t)974 5085 y Fo(+)82 b Fm(J)8 b Fo(\()p Fm( )22 b Fj(\000)c Fm(\021)s(;)1461 5064 y Fo(~)1447 5085 y(\001)p Fm( )k Fj(\000)1689 5064 y Fo(~)1675 5085 y(\001)p Fm(\021)s Fo(\))d(+)f Fm(J)8 b Fo(\()p Fm(\021)s(;)2103 5064 y Fo(~)2089 5085 y(\001)q Fm( )21 b Fj(\000)2331 5064 y Fo(~)2317 5085 y(\001)p Fm(\021)s Fo(\))e(+)f Fm(J)8 b Fo(\()p Fm( )22 b Fj(\000)c Fm(\021)s(;)2904 5064 y Fo(~)2890 5085 y(\001)p Fm(\021)s Fo(\))h(+)f Fm(J)8 b Fo(\()p Fm(\021)s(;)3319 5064 y Fo(~)3304 5085 y(\001)q Fm(\021)s Fo(\))974 5216 y(+)82 b Fm(\014)t Fo(\()p Fm( )22 b Fj(\000)c Fm(\021)s Fo(\))1439 5228 y Fl(x)1500 5216 y Fo(+)g Fm(\014)t(\021)1675 5228 y Fl(x)1740 5216 y Fo(=)23 b Fm(\027)1888 5195 y Fo(~)1874 5216 y(\001)1957 5195 y(~)1944 5216 y(\001\()p Fm( )f Fj(\000)c Fm(\021)s Fo(\))p eop %%Page: 5 5 5 4 bop 1092 226 a Fp(THE)29 b(STOCHASTIC)h(3D)f(QUASIGEOSTR)n(OPHIC)g (FLO)n(WS)603 b(5)456 425 y Fo(where)21 b Fm(\021)26 b Fo(is)c(assumed)g(to)g(b)r(e)h(su\016cien)n(tly)f(regular.)34 b(The)22 b(di\013erence)h Fm( )11 b Fj(\000)d Fm(\021)25 b Fo(satis\014es)c(the)i(same)456 526 y(b)r(oundary)i(condition)g(as)g Fm(\030)30 b Fo(in)d(\(3\).)36 b(Hence)26 b(b)n(y)f(the)i(formal)e (substitution)h Fm(v)g Fo(=)3004 505 y(~)2990 526 y(\001)p Fm( )18 b Fj(\000)3224 505 y Fo(~)3211 526 y(\001)p Fm(\021)29 b Fo(w)n(e)456 626 y(can)e(write)1027 766 y Fm(v)1067 778 y Fl(t)1115 766 y Fo(+)18 b Fm(J)8 b Fo(\()p Fm(G)p Fo(\()p Fm(v)s Fo(\))p Fm(;)14 b(v)s Fo(\))21 b(+)d Fm(J)8 b Fo(\()p Fm(\021)s(;)14 b(v)s Fo(\))19 b(+)f Fm(J)8 b Fo(\()p Fm(G)p Fo(\()p Fm(v)s Fo(\))p Fm(;)2326 745 y Fo(~)2311 766 y(\001)r Fm(\021)s Fo(\))18 b(+)g Fm(J)8 b Fo(\()p Fm(\021)s(;)2741 745 y Fo(~)2726 766 y(\001)q Fm(\021)s Fo(\))1115 910 y(+)18 b Fm(\014)t(G)p Fo(\()p Fm(v)s Fo(\))1421 922 y Fl(x)1483 910 y Fo(+)g Fm(\014)t(\021)1658 922 y Fl(x)1724 910 y Fo(=)k Fm(\027)1871 889 y Fo(~)1857 910 y(\001)q Fm(v)s(:)456 1055 y Fo(The)k(problem)g(\(2\))g (satis\014es)g(only)g(standard)f(b)r(oundary)g(conditions)h(with)h (resp)r(ect)f(to)g Fm(\021)k Fo(but)456 1160 y(not)25 b(with)i(resp)r(ect)e(to)1185 1139 y(~)1171 1160 y(\001)p Fm(\021)s Fo(.)36 b(T)-7 b(o)26 b(obtain)f(an)h(equation)f(with)h (standard)f(b)r(oundary)g(conditions)456 1261 y(w)n(e)34 b(need)h(the)h(transformation)d Fm(u)i Fo(=)g Fm(v)26 b Fo(+)1861 1240 y(~)1847 1261 y(\001)q Fm(\021)38 b Fo(assuming)c(formally)g(that)h Fm(\021)k Fo(is)34 b(su\016cien)n(tly) 456 1361 y(regular.)f(Then)23 b Fm(u)g Fo(ful\014lls)g(p)r(erio)r(dic)g (b)r(oundary)f(conditions)g(in)h Fm(x;)14 b(y)26 b Fo(directions,)e (and)e(satis\014es)456 1460 y(the)28 b(follo)n(wing)e(random)h(ev)n (olution:)683 1602 y Fm(u)731 1614 y Fl(t)778 1602 y Fo(+)18 b Fm(J)8 b Fo(\()p Fm(G)p Fo(\()p Fm(u)p Fo(\))p Fm(;)14 b(u)p Fo(\))19 b(+)f Fm(J)8 b Fo(\()p Fm(\021)22 b Fj(\000)c Fm(G)p Fo(\()1687 1581 y(~)1672 1602 y(\001)q Fm(\021)s Fo(\))p Fm(;)c(u)p Fo(\))19 b(+)f Fm(\014)t(G)p Fo(\()p Fm(u)p Fo(\))2265 1614 y Fl(x)2326 1602 y Fj(\000)g Fm(\014)t Fo(\()p Fm(G)p Fo(\()2604 1581 y(~)2589 1602 y(\001)q Fm(\021)s Fo(\)\))2767 1614 y Fl(x)2828 1602 y Fj(\000)g Fm(\021)2952 1614 y Fl(x)2994 1602 y Fo(\))23 b(=)g Fm(\027)3197 1581 y Fo(~)3183 1602 y(\001)p Fm(u:)-2867 b Fo(\(4\))456 1743 y(This)27 b(equation)g(will)h(b)r(e)g(treated)f(in) h(the)g(rest)f(of)h(the)g(pap)r(er.)456 1995 y Fu(Remark)i(2.1.)41 b Fo(T)-7 b(o)36 b(obtain)g(a)g(solution)g(of)h(the)g(original)e (equation)h(all)g(rev)n(erse)f(transfor-)456 2094 y(mations)f(mak)n(e)f (sense)h(if)h Fm(\021)j Fo(is)c(con)n(tained)g(in)g(a)h(Sob)r(olev)e (space)h(of)g(second)g(order.)56 b(Hence)456 2194 y(w)n(e)31 b(can)h(\014nd)g(a)f(solution)h(of)f(this)i(equation)e(con)n(tained)g 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y Fm(H)29 b Fj(\032)23 b Fm(V)709 3208 y Ff(0)732 3238 y Fo(.)37 b(Since)28 b(this)f(equation)g(is)g(giv) n(en)g(with)h(resp)r(ect)f(to)g(p)r(erio)r(dic)g(b)r(oundary)g (conditions)456 3338 y(in)g(horizon)n(tal)f(directions)h(the)h(space)f Fm(H)35 b Fo(is)27 b(de\014ned)h(to)g(b)r(e)1320 3495 y(_)1301 3516 y Fm(L)1358 3528 y Fp(2)1394 3516 y Fo(\()p Fm(O)r Fo(\))d(=)d Fj(f)p Fm(u)g Fj(2)i Fm(L)1883 3528 y Fp(2)1920 3516 y Fo(\()p Fm(O)r Fo(\))p Fm(;)2110 3403 y Fk(Z)2156 3592 y Fl(O)2226 3516 y Fm(udO)i Fo(=)c(0)p Fj(g)p Fm(:)456 3711 y Fo(The)30 b(inner)g(pro)r(duct)g(on)g Fm(H)37 b Fo(is)30 b(denoted)g(b)n(y)g(\()p Fj(\001)p Fm(;)14 b Fj(\001)p Fo(\).)45 b(Let)31 b Fm(H)2345 3681 y Fl(k)2385 3711 y Fm(;)d(k)i Fj(2)e Fh(N)40 b Fo(b)r(e)31 b(the)f(usual)g(Sob)r(olev)456 3811 y(space)d(consisting)g(of)i (functions)f(with)h(square)e(in)n(tegrable)g(deriv)-5 b(ativ)n(e)27 b(up)h(to)h Fm(k)s Fo(-th)f(order.)37 b(If)456 3910 y Fm(k)30 b Fo(is)d(not)g(an)g(in)n(teger)f(these)h(spaces)g(are)f 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Fp(2)1717 1807 y Fo(\))p Fi(.)44 b(Then)32 b(the)f(3D)h(QGE)f(e)l(quation)h(\(5\))g (or)f(\(4\))h(has)g(a)456 1907 y(unique)d(solution)h Fm(u)p Fo(\()p Fm(t)p Fo(\))23 b Fj(2)g Fm(L)1337 1919 y Fp(2)p Fl(;loc)1478 1907 y Fo(\(0)p Fm(;)14 b Fj(1)p Fo(;)g Fm(V)k Fo(\))h Fj(\\)g Fm(C)6 b Fo(\([0)p Fm(;)14 b Fj(1)p Fo(\);)g Fm(H)7 b Fo(\))p Fi(.)456 2067 y Fu(Remark)30 b(3.10.)40 b Fo(i\))29 b(On)g(accoun)n(t)f(of)h(Lemma)f(3.6)g(w)n(e)g (can)h(pro)n(v)n(e)e(that)i Fm(u)p Fo(\()p Fm(t)p Fo(\))g(dep)r(ends)g (con-)456 2167 y(tin)n(uously)e(on)g(the)h(initial)g(conditions)f Fm(x)d Fj(2)f Fm(H)7 b Fo(.)37 b(This)27 b(will)h(b)r(e)g(used)g(later) f(on.)456 2266 y(ii\))33 b(Due)g(to)f(Lemma)g(3.4)g(for)g Fm(t)f(>)f Fo(0)i(and)h(a)f(b)r(ounded)h(set)f(of)g(initial)h (conditions)f(in)h Fm(H)39 b Fo(the)456 2366 y(image)26 b(with)h(resp)r(ect)g(to)f(the)h(solution)g(op)r(erator)e Fm(u)2105 2378 y Fp(0)2165 2366 y Fj(!)e Fm(u)p Fo(\()p Fm(t)p Fo(\))k(is)f(b)r(ounded)h(in)h Fm(V)18 b Fo(.)37 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y(A)n(t)f(\014rst)g(w)n(e)g(consider)g(the)h(random)e(part)h(of)g(the)h (b)r(oundary)f(conditions.)36 b(Let)26 b Fm(W)38 b Fo(b)r(e)27 b(a)f(con-)456 4808 y(tin)n(uous)19 b(temp)r(oral)g(Wiener)h(pro)r (cess)f(with)h(v)-5 b(alues)20 b(in)g(a)f(Hilb)r(ert)i(space)e Fm(U)31 b Fo(=)23 b Fm(H)2965 4778 y Fp(2+)3059 4756 y Fg(1)p 3059 4765 V 3059 4798 a(2)3101 4808 y Fo(\()p Fm(O)3196 4820 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(2)p Fl(\031)3354 4808 y Fo(\))s Fj(\\)475 4896 y Fo(_)456 4917 y Fm(L)513 4929 y Fp(2)549 4917 y Fo(\()p Fm(O)644 4929 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(2)p Fl(\031)802 4917 y Fo(\))43 b(This)f(Wiener)h(pro)r(cess)e(is)h(de\014ned)h(for)f (p)r(ositiv)n(e)g(and)g(negativ)n(e)g(times;)50 b(see)456 5016 y(Arnold)37 b([1])h(P)n(age)e(547.)66 b(The)38 b(co)n(v)-5 b(ariance)36 b(op)r(erator)g(of)i Fm(W)49 b Fo(is)38 b(denoted)g(b)n(y)g Fm(Q)p Fo(,)i(whic)n(h)e(is)456 5116 y(a)31 b(p)r(ositiv)n(e)h(symmetric)g(linear)g(op)r(erator)e(on)i Fm(U)41 b Fo(with)33 b(\014nite)g(trace.)50 b(The)32 b(dynamics)g(of)g Fm(W)456 5216 y Fo(is)26 b(giv)n(en)f(b)n(y)g(the)i Fi(metric)h(dynamic)l(al)j(system)26 b Fo(consisting)f(of)h(a)f (probabilit)n(y)g(space)h(\(\012)p Fm(;)14 b Fj(F)8 b Fm(;)27 b Fh(P)p Fo(\))p eop %%Page: 10 10 10 9 bop 456 238 a Fp(10)705 b(JINQIA)n(O)29 b(DUAN)g(AND)g(BJ)2028 223 y(\177)2019 238 y(ORN)g(SCHMALFUSS)456 425 y Fo(and)34 b(a)g(\015o)n(w)f Fm(\022)r Fo(,)k(\(\012)p Fm(;)14 b Fj(F)8 b Fm(;)14 b Fh(P)p Fm(;)g(\022)r Fo(\))33 b(where)h Fh(P)f Fo(is)h(the)h(Wiener)f(measure)g(with)g(co)n(v)-5 b(ariance)33 b Fm(Q)h Fo(and)456 525 y Fm(\022)25 b Fo(=)e(\()p Fm(\022)679 537 y Fl(t)708 525 y Fo(\))740 537 y Fl(t)p Ff(2)p Fc(R)884 525 y Fo(is)k(the)h(\015o)n(w)f(of)h(the)g(Wiener)f (shift:)1125 660 y Fm(W)12 b Fo(\()p Fj(\001)p Fm(;)i(\022)1346 672 y Fl(t)1375 660 y Fm(!)s Fo(\))23 b(=)g Fm(W)12 b Fo(\()p Fj(\001)18 b Fo(+)g Fm(t;)c(!)s Fo(\))19 b Fj(\000)f Fm(W)12 b Fo(\()p Fm(t;)i(!)s Fo(\))83 b(for)27 b Fm(t)c Fj(2)g Fh(R)p 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Fm(dv)p 1440 5140 87 4 v 1447 5216 a(dt)1555 5159 y Fo(+)18 b Fm(\027)5 b(Av)26 b Fo(=)d(0)p Fm(;)180 b(v)s Fo(\(0\))23 b(=)g Fm(g)s(:)-2015 b Fo(\(9\))p eop %%Page: 11 11 11 10 bop 1092 226 a Fp(THE)29 b(STOCHASTIC)h(3D)f(QUASIGEOSTR)n(OPHIC) g(FLO)n(WS)570 b(11)456 425 y Fo(w)n(e)34 b(can)g(di\013eren)n(tiate)h (the)g(the)g(solution)f(with)h(resp)r(ect)g(to)g Fm(D)2500 437 y Fl(x)2541 425 y Fo(.)58 b(The)35 b(co)r(e\016cien)n(ts)f(of)h (the)456 525 y(equation)27 b(are)f(indep)r(enden)n(t)j(of)e Fm(x)p Fo(.)38 b(The)28 b(function)g Fm(D)2171 537 y Fl(x)2212 525 y Fm(S)5 b Fo(\()p Fm(t)p Fo(\))p Fm(g)31 b Fo(ful\014lls)d(the)g(same)f(equation)g(as)456 624 y Fm(S)5 b Fo(\()p Fm(t)p Fo(\))p Fm(g)31 b Fo(but)e(with)h(deriv)n(ed) e(initial)h(condition.)39 b(Hence)29 b(w)n(e)g(ha)n(v)n(e)e Fm(D)2586 636 y Fl(x)2628 624 y Fm(S)5 b Fo(\()p Fm(t)p Fo(\))p Fm(g)27 b Fo(=)e Fm(S)5 b Fo(\()p Fm(t)p Fo(\))p Fm(D)3154 636 y Fl(x)3196 624 y Fm(g)s Fo(.)39 b(The)456 724 y(regularit)n(y)33 b(of)j 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Fm(;)h(D)1099 1556 y Fl(y)1139 1544 y Fm(;)h(D)1259 1556 y Fl(y)r(y)1334 1544 y Fm(;)14 b Fj(\001)g(\001)g(\001)g Fo(.)456 1644 y(Similar)23 b(considerations)e(allo)n(w)i(us)g(to)h(pro)n(v)n(e)e(the) i(estimates)f(for)g(deriv)-5 b(ativ)n(es)22 b Fm(D)3006 1656 y Fl(z)r(x)3082 1644 y Fm(;)14 b Fj(\001)g(\001)g(\001)27 b Fm(;)14 b(D)3349 1656 y Fl(z)r(z)3421 1644 y Fo(.)456 1747 y(In)23 b(con)n(trast)e(to)i(the)g(ab)r(o)n(v)n(e)f(form)n(ulae)g (a)g(deriv)-5 b(ativ)n(e)23 b Fm(D)2184 1759 y Fl(z)2245 1747 y Fo(c)n(hanges)e Fm(A)i Fo(to)g(an)g(op)r(erator)3192 1726 y(^)3170 1747 y Fm(A)g Fo(giv)n(en)456 1850 y(in)g(\(9\))g(with)g (semigroup)1268 1829 y(^)1254 1850 y Fm(S)t Fo(\()p Fm(t)p Fo(\).)36 b(The)23 b(assumption)f(\(8)h(ensures)f(that)2635 1829 y(^)2621 1850 y Fm(S)5 b Fo(\()p Fm(t)p Fo(\))23 b(is)g(of)g(negativ)n(e)e(t)n(yp)r(e)456 1950 y(with)28 b(our)f(usual)g(b)r(oundary)g(conditions.)36 b(W)-7 b(e)28 b(get)456 2264 y Fj(k)p Fm(D)567 2276 y Fl(z)604 2264 y Fm(AS)5 b Fo(\()p 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b(that)h(if)f(w)n(e)g(di\013eren)n (tiate)456 2542 y(the)f(co)r(e\016cien)n(ts)f(of)g(\(9\))h(t)n(wice)f (in)h(z)g(direction)f(one)g(term)g(without)h(a)g(second)e Fm(z)35 b Fo(deriv)-5 b(ativ)n(e)456 2641 y(app)r(ears.)33 b(T)-7 b(o)22 b(tak)n(e)f(this)h(term)f(in)n(to)h(accoun)n(t)f(w)n(e)g (need)h(the)g(system)f(of)h(di\013eren)n(tial)f(op)r(erators)456 2741 y(in)27 b(\(8\).)456 2840 y(T)-7 b(o)27 b(obtain)g(the)h(second)f (conclusion)g(w)n(e)g(use)g(the)h(estimate)g(for)f Fm(\014)g Fj(2)d Fo(\(0)p Fm(;)2791 2808 y Fp(1)p 2791 2822 34 4 v 2791 2869 a(4)2834 2840 y Fo(\))901 3056 y Fj(k)p Fm(AS)5 b Fo(\()p Fm(t)p Fo(\))p Fm(g)s Fj(k)1240 3022 y Fp(2)1240 3081 y Fl(D)r Fp(\()p Fl(A)1372 3064 y Fe(\014)1410 3081 y Fp(\))1463 3056 y Fo(=)23 b Fj(k)p Fm(A)1655 3022 y Fl(\014)1699 3056 y Fm(AS)5 b Fo(\()p Fm(t)p Fo(\))p Fm(g)s Fj(k)1996 3022 y Fp(2)1996 3077 y Fl(H)2082 3056 y Fj(\024)2223 3000 y Fm(C)2288 2970 y Fp(2)p 2180 3037 189 4 v 2180 3113 a Fm(t)2210 3089 y Fp(2)p Ff(\000)p Fp(2)p 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y(an)c(Ornstein-Uhlen)n(b)r(ec) n(k)f(pro)r(cess)987 3985 y Fm(\021)1028 3997 y Fp(1)1066 3985 y Fo(\()p Fj(\001)p Fm(;)14 b(!)s Fo(\))23 b(=)f Fm(A)1431 3872 y Fk(Z)1515 3893 y Ff(\001)1478 4061 y Fp(0)1552 3985 y Fm(S)5 b Fo(\()p Fj(\001)19 b(\000)f Fm(\034)9 b Fo(\))p Fm(d)1905 3964 y Fo(~)1885 3985 y Fm(G)q(W)j Fo(\()p Fm(\034)5 b(;)14 b(!)s Fo(\))23 b Fj(2)h Fm(L)2397 3997 y Fp(2)p Fl(;loc)2537 3985 y Fo(\(0)p Fm(;)14 b Fj(1)p Fo(;)g Fm(H)2844 3951 y Fp(2)2881 3985 y Fo(\))456 4187 y(solving)20 b(\(7\))i(with)h(initial)f(condition)f (zero)g(at)g Fm(t)i Fo(=)g(0.)35 b Fm(\021)2180 4199 y Fp(1)2217 4187 y Fo(\()p Fm(t;)14 b(!)s Fo(\))22 b(is)g(p)r(erio)r (dic)f(in)h Fm(x;)29 b(y)24 b Fo(directions.)456 4348 y(No)n(w)j(w)n(e)g(are)g(going)f(to)h(sho)n(w)g(that)h Fm(\021)1655 4360 y Fp(1)1692 4348 y Fo(\()p Fj(\001)p Fm(;)14 b(!)s Fo(\))28 b(has)f(con)n(tin)n(uous)g(paths)g(in)h Fm(H)7 b Fo(.)456 4509 y Fu(Lemma)29 b(4.3.)40 b Fi(Ther)l(e)31 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y(ORN)g(SCHMALFUSS)456 425 y Fo(for)e(some)g Fm(\015)g(>)c Fo(0.)36 b(The)28 b(left)g(hand)g(side)f(is)h(equal)f(to) 1334 521 y Ff(1)1307 546 y Fk(X)1314 723 y Fl(i)p Fp(=1)1441 625 y Fm(\025)1489 590 y Fp(2)1489 645 y Fl(i)1527 625 y Fj(k)p Fm(Q)1645 568 y Fg(1)p 1644 577 29 3 v 1644 610 a(2)1705 604 y Fo(~)1686 625 y Fm(G)1751 590 y Ff(\003)1790 625 y Fm(e)1829 637 y Fl(i)1856 625 y Fj(k)1898 590 y Fp(2)1898 645 y Fl(H)1974 512 y Fk(Z)2057 532 y Fl(t)2020 700 y Fp(0)2100 625 y Fm(e)2139 590 y Ff(\000)p Fp(2)p Fl(\025)2263 598 y Fe(i)2290 590 y Fl(\034)2331 625 y Fm(\034)2376 590 y Ff(\000)p Fl(\015)2471 625 y Fm(d\034)5 b(;)-2122 b Fo(\(11\))456 839 y(see)30 b(Da)h(Prato)e(and)i(Zab)r(czyk) f([23)o(])i(Prop)r(osition)d(13.2.4)g(where)h(\()p Fm(e)2630 851 y Fl(i)2658 839 y Fo(\))h(is)g(orthonormal)e(base)456 939 y(in)n(tro)r(duced)e(ab)r(o)n(v)n(e.)35 b(F)-7 b(or)27 b(su\016cien)n(tly)h(small)f Fm(\015)h(>)23 b Fo(0)k(w)n(e)g(ha)n(v)n (e)f(a)i Fm(c)p Fo(\()p Fm(\015)5 b Fo(\))23 b Fm(>)f Fo(0:)1193 1134 y Fm(\025)1241 1094 y Ff(\000)p Fp(2)p Fl(\014)1241 1157 y(i)1385 1021 y Fk(Z)1468 1042 y Fl(t)1431 1210 y Fp(0)1511 1134 y Fm(\034)1556 1100 y Ff(\000)p Fl(\015)1651 1134 y Fm(e)1690 1100 y Ff(\000)p Fp(2)p Fl(\025)1814 1108 y Fe(i)1840 1100 y Fl(\034)1882 1134 y Fm(d\034)33 b Fj(\024)22 b Fm(c)p Fo(\()p Fm(\015)5 b Fo(\))2243 1021 y Fk(Z)2326 1042 y Fl(t)2289 1210 y Fp(0)2369 1134 y Fm(e)2408 1100 y Ff(\000)p Fp(2)p Fl(\025)2532 1108 y Fe(i)2559 1100 y Fl(\034)2600 1134 y Fm(d\034)g(:)456 1318 y Fo(Hence)27 b(\(11\))h(can)f(b)r(e)h(estimated)g(b)n(y)565 1419 y Ff(1)538 1444 y Fk(X)544 1621 y Fl(i)p Fp(=1)658 1523 y Fm(\025)706 1488 y Fp(2)706 1543 y Fl(i)743 1523 y Fj(k)p Fm(Q)861 1466 y Fg(1)p 861 1475 V 861 1508 a(2)922 1502 y Fo(~)903 1523 y Fm(G)968 1488 y Ff(\003)1006 1523 y Fm(e)1045 1535 y Fl(i)1073 1523 y Fj(k)1115 1488 y Fp(2)1115 1543 y Fl(H)1191 1410 y Fk(Z)1274 1430 y Fl(t)1237 1598 y Fp(0)1317 1523 y Fm(\034)1362 1488 y Ff(\000)p Fl(\015)1457 1523 y Fm(e)1496 1488 y Ff(\000)p Fp(2)p Fl(\025)1620 1496 y Fe(i)1647 1488 y Fl(\034)1688 1523 y Fm(d\034)681 1799 y Fj(\024)22 b Fm(c)p Fo(\()p Fm(\015)5 b Fo(\))957 1695 y Ff(1)930 1720 y Fk(X)937 1897 y Fl(i)p Fp(=1)1064 1799 y Fm(\025)1112 1759 y Fp(2+2)p Fl(\014)1112 1822 y(i)1275 1799 y Fj(k)p Fm(Q)1393 1743 y Fg(1)p 1392 1752 V 1392 1785 a(2)1453 1778 y Fo(~)1434 1799 y Fm(G)1499 1765 y Ff(\003)1537 1799 y Fm(e)1576 1811 y Fl(i)1604 1799 y Fj(k)1646 1765 y Fp(2)1646 1820 y Fl(H)1722 1686 y Fk(Z)1805 1707 y Fl(t)1768 1875 y Fp(0)1848 1799 y Fm(e)1887 1765 y Ff(\000)p Fp(2)p Fl(\025)2011 1773 y Fe(i)2037 1765 y Fl(\034)2079 1799 y Fm(d\034)33 b Fo(=)23 b Fm(c)p Fo(\()p Fm(\015)5 b Fo(\))2441 1686 y Fk(Z)2523 1707 y Fl(t)2487 1875 y Fp(0)2567 1799 y Fj(k)p Fm(AS)g Fo(\()p Fm(\034)k Fo(\))p Fj(k)2878 1765 y Fp(2)2878 1824 y Ff(L)2924 1832 y Fg(2)2956 1824 y Fp(\()p Fl(U)o(;D)r Fp(\()p Fl(A)3178 1807 y Fe(\014)3217 1824 y Fp(\)\))3273 1799 y Fm(d\034)456 2009 y Fo(whic)n(h)27 b(is)h(\014nite)g(b)n(y)f (Lemma)g(4.1.)p 3384 2009 4 57 v 3388 1957 50 4 v 3388 2009 V 3437 2009 4 57 v 456 2163 a(W)-7 b(e)28 b(no)n(w)f(are)f(lo)r (oking)h(for)g(a)g(stationary)f(pro)r(cess)g(solving)h(\(7\).)456 2314 y Fu(Lemma)i(4.4.)40 b Fi(Ther)l(e)31 b(exists)e(a)i(r)l(andom)f (variable)i Fm(\021)2177 2326 y Fp(1)2238 2314 y Fj(2)24 b Fm(H)36 b Fi(such)30 b(that)g Fm(t)23 b Fj(!)h Fm(\021)2982 2326 y Fp(1)3019 2314 y Fo(\()p Fm(\022)3090 2326 y Fl(t)3120 2314 y Fm(!)s Fo(\))29 b Fi(solves)456 2414 y(\(7\).)456 2565 y(Pr)l(o)l(of.)43 b Fo(By)32 b(the)h(F)-7 b(ourier)31 b(metho)r(d)i(w)n(e)f(ha)n(v)n(e)f Fj(k)p Fm(S)5 b Fo(\()p Fm(t)p Fo(\))p Fj(k)2155 2535 y Fp(2)2155 2588 y Fl(H)q(;H)2324 2565 y Fj(\024)31 b Fm(e)2459 2535 y Ff(\000)p Fp(2)p Fl(\025)2583 2543 y Fg(1)2615 2535 y Fl(t)2644 2565 y Fo(.)52 b(W)-7 b(e)33 b(obtain)f(for)g Fm(n)3312 2577 y Fp(1)3380 2565 y Fm(<)456 2665 y(n)506 2677 y Fp(2)543 2665 y Fm(;)27 b(n)643 2677 y Fp(1)704 2665 y Fj(!)c(1)1009 2799 y Fh(E)65 b Fo(sup)1078 2872 y Fl(s)p Ff(2)p Fp([0)p Fl(;)p Fp(1])1292 2799 y Fj(k)p Fm(\021)1375 2811 y Fp(1)1412 2799 y Fo(\()p Fm(s)18 b Fo(+)g Fm(n)1634 2811 y Fp(1)1671 2799 y Fm(;)c(\022)1747 2811 y Ff(\000)p Fl(n)1840 2819 y Fg(1)1877 2799 y Fm(!)s Fo(\))k Fj(\000)g Fm(\021)2106 2811 y Fp(1)2144 2799 y Fo(\()p Fm(s)g Fo(+)g Fm(n)2366 2811 y Fp(2)2403 2799 y Fm(;)c(\022)2479 2811 y Ff(\000)p Fl(n)2572 2819 y Fg(2)2609 2799 y Fm(!)s Fo(\))p Fj(k)2738 2765 y Fp(2)2738 2819 y Fl(H)1087 3057 y Fo(=)23 b Fh(E)66 b Fo(sup)1238 3131 y Fl(s)q Ff(2)p Fp([0)p Fl(;)p Fp(1])1458 3057 y Fj(k)p Fm(A)1576 2944 y Fk(Z)1658 2965 y Fl(n)1699 2973 y Fg(1)1732 2965 y Fp(+)p Fl(s)1621 3133 y Fp(0)1832 3057 y Fm(S)5 b Fo(\()p Fm(n)1970 3069 y Fp(1)2026 3057 y Fo(+)18 b Fm(s)g Fj(\000)g Fm(\034)9 b Fo(\))p Fm(d)2389 3036 y Fo(~)2369 3057 y Fm(G)q(W)j Fo(\()p Fm(\034)5 b(;)14 b(\022)2674 3069 y Ff(\000)p Fl(n)2767 3077 y Fg(1)2804 3057 y Fm(!)s Fo(\))1083 3315 y Fj(\000)k Fm(A)1242 3202 y Fk(Z)1325 3223 y Fl(n)1366 3231 y Fg(2)1398 3223 y Fp(+)p Fl(s)1288 3391 y Fp(0)1499 3315 y Fm(S)5 b Fo(\()p Fm(n)1637 3327 y Fp(2)1692 3315 y Fo(+)18 b Fm(s)h Fj(\000)f Fm(\034)9 b Fo(\))p Fm(d)2056 3294 y Fo(~)2036 3315 y Fm(G)q(W)j Fo(\()p Fm(\034)5 b(;)14 b(\022)2341 3327 y Ff(\000)p Fl(n)2434 3335 y Fg(2)2471 3315 y Fm(!)s Fo(\))p Fj(k)2600 3281 y Fp(2)2600 3336 y Fl(H)456 3080 y Fo(\(12\))456 3499 y(W)-7 b(e)29 b(can)f(con)n(tin)n(ue)g(b)n(y)g(a)g (simple)h(in)n(tegral)f(substitution,)h(the)g(\()p Fm(\022)2537 3511 y Fl(t)2566 3499 y Fo(\))2598 3511 y Fl(t)p Ff(2)p Fc(R)2715 3499 y Fo(-in)n(v)-5 b(ariance)27 b(of)h Fh(P)g Fo(and)456 3599 y(the)g(semigroup)e(prop)r(ert)n(y)g(of)i Fm(S)489 3779 y Fo(=)22 b Fh(E)66 b Fo(sup)640 3852 y Fl(s)p Ff(2)p Fp([0)p Fl(;)p Fp(1])859 3779 y Fj(k)p Fm(A)977 3666 y Fk(Z)1059 3686 y Fl(s)1022 3854 y Ff(\000)p Fl(n)1115 3862 y Fg(1)1166 3779 y Fm(S)5 b Fo(\()p Fm(s)18 b Fj(\000)g Fm(\034)9 b Fo(\))p Fm(d)1534 3758 y Fo(~)1514 3779 y Fm(G)r(W)j Fo(\()p Fm(\034)5 b(;)14 b(!)s Fo(\))k Fj(\000)g Fm(A)2045 3666 y Fk(Z)2128 3686 y Fl(s)2091 3854 y Ff(\000)p Fl(n)2184 3862 y Fg(2)2235 3779 y Fm(S)5 b Fo(\()p Fm(s)18 b Fj(\000)g Fm(\034)9 b Fo(\))p Fm(d)2603 3758 y Fo(~)2583 3779 y Fm(G)r(W)j Fo(\()p Fm(\034)5 b(;)14 b(!)s Fo(\))p Fj(k)2979 3744 y Fp(2)2979 3799 y Fl(H)489 4038 y Fo(=)22 b Fh(E)66 b Fo(sup)640 4112 y Fl(s)p Ff(2)p Fp([0)p Fl(;)p Fp(1])859 4038 y Fj(k)p Fm(A)977 3925 y Fk(Z)1059 3946 y Ff(\000)p Fl(n)1152 3954 y Fg(1)1022 4114 y Ff(\000)p Fl(n)1115 4122 y Fg(2)1203 4038 y Fm(S)5 b Fo(\()p Fm(s)18 b Fj(\000)g Fm(\034)9 b Fo(\))p Fm(d)1571 4017 y Fo(~)1551 4038 y Fm(G)r(W)j Fj(k)1750 4004 y Fp(2)1750 4059 y Fl(H)1835 4038 y Fo(=)23 b Fh(E)65 b Fo(sup)1986 4112 y Fl(s)p Ff(2)p Fp([0)p Fl(;)p Fp(1])2205 4038 y Fj(k)p Fm(A)2323 3925 y Fk(Z)2406 3946 y Fp(0)2369 4114 y Ff(\000)p Fl(n)2462 4122 y Fg(2)2494 4114 y Fp(+)p Fl(n)2586 4122 y Fg(1)2637 4038 y Fm(S)5 b Fo(\()p Fm(n)2775 4050 y Fp(1)2830 4038 y Fo(+)18 b Fm(s)h Fj(\000)f Fm(\034)9 b Fo(\))p Fm(d)3194 4017 y Fo(~)3174 4038 y Fm(G)q(W)j Fj(k)3372 4004 y Fp(2)3372 4059 y Fl(H)489 4298 y Fj(\024)59 b Fo(sup)576 4371 y Fl(s)p Ff(2)p Fp([0)p Fl(;)p Fp(1])790 4298 y Fm(e)829 4264 y Ff(\000)p Fp(2)p Fl(\025)953 4272 y Fg(1)985 4264 y Fp(\()p Fl(n)1052 4272 y Fg(1)1084 4264 y Fp(+)p Fl(s)p Fp(\))1197 4298 y Fh(E)8 b Fj(k)p Fm(A)1364 4185 y Fk(Z)1453 4205 y Fp(0)1416 4373 y Ff(\000)p Fl(n)1509 4381 y Fg(1)1541 4373 y Fp(+)p Fl(n)1633 4381 y Fg(2)1684 4298 y Fm(S)d Fo(\()p Fj(\000)p Fm(\034)k Fo(\))p Fm(d)1976 4277 y Fo(~)1957 4298 y Fm(G)q(W)j Fo(\()p Fm(\034)5 b(;)14 b(!)s Fo(\))p Fj(k)2352 4264 y Fp(2)2352 4318 y Fl(H)489 4545 y Fj(\024)22 b Fm(e)615 4511 y Ff(\000)p Fp(2)p Fl(\025)739 4519 y Fg(1)772 4511 y Fl(n)813 4519 y Fg(1)863 4432 y Fk(Z)946 4452 y Ff(1)909 4621 y Fp(0)1030 4545 y Fj(k)p Fm(AS)5 b Fo(\()p Fm(\034)k Fo(\))1319 4524 y(~)1299 4545 y Fm(G)q(Q)1441 4488 y Fg(1)p 1441 4497 29 3 v 1441 4531 a(2)1483 4545 y Fj(k)1525 4511 y Fp(2)1525 4567 y Ff(L)1571 4575 y Fg(2)1603 4567 y Fp(\()p Fl(U)o(;H)t Fp(\))1805 4545 y Fj(\024)23 b Fm(C)6 b(e)1997 4511 y Ff(\000)p Fp(2)p Fl(\025)2121 4519 y Fg(1)2153 4511 y Fl(n)2194 4519 y Fg(1)2231 4545 y Fm(:)456 4729 y Fo(The)32 b(\014niteness)h(of)g(the)g(last)f(in)n(tegral)f(follo)n(ws)h(b)n(y)g (Lemma)h(4.1)e(Hence)i Fm(\021)2845 4741 y Fp(1)2883 4729 y Fo(\()p Fj(\001)22 b Fo(+)f Fm(n;)14 b(\022)3172 4741 y Ff(\000)p Fl(n)3269 4729 y Fm(!)s Fo(\))33 b(is)456 4829 y Fm(L)513 4841 y Fp(2)549 4829 y Fo(-con)n(v)n(ergen)n(t)26 b(with)j(resp)r(ect)f(to)h Fm(C)6 b Fo(\([0)p Fm(;)14 b Fo(1];)g Fm(H)7 b Fo(\).)39 b(This)28 b(con)n(v)n(ergence)e(is)j(exp) r(onen)n(tially)e(fast.)456 4928 y(Via)e(Lemma)g(4.3)f(and)h(the)h (semigroup)e(prop)r(ert)n(y)g(the)i(exp)r(onen)n(tial)e(con)n(v)n (ergence)f(to)i(zero)g(of)1209 5119 y Fh(E)65 b Fo(sup)1272 5193 y Fl(s)p Ff(2)p Fp([0)p Fl(;)p Fp(1])1491 5119 y Fj(k)p Fm(A)1609 5006 y Fk(Z)1692 5027 y Ff(\000)p Fl(n)p Fp(+)p Fl(s)1655 5195 y Ff(\000)p Fl(n)1885 5119 y Fm(S)5 b Fo(\()p Fm(s)18 b Fj(\000)g Fm(\034)9 b Fo(\))p Fm(d)2253 5098 y Fo(~)2233 5119 y Fm(G)r(W)j Fo(\()p Fm(\034)5 b(;)14 b(!)s Fo(\))p Fj(k)2629 5085 y Fp(2)2629 5140 y Fl(H)456 5119 y Fo(\(13\))p eop %%Page: 13 13 13 12 bop 1092 226 a Fp(THE)29 b(STOCHASTIC)h(3D)f(QUASIGEOSTR)n(OPHIC) g(FLO)n(WS)570 b(13)456 425 y Fo(for)27 b Fm(n)c Fj(!)g(1)k Fo(is)h(easily)f(seen.)456 525 y(By)k(the)h(Borel)e(and)h(Can)n(telli)g (Lemma)h(there)f(exists)g(a)g(set)h(\012)2458 495 y Fp(0)2524 525 y Fj(\032)d Fo(\012)j(of)f(full)h(measure)f(suc)n(h)456 624 y(that)25 b(for)g Fm(!)h Fj(2)d Fo(\012)974 594 y Fp(0)1036 624 y Fo(w)n(e)i(ha)n(v)n(e)f(the)i(ab)r(o)n(v)n(e)e(con)n(v) n(ergences)e Fm(!)s Fo(-wise.)35 b(The)26 b(same)e(remains)h(true)g(on) 456 724 y(\012)516 694 y Fl(j)581 724 y Fo(if)30 b(w)n(e)g(replace)f Fm(!)k Fo(b)n(y)d Fm(\022)1310 736 y Fl(j)1345 724 y Fm(!)i Fo(for)e Fm(j)i Fj(2)27 b Fh(Z)p Fo(.)38 b(The)30 b(set)2135 662 y Fk(T)2205 749 y Fl(j)2253 724 y Fo(\012)2313 694 y Fl(j)2378 724 y Fo(is)g(\()p Fm(\022)2535 736 y Fl(t)2565 724 y Fo(\))2597 736 y Fl(t)p Ff(2)p Fc(Z)2710 724 y Fo(-in)n(v)-5 b(arian)n(t)28 b(and)i(has)g(a)456 826 y(full)e(measure.)456 926 y(W)-7 b(e)37 b(denote)h(b)n(y)f Fm(\021)1052 938 y Fp(1)1089 926 y Fo(\()p Fm(!)s Fo(\))h(the)f Fm(!)s Fo(-wise)g(limit)h(of)f Fm(\021)2024 938 y Fp(1)2062 926 y Fo(\(0)24 b(+)h Fm(n;)14 b(\022)2376 938 y Ff(\000)p Fl(n)2473 926 y Fm(!)s Fo(\))37 b(whic)n(h)g(exists)g(on)3208 864 y Fk(T)3278 951 y Fl(j)3326 926 y Fo(\012)3386 896 y Fl(j)3421 926 y Fo(.)456 1039 y(T)-7 b(o)31 b(see)g(that)h Fm(\021)945 1051 y Fp(1)983 1039 y Fo(\(0)20 b(+)h Fm(n;)14 b(\022)1289 1051 y Ff(\000)p Fl(n)1386 1039 y Fm(\022)1426 1044 y Fp(\026)1425 1055 y Fl(t)1454 1039 y Fm(!)s Fo(\))32 b(is)f(con)n(v)n(erging)e(on)2197 977 y Fk(T)2266 1064 y Fl(j)2315 1039 y Fo(\012)2375 1009 y Fl(j)2441 1039 y Fo(w)n(e)j(can)f(restrict)g(ourselv)n(es)f(to)457 1131 y(\026)456 1146 y Fm(t)23 b Fj(2)g Fo([0)p Fm(;)14 b Fo(1].)35 b(In)24 b(particular,)f(if)1389 1131 y(\026)1388 1146 y Fm(t)g Fj(2)g Fo(\()p Fj(\000)p Fo(1)p Fm(;)14 b Fo(0\),)24 b Fm(\022)1856 1151 y Fp(\026)1855 1162 y Fl(t)1884 1146 y Fm(!)j Fo(can)c(b)r(e)h(written)g(as)f Fm(\022)2644 1161 y Fp(1+)2729 1150 y(\026)2728 1161 y Fl(t)2758 1146 y Fm(\022)2797 1158 y Ff(\000)p Fp(1)2886 1146 y Fm(!)s(;)36 b Fo(1)11 b(+)3130 1131 y(\026)3129 1146 y Fm(t)22 b Fj(2)i Fo(\(0)p Fm(;)14 b Fo(1\))456 1246 y(for)584 1230 y(\026)583 1246 y Fm(t)23 b Fj(2)g Fo(\()p Fj(\000)p Fo(1)p Fm(;)14 b Fo(0\).)36 b(W)-7 b(e)28 b(can)f(write)601 1459 y Fm(\021)642 1471 y Fp(1)680 1459 y Fo(\()p Fm(n;)14 b(\022)838 1475 y Ff(\000)p Fl(n)p Fp(+)983 1464 y(\026)982 1475 y Fl(t)1011 1459 y Fm(!)s Fo(\))23 b(=)g Fm(A)1285 1346 y Fk(Z)1369 1356 y Fp(\026)1368 1367 y Fl(t)1331 1535 y Ff(\000)p Fl(n)p Fp(+)1476 1524 y(\026)1475 1535 y Fl(t)1518 1459 y Fm(S)5 b Fo(\()1607 1444 y(\026)1606 1459 y Fm(t)18 b Fj(\000)g Fm(\034)9 b Fo(\))p Fm(d)1877 1438 y Fo(~)1857 1459 y Fm(G)r(W)j Fo(\()p Fm(\034)5 b(;)14 b(!)s Fo(\))1121 1712 y(=)23 b Fm(A)1285 1599 y Fk(Z)1369 1608 y Fp(\026)1368 1619 y Fl(t)1331 1787 y Ff(\000)p Fl(n)1442 1712 y Fm(S)5 b Fo(\()1531 1696 y(\026)1530 1712 y Fm(t)18 b Fj(\000)g Fm(\034)9 b Fo(\))p Fm(d)1801 1691 y Fo(~)1781 1712 y Fm(G)r(W)j Fo(\()p Fm(\034)5 b(;)14 b(!)s Fo(\))k Fj(\000)g Fm(A)2312 1599 y Fk(Z)2395 1619 y Ff(\000)p Fl(n)p Fp(+)2540 1608 y(\026)2539 1619 y Fl(t)2359 1787 y Ff(\000)p Fl(n)2582 1712 y Fm(S)5 b Fo(\()2671 1696 y(\026)2670 1712 y Fm(t)19 b Fj(\000)f Fm(\034)9 b Fo(\))p Fm(d)2942 1691 y Fo(~)2922 1712 y Fm(G)q(W)j Fo(\()p Fm(\034)5 b(;)14 b(!)s Fo(\))p Fm(:)456 1908 y Fo(By)36 b(the)i(ab)r(o)n(v)n(e)d(considerations)h(the) h(\014rst)g(term)g(on)f(the)i(righ)n(t)e(hand)h(side)g(is)g(con)n(v)n (ergen)n(t)456 2008 y(and)c(the)h(second)f(term)h(tends)g(to)f(zero)g (b)n(y)g(\(13\))h(using)f(the)h(semigroup)f(prop)r(ert)n(y)-7 b(.)54 b(Let)3394 1987 y(\026)3384 2008 y(\012)456 2108 y(b)r(e)30 b(the)h(set)g(of)f Fm(!)s Fo('s)g(suc)n(h)g(that)h Fm(\021)1502 2120 y Fp(1)1540 2108 y Fo(\()p Fm(n;)14 b(\022)1698 2120 y Ff(\000)p Fl(n)1795 2108 y Fm(!)s Fo(\))30 b(con)n(v)n(erges)e Fm(!)s Fo(-wise.)45 b(W)-7 b(e)30 b(ha)n(v)n(e)g(for)g Fm(t)d Fj(2)i Fh(R)36 b Fo(that)456 2207 y Fm(\022)495 2219 y Fl(t)533 2186 y Fo(\026)524 2207 y(\012)30 b Fj(\033)708 2145 y Fk(T)777 2232 y Fl(j)826 2207 y Fo(\012)886 2177 y Fl(j)921 2207 y Fo(.)49 b(Hence)32 b(the)g(\()p Fm(\022)1462 2219 y Fl(t)1492 2207 y Fo(\))1524 2219 y Fl(t)p Ff(2)p Fc(R)1640 2207 y Fo(-in)n(v)-5 b(arian)n(t)30 b(set)2154 2145 y Fk(T)2223 2232 y Fl(t)p Ff(2)p Fc(R)2353 2207 y Fm(\022)2392 2219 y Fl(t)2431 2186 y Fo(\026)2421 2207 y(\012)i(con)n(tains)f(a)g(set)h(of)g(measure)456 2310 y(one.)47 b(W)-7 b(e)31 b(can)g(construct)g(a)g(new)g(probabilit)n (y)f(space)h(if)g(w)n(e)g(consider)f(the)i(trace)e(of)h Fj(F)40 b Fo(and)456 2409 y(the)34 b(restriction)f(of)h Fh(P)f Fo(to)h(this)h(new)f(set)1785 2347 y Fk(T)1854 2434 y Fl(t)1897 2409 y Fm(\022)1936 2421 y Fl(t)1975 2388 y Fo(\026)1966 2409 y(\012.)56 b(W)-7 b(e)35 b(will)f(use)g(for)g (this)g(new)h(probabilit)n(y)456 2509 y(space)26 b(the)i(old)g (notations.)456 2608 y(T)-7 b(o)20 b(see)h(that)h Fm(t)h Fj(!)g Fm(\021)1073 2620 y Fp(1)1110 2608 y Fo(\()p Fm(\022)1181 2620 y Fl(t)1211 2608 y Fm(!)s Fo(\))e(solv)n(es)e(\(7\))j(w)n(e)e (consider)h Fm(\021)2150 2620 y Fp(1)2187 2608 y Fo(\()p Fm(\022)2258 2620 y Fl(t)2288 2608 y Fm(!)s Fo(\))6 b Fj(\000)g Fm(S)f Fo(\()p Fm(t)p Fo(\))p Fm(\021)2643 2620 y Fp(1)2679 2608 y Fo(\()p Fm(!)s Fo(\))21 b(whic)n(h)g(is)g(a)g (v)n(ersion)456 2720 y(of)27 b Fm(A)626 2653 y Fk(R)682 2673 y Fl(t)665 2749 y Fp(0)725 2720 y Fm(S)5 b Fo(\()p Fm(t)18 b Fj(\000)g Fm(\034)9 b Fo(\))p Fm(d)1084 2699 y Fo(~)1064 2720 y Fm(G)r(W)39 b Fo(suc)n(h)27 b(that)h(b)n(y)g(the)g (v)-5 b(ariation)26 b(of)i(constan)n(ts)e(form)n(ulae)1078 2934 y Fm(S)5 b Fo(\()p Fm(t)p Fo(\))p Fm(\021)1269 2946 y Fp(1)1307 2934 y Fo(\()p Fm(!)s Fo(\))18 b(+)g Fm(A)1603 2821 y Fk(Z)1686 2841 y Fl(t)1649 3009 y Fp(0)1729 2934 y Fm(S)5 b Fo(\()p Fm(t)19 b Fj(\000)f Fm(\034)9 b Fo(\))p Fm(d)2089 2913 y Fo(~)2069 2934 y Fm(G)q(W)j Fo(\()p Fm(\034)5 b(;)14 b(!)s Fo(\))24 b(=)e Fm(\021)2574 2946 y Fp(1)2612 2934 y Fo(\()p Fm(\022)2683 2946 y Fl(t)2712 2934 y Fm(!)s Fo(\))p Fm(;)456 3128 y Fo(see)27 b(Da)g(Prato)f(and)i (Zab)r(cyk)f([23)o(])h(\(13.2.13\).)p 3384 3128 4 57 v 3388 3076 50 4 v 3388 3128 V 3437 3128 4 57 v 456 3297 a Fu(Remark)i(4.5.)41 b Fo(Using)21 b(the)g(tec)n(hnique)g(for)g(the)h (pro)r(of)e(of)i(the)f(last)g(lemma)g(w)n(e)g(can)g(also)f(sho)n(w)456 3396 y(that)30 b(there)g(exists)g(a)g(\()p Fm(\022)1228 3408 y Fl(t)1257 3396 y Fo(\))1289 3408 y Fl(t)p Ff(2)p Fc(R)1405 3396 y Fo(-in)n(v)-5 b(arian)n(t)29 b(set)h(of)h(full)f (measure)g(suc)n(h)g(that)g Fm(t)d Fj(!)h Fm(\021)3069 3408 y Fp(1)3106 3396 y Fo(\()p Fm(\022)3177 3408 y Fl(t)3207 3396 y Fm(!)s Fo(\))i(has)456 3496 y(paths)d(in)h Fm(L)836 3508 y Fp(2)p Fl(;loc)976 3496 y Fo(\(0)p Fm(;)14 b Fj(1)p Fo(;)g Fm(H)1283 3466 y Fp(2)1320 3496 y Fo(\).)37 b(Since)28 b(the)g(distribution)g(of)f Fm(\021)2360 3508 y Fp(1)2425 3496 y Fo(is)h(a)f(Gau\031)g(measure)g(it)h(can)f(b)r(e)456 3596 y(deriv)n(ed)g(that)i(the)h(stationary)d(pro)r(cess)g Fm(\021)1800 3608 y Fp(1)1838 3596 y Fo(\()p Fm(\022)1909 3608 y Ff(\001)1933 3596 y Fo(\))i(has)f(tra)5 b(jectories)27 b(in)i Fm(L)2736 3608 y Fp(4)p Fl(;loc)2876 3596 y Fo(\(0)p Fm(;)14 b Fj(1)p Fo(;)g Fm(H)3183 3566 y Fp(2)3239 3596 y Fj(\\)19 b Fm(H)7 b Fo(\),)456 3695 y(see)27 b(Da)g(Prato)f(and)i (Zab)r(czyk)f([21)o(])g(Corollary)f(2.17.)456 3852 y(W)-7 b(e)38 b(no)n(w)f(in)n(tro)r(duce)h(the)g(term)g Fi(temp)l(er)l(e)l(d)h (r)l(andom)h(variable)p Fo(.)70 b(A)38 b(random)f(v)-5 b(ariable)37 b Fm(\030)42 b Fo(is)456 3952 y(called)27 b Fi(temp)l(er)l(e)l(d)h Fo(if)g(this)g(random)e(v)-5 b(ariable)27 b(has)g(a)g(sub)r(exp)r(onen)n(tial)h(gro)n(wth:)1491 4154 y(lim)14 b(sup)1475 4224 y Fl(t)p Ff(!1)p Fl(;t)p Ff(2)p Fc(T)1785 4098 y Fo(log)1892 4061 y Fp(+)1947 4098 y Fo(\()p Fm(\030)t Fo(\()p Fm(\022)2090 4110 y Fl(t)2120 4098 y Fm(!)s Fo(\)\))p 1785 4135 455 4 v 1974 4211 a Fj(j)p Fm(t)p Fj(j)2272 4154 y Fo(=)23 b(0)p Fm(:)456 4358 y Fo(In)i(the)g(case)f(of)h(an)f(ergo)r(dic)g(measure)g Fh(P)p Fo(,)g(it)h(is)g(w)n(ell)f(kno)n(wn)h(that)g(there)f(is)h(only)f (one)h(alterna-)456 4457 y(tiv)n(e,)d(i.e.,)h(the)f(lim)14 b(sup)22 b(in)f(the)h(last)f(form)n(ula)g(is)g(+)p Fj(1)p Fo(;)j(see)d(Arnold)g([1)o(])h(P)n(age)e(165,)h(Prop)r(osition)456 4557 y(4.1.3.)456 4714 y Fu(Lemma)29 b(4.6.)40 b Fi(Ther)l(e)c(exists)f (a)g Fo(\()p Fm(\022)1609 4726 y Fl(t)1639 4714 y Fo(\))1671 4726 y Fl(t)p Ff(2)p Fc(R)1787 4714 y Fi(-invariant)h(set)e Fo(\012)2376 4726 y Fp(1)2448 4714 y Fi(of)i(ful)t(l)g Fh(P)p Fi(-me)l(asur)l(e)d(such)i(that)456 4814 y(for)30 b(any)g Fm(!)c Fj(2)d Fo(\012)963 4826 y Fp(1)1030 4814 y Fi(,)31 b Fm(\021)1127 4826 y Fp(1)1164 4814 y Fo(\()p Fm(!)s Fo(\))f Fi(is)g(wel)t(l)h(de\014ne)l(d)f(and)1505 5020 y Fo(lim)1458 5070 y Fl(t)p Ff(!\0061)1691 4964 y Fo(log)1799 4927 y Fp(+)1868 4964 y Fj(k)p Fm(\021)1951 4976 y Fp(1)1988 4964 y Fo(\()p Fm(\022)2059 4976 y Fl(t)2088 4964 y Fm(!)s Fo(\))p Fj(k)2217 4976 y Fl(H)p 1691 5001 589 4 v 1970 5077 a Fm(t)2313 5020 y Fo(=)22 b(0)456 5199 y Fi(which)31 b(me)l(ans)f(that)f(the)h Fm(H)37 b Fi(norm)29 b(of)i Fm(\021)1717 5211 y Fp(1)1784 5199 y Fi(is)f(temp)l(er)l(e)l(d.)p eop %%Page: 14 14 14 13 bop 456 238 a Fp(14)705 b(JINQIA)n(O)29 b(DUAN)g(AND)g(BJ)2028 223 y(\177)2019 238 y(ORN)g(SCHMALFUSS)456 425 y Fi(Pr)l(o)l(of.)43 b Fo(On)27 b(accoun)n(t)g(of)g(\(12\))h(w)n(e)f(ha)n(v)n(e)1491 566 y Fh(E)66 b Fo(sup)1555 640 y Fl(s)p Ff(2)p Fp([0)p Fl(;)p Fp(1])1774 566 y Fj(k)p Fm(\021)1857 578 y Fp(1)1894 566 y Fo(\()p Fm(\022)1965 578 y Fl(s)2001 566 y Fm(!)s Fo(\))p Fj(k)2130 578 y Fl(H)2215 566 y Fm(<)23 b Fj(1)p Fm(:)456 775 y Fo(Hence)35 b(b)n(y)g(the)h(Birkho\013)e(ergo)r(dic)g (theorem,)j(w)n(e)e(ha)n(v)n(e)f(a)h(\()p Fm(\022)2442 787 y Fl(t)2471 775 y Fo(\))2503 787 y Fl(t)p Ff(2)p Fc(Z)2616 775 y Fo({in)n(v)-5 b(arian)n(t)34 b(set)h(\012)3211 787 y Fp(1)3284 775 y Fj(\032)g Fo(\012)456 875 y(suc)n(h)27 b(that)1361 1054 y(lim)1306 1104 y Fl(n)p Ff(!\0061)1556 987 y Fo(sup)1681 1007 y Fl(s)p Ff(2)p Fp([0)p Fl(;)p Fp(1])1898 987 y Fj(k)p Fm(\021)1981 999 y Fp(1)2018 987 y Fo(\()p Fm(\022)2089 999 y Fl(s)p Fp(+)p Fl(n)2217 987 y Fm(!)s Fo(\))p Fj(k)2346 999 y Fl(H)p 1556 1035 853 4 v 1957 1111 a Fm(n)2441 1054 y Fo(=)c(0)p Fm(:)456 1226 y Fo(On)k(the)h(other)f(hand)h(for)f Fm(t)c(>)f Fo(0:)1316 1368 y Fj(k)p Fm(\021)1399 1380 y Fp(1)1436 1368 y Fo(\()p Fm(\022)1507 1380 y Fl(t)1537 1368 y Fm(!)s Fo(\))p Fj(k)1666 1380 y Fl(H)1751 1368 y Fj(\024)60 b Fo(sup)1839 1441 y Fl(s)p Ff(2)p Fp([0)p Fl(;)p Fp(1])2052 1368 y Fj(k)p Fm(\021)2135 1380 y Fp(1)2172 1368 y Fo(\()p Fm(\022)2243 1383 y Fl(s)p Fp(+[)p Fl(t)p Fp(])2392 1368 y Fm(!)s Fo(\))p Fj(k)2521 1380 y Fl(H)456 1572 y Fo(suc)n(h)27 b(that)1582 1740 y(lim)1535 1790 y Fl(t)p Ff(!\0061)1768 1684 y Fj(k)p Fm(\021)1851 1696 y Fp(1)1888 1684 y Fo(\()p Fm(\022)1959 1696 y Fl(t)1988 1684 y Fm(!)s Fo(\))p Fj(k)2117 1696 y Fl(H)p 1768 1721 412 4 v 1959 1797 a Fm(t)2213 1740 y Fo(=)c(0)p Fm(:)456 1917 y Fo(and)k(similarly)g(for)g Fm(t)c(<)f Fo(0,)28 b(whic)n(h)f(is)h(su\016cien)n(t)f(for)h(the)f(\()p Fm(\022)2327 1929 y Fl(t)2357 1917 y Fo(\))2389 1929 y Fl(t)p Ff(2)p Fc(R)2505 1917 y Fo({in)n(v)-5 b(ariance)26 b(of)i(\012)3094 1929 y Fp(1)3131 1917 y Fo(.)p 3384 1917 4 57 v 3388 1864 50 4 v 3388 1917 V 3437 1917 4 57 v 456 2076 a(Collecting)f(our)f(calculations)h(w)n(e)g(get)456 2233 y Fu(Theorem)j(4.7.)40 b Fi(L)l(et)34 b Fm(Q)g Fi(b)l(e)g(a)h (line)l(ar)g(op)l(er)l(ator)g(on)g Fm(U)40 b Fo(=)31 b Fm(H)2455 2181 y Fg(5)p 2455 2190 29 3 v 2455 2223 a(2)2497 2233 y Fo(\()p Fm(O)2592 2245 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(2)p Fl(\031)2750 2233 y Fo(\))22 b Fj(\\)2900 2212 y Fo(_)2881 2233 y Fm(L)2938 2245 y Fp(2)2975 2233 y Fo(\()p Fm(O)3070 2245 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(2)p Fl(\031)3227 2233 y Fo(\))35 b Fi(with)456 2333 y(\014nite)21 b(tr)l(ac)l(e.)35 b(Then)22 b(ther)l(e)g(exists)f(a)h(temp)l(er)l(e)l(d)f(r)l(andom)h(variable)i Fm(\021)2561 2345 y Fp(1)2619 2333 y Fi(and)e(a)g Fo(\()p Fm(\022)2907 2345 y Fl(t)2937 2333 y Fo(\))2969 2345 y Fl(t)p Ff(2)p Fc(R)3085 2333 y Fi(-invariant)456 2432 y(set)29 b(of)i(ful)t(l)f(me)l(asur)l(e)f(such)h(that)1029 2574 y Fm(t)23 b Fj(!)g Fm(\021)1229 2586 y Fp(1)1266 2574 y Fo(\()p Fm(\022)1337 2586 y Fl(t)1367 2574 y Fm(!)s Fo(\))g Fj(2)g Fm(C)6 b Fo(\([0)p Fm(;)14 b Fj(1)p Fo(\);)g Fm(H)7 b Fo(\))19 b Fj(\\)g Fm(L)2164 2586 y Fp(4)p Fl(;loc)2304 2574 y Fo(\(0)p Fm(;)14 b Fj(1)p Fo(;)g Fm(H)2611 2539 y Fp(2)2666 2574 y Fj(\\)19 b Fm(H)7 b Fo(\))p Fm(:)456 2715 y(\021)497 2727 y Fp(1)534 2715 y Fo(\()p Fm(!)s Fo(\))30 b Fi(is)g(p)l(erio)l(dic)i(in)e Fm(x;)e(y)k Fi(dir)l(e)l(ctions.)456 2868 y Fo(In)37 b(a)f(similar)h(manner)f(w)n (e)h(can)g(consider)f(\(7\))h(with)g(time-p)r(erio)r(dic)g(b)r(oundary) g(condition)456 2968 y(represen)n(ting)26 b(the)i(impact)f(of)h(the)g (earth's)f(rotation)f(on)i(the)g(\015uid)g(:)848 3148 y Fm(\021)889 3160 y Fp(2)927 3168 y Fl(t)979 3148 y Fo(=)1081 3127 y(~)1067 3148 y(\001)p Fm(\021)1177 3160 y Fp(2)1214 3148 y Fm(;)1284 3091 y(@)5 b(\021)1374 3103 y Fp(2)p 1284 3128 128 4 v 1299 3205 a Fm(@)g(n)1444 3148 y Fo(=)23 b(0)p Fm(;)37 b Fo(on)27 b Fm(O)1812 3160 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(0)1928 3148 y Fm(;)2058 3091 y(@)5 b(\021)2148 3103 y Fp(2)p 2058 3128 V 2073 3205 a Fm(@)g(n)2218 3148 y Fo(=)23 b Fm(u)2354 3160 y Fp(0)2405 3148 y Fo(sin\(2)p Fm(\031)s(t)p Fo(\))g(on)k Fm(O)2894 3160 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(2)p Fl(\031)456 3148 y Fo(\(14\))456 3334 y(for)h Fm(u)632 3346 y Fp(0)694 3334 y Fj(2)e Fm(H)860 3281 y Fg(1)p 860 3290 29 3 v 860 3323 a(2)922 3334 y Fj(\\)1016 3313 y Fo(_)997 3334 y Fm(L)1054 3346 y Fp(2)1090 3334 y Fo(\()p Fm(O)1185 3346 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(2)p Fl(\031)1343 3334 y Fo(\))j(and)g(\(spatial\))g(p)r(erio)r(dic)g(b)r (oundary)f(conditions)g(on)h(the)g(other)456 3433 y(faces.)36 b(W)-7 b(e)28 b(obtain)f(without)h(pro)r(of)f(the)h(follo)n(wing)f(:) 456 3590 y Fu(Lemma)i(4.8.)40 b Fi(Supp)l(ose)i(that)f Fm(u)1541 3602 y Fp(0)1621 3590 y Fj(2)j Fm(H)1805 3538 y Fg(1)p 1805 3547 V 1805 3580 a(2)1848 3590 y Fo(\()p Fm(O)1943 3602 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(2)p Fl(\031)2100 3590 y Fo(\))27 b Fj(\\)2260 3569 y Fo(_)2241 3590 y Fm(L)2298 3602 y Fp(2)2335 3590 y Fo(\()p Fm(O)2430 3602 y Ff(\001)p Fl(;)p Ff(\001)p Fl(;)p Fp(2)p Fl(\031)2588 3590 y Fo(\))p Fi(.)73 b(Then)41 b(ther)l(e)g(exists)g(a)456 3690 y(c)l(ontinuous)29 b(p)l(erio)l(dic)i(solution)1677 3831 y Fm(t)23 b Fj(!)g Fm(\021)1877 3843 y Fp(2)1914 3831 y Fo(\()p Fm(t)p Fo(\))h Fj(2)f Fm(H)2186 3797 y Fp(2)456 3972 y Fi(which)31 b(satis\014es)f(\(14\).)39 b(In)30 b(p)l(articular,)h Fm(\021)1772 3984 y Fp(2)1809 3972 y Fo(\()p Fm(t)p Fo(\))g Fi(is)f(also)g(p)l(erio)l(dic)i(in)e Fm(x)p Fi(-)p Fm(y)j Fi(dir)l(e)l(ctions.)456 4132 y Fo(The)27 b(pro)r(of)g(is)h(based)f(on)g(the)h(prop)r(erties)f(of)g (the)h(op)r(erator)2398 4111 y(~)2379 4132 y Fm(G)p Fo(.)456 4331 y(Let)j Fm(\022)649 4301 y Fp(2)715 4331 y Fo(=)d(\()p Fm(\022)881 4301 y Fp(2)919 4331 y Fo(\))951 4343 y Fl(t)p Ff(2)p Fc(R)1098 4331 y Fo(b)r(e)k(the)f(shift)h(op)r(erator)d Fm(\022)1931 4301 y Fp(2)1929 4351 y Fl(t)1969 4331 y Fm(f)9 b Fo(\()p Fj(\001)p Fo(\))29 b(=)f Fm(f)9 b Fo(\()p Fm(t)21 b Fo(+)f Fj(\001)p Fo(\))31 b(for)g(appropriate)e(functions)456 4430 y Fm(f)9 b Fo(.)36 b(W)-7 b(e)28 b(consider)f(the)h(h)n(ull)f(of)h Fm(u)1481 4442 y Fp(0)1532 4430 y Fo(sin)13 b Fm(t)28 b Fo(with)g(resp)r(ect)f(to)h Fm(\022)2321 4400 y Fp(2)2358 4430 y Fo(:)633 4584 y(\012)693 4596 y Fp(2)753 4584 y Fo(=)851 4505 y Fk([)841 4683 y Fl(t)p Ff(2)p Fc(R)967 4584 y Fm(\022)1008 4549 y Fp(2)1006 4604 y Fl(t)1045 4584 y Fo(\()p Fm(u)1125 4596 y Fp(0)1176 4584 y Fo(sin\(2)p Fm(\031)s Fj(\001)p Fo(\)\))c(=)1610 4505 y Fk([)1600 4683 y Fl(t)p Ff(2)p Fc(R)1726 4584 y Fm(u)1774 4596 y Fp(0)1825 4584 y Fo(sin\(2)p Fm(\031)s Fo(\()p Fm(t)18 b Fo(+)g Fj(\001)p Fo(\)\))24 b(=)2487 4505 y Fk([)2413 4687 y Fl(t)p Ff(2)p Fp([0)p Fl(;)p Fp(2)p Fl(\031)r Fp(\))2668 4584 y Fm(u)2716 4596 y Fp(0)2766 4584 y Fo(sin\(2)p Fm(\031)s Fo(\()p Fm(t)19 b Fo(+)f Fj(\001)p Fo(\)\))p Fm(:)456 4817 y Fo(Summarizing,)26 b(w)n(e)i(ha)n(v)n(e)e(found)h(a)g (pro)r(cess)g Fm(\021)f Fo(=)d Fm(\021)2082 4829 y Fp(1)2137 4817 y Fo(+)18 b Fm(\021)2261 4829 y Fp(2)2326 4817 y Fo(whic)n(h)27 b(will)h(serv)n(e)e(as)g(a)h(mo)r(del)h(for)456 4917 y(the)g(p)r(erturbation)f(on)g(the)h(o)r(cean)f(surface.)456 5116 y(After)37 b(these)g(preparations)e(w)n(e)i(can)g(in)n(tro)r(duce) f(a)h(nonautonomous/random)d(dynamical)456 5216 y(system.)i(Let)28 b Fm(\022)i Fo(b)r(e)e(a)f(\015o)n(w)g(on)g(a)h(set)f(\012)h(\(suc)n(h) f(that)h(\(6\))g(is)g(ful\014lled\).)p eop %%Page: 15 15 15 14 bop 1092 226 a Fp(THE)29 b(STOCHASTIC)h(3D)f(QUASIGEOSTR)n(OPHIC) g(FLO)n(WS)570 b(15)456 425 y Fu(De\014nition)30 b(4.9.)40 b Fi(A)32 b(nonautonomous)g(dynamic)l(al)i(system)e Fm(\036)h Fi(on)g(a)f(phase)i(sp)l(ac)l(e)f Fm(H)39 b Fi(with)456 525 y(r)l(esp)l(e)l(ct)29 b(to)h(a)g(\015ow)g Fm(\022)i Fi(is)e(a)g(mapping)1557 670 y Fm(\036)23 b Fo(:)37 b Fh(T)1745 636 y Fp(+)1818 670 y Fj(\002)18 b Fo(\012)h Fj(\002)f Fm(H)29 b Fj(!)23 b Fm(H)456 816 y Fi(ful\014l)t(ling)30 b(the)g(c)l(o)l(cycle)h(pr)l(op)l(erty)1068 962 y Fm(\036)p Fo(\()p Fm(t)19 b Fo(+)f Fm(\034)5 b(;)14 b(!)s(;)g Fj(\001)p Fo(\))23 b(=)g Fm(\036)p Fo(\()p Fm(t;)14 b(\022)1804 974 y Fl(\034)1846 962 y Fm(!)s(;)g(\036)p Fo(\()p Fm(\034)5 b(;)14 b(!)s(;)g Fj(\001)p Fo(\)\))85 b Fi(for)31 b Fm(t;)c(\034)33 b Fj(2)24 b Fh(T)2778 927 y Fp(+)1199 1091 y Fm(\036)p Fo(\(0)p Fm(;)14 b(!)s(;)g Fj(\001)p Fo(\))23 b(=)g(id)1686 1103 y Fl(H)456 1245 y Fi(for)30 b Fm(!)c Fj(2)e Fo(\012)30 b Fi(and)g Fm(t;)e(\034)33 b Fj(2)24 b Fh(T)1281 1215 y Fp(+)1335 1245 y Fi(.)40 b(Supp)l(ose)30 b(that)g(the)g(\015ow)g Fm(\022)i Fi(is)f(c)l(arrie)l(d)g(by)f(a)g(metric)h(dynamic)l(al)456 1345 y(system)21 b(and)i Fm(\036)f Fi(is)g(supp)l(ose)l(d)h(to)e(b)l(e) h(me)l(asur)l(able)h(then)e Fm(\036)h Fi(forms)h(a)f(r)l(andom)h (dynamic)l(al)g(system.)456 1501 y Fo(This)k(de\014nition)h(is)g(due)f (to)h(Arnold)f([1])h(or)e(Klo)r(eden)h(et.)38 b(al.)e([15)o(])456 1657 y Fu(Remark)30 b(4.10.)40 b Fo(W)-7 b(e)42 b(no)n(w)f(consider)f (the)i(\015o)n(w)f Fm(\022)49 b Fo(=)c(\()p Fm(\022)2352 1627 y Fp(1)2390 1657 y Fm(;)14 b(\022)2468 1627 y Fp(2)2506 1657 y Fo(\))42 b(on)f(\012)46 b(=)g(\012)2986 1627 y Fp(1)3051 1657 y Fj(\002)27 b Fo(\012)3203 1627 y Fp(2)3240 1657 y Fo(.)79 b(By)456 1756 y(the)38 b(global)e(forw)n(ard)g (existence)h(and)h(uniqueness)f(of)h(the)g(solution)f(of)h(\(5\))f(for) g Fm(\021)s Fo(\()p Fm(\022)3223 1768 y Fl(t)3253 1756 y Fm(!)s Fo(\))j(=)456 1856 y Fm(\021)497 1868 y Fp(1)534 1856 y Fo(\()p Fm(\022)607 1826 y Fp(1)605 1876 y Fl(t)645 1856 y Fm(!)697 1868 y Fp(1)734 1856 y Fo(\))14 b(+)g Fm(\021)900 1868 y Fp(2)937 1856 y Fo(\()p Fm(\022)1010 1826 y Fp(2)1008 1876 y Fl(t)1048 1856 y Fm(!)1100 1868 y Fp(2)1136 1856 y Fo(\))p Fm(;)28 b(!)e Fo(=)d(\()p Fm(!)1469 1868 y Fp(1)1506 1856 y Fm(;)14 b(!)1595 1868 y Fp(2)1632 1856 y Fo(\))23 b Fj(2)h Fo(\012,)h(the)h(solution)f(op)r (erator)e(of)j(\(5\),)g(whic)n(h)f(maps)g(an)456 1956 y(initial)33 b(condition)f Fm(u)1118 1968 y Fp(0)1186 1956 y Fj(2)g Fm(H)40 b Fo(and)32 b(a)h(sample)f(p)r(oin)n(t)h Fm(!)i Fo(to)e(the)g(solution)f(at)h(time)g Fm(t)p Fo(,)h(has)e(the)456 2056 y(co)r(cycle)20 b(prop)r(ert)n(y)f(for)i Fm(H)29 b Fo(=)1393 2035 y(_)1374 2056 y Fm(L)1431 2068 y Fp(2)1468 2056 y Fo(\()p Fm(O)r Fo(\).)36 b(W)-7 b(e)21 b(will)g(denote)g(this)g (solution)f(op)r(erator)f(b)n(y)h Fm(\036)p Fo(\()p Fm(t;)14 b(!)s(;)g(x)p Fo(\).)456 2156 y(Note)28 b(that)g(b)n(y)f(the)i(p)r (erio)r(dicit)n(y)e(of)h Fm(\021)1647 2168 y Fp(2)1713 2156 y Fo(the)g(restriction)f(of)h Fm(\036)h Fo(on)e Fh(Z)22 b Fo(is)27 b(a)h Fi(r)l(andom)j(dynamic)l(al)456 2256 y(system)22 b Fo(for)g(an)n(y)g(\014xed)h Fm(!)1242 2268 y Fp(2)1302 2256 y Fj(2)h Fo(\012)1441 2268 y Fp(2)1478 2256 y Fo(.)35 b(Indeed,)24 b Fm(\022)1864 2268 y Fl(i)1892 2256 y Fm(!)h Fo(=)e(\()p Fm(\022)2130 2226 y Fp(1)2128 2277 y Fl(i)2168 2256 y Fm(!)2220 2268 y Fp(1)2257 2256 y Fm(;)14 b(!)2346 2268 y Fp(2)2383 2256 y Fo(\))23 b(whic)n(h)f(can)h (b)r(e)g(iden)n(ti\014ed)g(with)456 2355 y Fm(\022)497 2325 y Fp(1)562 2355 y Fo(and)k(whic)n(h)h(lea)n(v)n(es)e Fh(P)g Fo(in)n(v)-5 b(arian)n(t)27 b(\()p Fh(P)f Fo(is)i(ergo)r(dic)e (with)i(resp)r(ect)g(to)f(\()p Fm(\022)2756 2325 y Fp(1)2754 2377 y Fl(i)2794 2355 y Fo(\))2826 2367 y Fl(i)p Ff(2)p Fc(Z)2937 2355 y Fo(\).)456 2511 y Fu(Remark)j(4.11.)40 b Fo(Another)g(opp)r(ortunit)n(y)f(to)h(get)g(an)f(example)h(for)f(a)h Fi(c)l(omplete)g Fo(random)456 2611 y(dynamical)27 b(system)h(w)n(ould) f(b)r(e)h(to)g(equip)g(\012)1872 2623 y Fp(2)1938 2611 y Fo(with)g(an)g(ergo)r(dic)f(measure.)36 b(But)29 b(in)f(con)n(trast) 456 2710 y(to)g(the)h(fact)g(that)g(the)g(daily)f(or)g(y)n(early)f (rotation)g(of)i(the)g(earth)f(is)g(w)n(ell)h(determined)g(suc)n(h)f(a) 456 2810 y(random)23 b(ansatz)g(w)n(ould)h(express)f(that)h(the)h(b)r (eginning)f(of)g(these)g(p)r(erio)r(ds)g(is)g(rather)f(random.)456 2966 y(The)c(main)h(result)f(of)g(this)h(article)f(is)g(the)h (existence)f(of)g(an)g(attractor)f(for)h(the)h(nonautonomous)456 3065 y(dynamical)31 b(system.)50 b(This)32 b(attractor)e(will)j (attract)e(random)g(sets)h(in)g(probabilit)n(y)-7 b(.)49 b(Before)456 3165 y(w)n(e)27 b(giv)n(e)f(the)i(main)g(theorem,)f(w)n(e) g(mak)n(e)g(some)g(basic)g(remarks)f(on)h Fi(r)l(andom)k(sets)p Fo(.)456 3265 y(Supp)r(ose)39 b(that)g Fm(H)46 b Fo(is)39 b(a)g(P)n(olish)e(space.)71 b(A)39 b(set)h(function)f Fm(!)45 b Fj(!)d Fm(D)r Fo(\()p Fm(!)s Fo(\))e(with)f(closed)g(and)456 3364 y(nonempt)n(y)29 b(images)f(is)h(called)f(a)h(closed)g(random)f (set)h(o)n(v)n(er)e(\(\012)2466 3376 y Fp(1)2504 3364 y Fm(;)14 b Fj(F)8 b Fm(;)14 b Fh(P)p Fo(\))28 b(if)h(and)g(only)g(if)h (there)456 3464 y(exists)d(a)g(coun)n(table)g(n)n(um)n(b)r(er)g(of)h (random)e(v)-5 b(ariables)1451 3609 y Fm(\030)1487 3621 y Fl(i)1538 3609 y Fo(:)23 b(\(\012)1676 3621 y Fp(1)1714 3609 y Fm(;)14 b Fj(F)8 b Fm(;)14 b Fh(P)p Fo(\))22 b Fj(!)h Fm(H)r(;)97 b(i)22 b Fj(2)i Fh(N)456 3755 y Fo(suc)n(h)j(that) 1646 3913 y Fm(D)r Fo(\()p Fm(!)s Fo(\))c(=)p 1947 3816 308 4 v 1956 3834 a Fk([)1947 4013 y Fl(i)p Ff(2)p Fc(N)2071 3913 y Fm(\030)2107 3925 y Fl(i)2135 3913 y Fo(\()p Fm(!)s Fo(\))456 4142 y(see)k(Castaing)f(and)i(V)-7 b(aladier)26 b([5])i(Chapter)f(3.)456 4241 y(A)40 b(random)f(set)h Fm(D)i Fo(is)e(called)f(temp)r(ered)h(if)h(the)f(random)f(v)-5 b(ariable)39 b(dist)2869 4253 y Fl(H)2932 4241 y Fo(\()p Fm(D)r Fo(\()p Fm(!)s Fo(\))p Fm(;)14 b Fj(f)p Fo(0)p Fj(g)p Fo(\))40 b(is)456 4341 y(temp)r(ered)28 b(where)1363 4487 y(dist)1497 4499 y Fl(H)1560 4487 y Fo(\()p Fm(A;)14 b(B)t Fo(\))24 b(=)i(sup)1902 4556 y Fl(a)p Ff(2)p Fl(A)2059 4487 y Fo(inf)2046 4541 y Fl(b)p Ff(2)p Fl(B)2187 4487 y Fj(k)p Fm(a)18 b Fj(\000)g Fm(b)p Fj(k)2452 4499 y Fl(H)2514 4487 y Fm(:)456 4681 y Fo(W)-7 b(e)28 b(no)n(w)f(de\014ne)g (the)h(term)g Fi(r)l(andom)i(attr)l(actor)p Fo(.)456 4837 y Fu(De\014nition)g(4.12.)40 b Fi(L)l(et)29 b Fm(\036)h Fi(b)l(e)f(a)h(r)l(andom)g(dynamic)l(al)h(system)e(over)h(the)g(metric) g(dynamic)l(al)456 4936 y(system)i Fm(\022)r Fi(.)47 b(A)32 b(temp)l(er)l(e)l(d)h(r)l(andom)g(set)g Fo(\()p Fm(A)p Fo(\()p Fm(!)s Fo(\)\))1970 4948 y Fl(!)r Ff(2)p Fp(\012)2143 4936 y Fi(with)g(c)l(omp)l(act)h(and)f(nonempty)f(images) 456 5036 y Fm(A)p Fo(\()p Fm(!)s Fo(\))e Fi(is)g(c)l(al)t(le)l(d)h(r)l (andom)f(attr)l(actor)g(if)1319 5182 y Fm(\036)p Fo(\()p Fm(t;)14 b(!)s(;)g(A)p Fo(\()p Fm(!)s Fo(\)\))24 b(=)e Fm(A)p Fo(\()p Fm(\022)2016 5194 y Fl(t)2046 5182 y Fm(!)s Fo(\))p Fm(;)184 b(t)23 b Fj(2)g Fh(T)2527 5147 y Fp(+)456 5182 y Fo(\(15\))p eop %%Page: 16 16 16 15 bop 456 238 a Fp(16)705 b(JINQIA)n(O)29 b(DUAN)g(AND)g(BJ)2028 223 y(\177)2019 238 y(ORN)g(SCHMALFUSS)456 425 y Fi(and)h(for)g(any)h (r)l(andom)f(temp)l(er)l(e)l(d)g(close)l(d)h(r)l(andom)f(set)f Fm(D)r Fo(\()p Fm(!)s Fo(\))p Fi(:)1119 601 y Fo(\()p Fh(P)p Fo(\))122 b(lim)1249 659 y Fl(t)p Ff(!1)p Fl(;t)p Ff(2)p Fc(T)1540 643 y Fg(+)1595 601 y Fo(dist)1729 613 y Fl(H)1792 601 y Fo(\()p 1824 529 463 4 v Fm(\036)p Fo(\()p Fm(t;)14 b(!)s(;)g(D)r Fo(\()p Fm(!)s Fo(\)\))q Fm(;)g(A)p Fo(\()p Fm(\022)2457 613 y Fl(t)2486 601 y Fm(!)s Fo(\)\))24 b(=)e(0)p Fm(:)-2325 b Fo(\(16\))456 819 y(In)32 b(the)i(next)e(section)h(w)n(e)f(will)h(sho)n(w)f(that)h (the)g(dynamical)f(system)g(generated)g(b)n(y)g(\(5\))h(re-)456 918 y(stricted)27 b(to)h Fh(T)23 b Fo(=)h Fh(Z)d Fo(has)27 b(suc)n(h)h(a)g(random)f(attractor.)36 b(Ho)n(w)n(ev)n(er,)26 b(b)r(ecause)i(of)f(time-p)r(erio)r(dic)456 1018 y(p)r(erturbation,)33 b(w)n(e)g(do)g(not)g(ha)n(v)n(e)e(a)i(random)f(dynamical)g(system)h 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Fp(1)1875 2802 y Fm(;)14 b(!)1964 2814 y Fp(2)2001 2802 y Fo(\)\))p Fm(;)g Fo(0\))32 b(has)g(a)f(sub)r(exp)r (onin)n(tially)h(gro)n(wth.)49 b(If)456 2902 y(w)n(e)34 b(consider)f Fm(D)j Fo(with)f(resp)r(ect)f(to)h(the)f(restriction)g(of) g(\()p Fm(\022)2342 2914 y Fl(t)2372 2902 y Fo(\))2404 2914 y Fl(t)p Ff(2)p Fc(R)2554 2902 y Fo(to)g Fh(Z)28 b Fo(family)35 b(of)f(sets)g Fm(D)j Fo(is)456 3001 y(supp)r(osed)27 b(to)g(b)r(e)h(temp)r(ered.)38 b(The)27 b(set)h(of)f(these)h(set)g (families)f(is)h(denoted)f(b)n(y)h Fj(D)r Fo(.)456 3201 y(The)f(follo)n(wing)g(theorem)g(states)g(the)h(existence)f(of)h (random)f(attractors.)456 3369 y Fu(Theorem)j(5.1.)40 b Fi(Supp)l(ose)27 b(that)f Fm(\022)1571 3339 y Fp(1)1635 3369 y Fi(b)l(e)g(a)h(metric)f(dynamic)l(al)i(system)f(and)f(supp)l (ose)h(that)f(for)456 3469 y Fm(!)508 3481 y Fp(1)583 3469 y Fj(2)39 b Fo(\012)737 3481 y Fp(1)774 3469 y Fm(;)28 b(t)38 b Fj(2)h Fh(Z)1048 3439 y Fp(+)1135 3469 y Fi(the)g(mappings)g Fm(\036)p Fo(\()p Fm(t;)14 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b(ha)n(v)n(e)e(used)h(that)h(the)g(pro)r(duct)f(of)h(t)n(w)n(o)f(temp)r (ered)g(random)456 2443 y(v)-5 b(ariables)30 b(is)i(temp)r(ered)g(whic) n(h)f(giv)n(es)g(the)h(general)e(con)n(v)n(ergence)g(conclusion)h(of)g (the)i(main)456 2543 y(theorem)27 b(4.13)f(since)h Fm(A)p Fo(\()p Fm(!)1303 2555 y Fp(1)1341 2543 y Fm(;)20 b Fo(\026)-48 b Fm(!)1430 2555 y Fp(2)1467 2543 y Fo(\))28 b(is)f(\()p Fm(\022)1681 2555 y Fl(t)1711 2543 y Fo(\))1743 2555 y Fl(t)p Ff(2)p Fc(Z)1856 2543 y Fo(-temp)r(ered.)p 3384 2543 4 57 v 3388 2490 50 4 v 3388 2543 V 3437 2543 4 57 v 1709 2733 a(6.)45 b Fn(Summar)-6 b(y)456 2882 y Fo(W)f(e)23 b(ha)n(v)n(e)e(studied)i(the)g(3D)g(baro)r(clinic)f (quasigeostrophic)e(\015o)n(w)i(mo)r(del)h(under)f(random)g(wind)456 2982 y(forcing)29 b(and)h(time-p)r(erio)r(dic)h(\015uctuations)f(on)h (\015uid)f(b)r(oundary)-7 b(,)31 b(i.e.,)h(on)e(the)h(in)n(terface)e(b) r(e-)456 3081 y(t)n(w)n(een)j(the)h(o)r(ceans)e(and)i(the)g (atmosphere.)51 b(The)32 b(time-p)r(erio)r(dic)h(\015uctuations)f(are)g (due)h(to)456 3181 y(p)r(erio)r(dic)20 b(rotation)f(of)h(the)g(earth)g (and)g(th)n(us)g(p)r(erio)r(dic)g(exp)r(osure)f(of)h(the)h(earth)f(to)g (the)g(solar)f(ra-)456 3281 y(diation.)38 b(W)-7 b(e)28 b(ha)n(v)n(e)f(established)h(the)g(w)n(ell-p)r(osedness)f(of)h(the)h (baro)r(clinic)e(quasigeostrophic)456 3380 y(\015o)n(w)i(mo)r(del)h(in) f(the)i(state)e(space)g(\(Theorem)g(3.9\),)h(and)f(w)n(e)h(ha)n(v)n(e)e (demonstrated)h(the)h(exis-)456 3480 y(tence)e(of)g(the)g(random)f (attractors)g(\(Theorem)g(4.13\),)g(again)g(in)h(the)h(state)f(space.) 37 b(W)-7 b(e)29 b(ha)n(v)n(e)456 3580 y(also)d(discussed)h(the)h (relev)-5 b(ance)27 b(of)g(our)g(results)g(to)h(climate)f(mo)r(deling.) 456 3763 y Fu(Ac)m(kno)m(wledgemen)m(t.)66 b Fo(A)38 b(part)g(of)g(this)g(w)n(ork)f(w)n(as)g(done)g(at)h(the)g(Ob)r(erw)n (olfac)n(h)f(Math-)456 3863 y(ematical)e(Researc)n(h)g(Institute)i (supp)r(orted)f(b)n(y)g Fi(V)-6 b(olkswagen)39 b(Stiftung)c Fo(while)i(the)g(authors)456 3963 y(w)n(ere)23 b(Researc)n(h)g(in)i(P)n (airs)e(F)-7 b(ello)n(ws.)35 b(This)25 b(w)n(ork)e(w)n(as)g(partly)h (supp)r(orted)h(b)n(y)f(the)h(NSF)g(Gran)n(t)456 4062 y(DMS-9973204.)1708 4253 y Fn(References)491 4385 y Ft([1])35 b(L.)23 b(Arnold.)g Fq(Random)k(Dynamic)l(al)f(Systems)p Ft(.)d(Springer,)g(New)h(Y)-6 b(ork,)23 b(1998.)491 4468 y([2])35 b(L.)c(Arnold.)f(Hasselmann's)g(programm)f(visited:)47 b(The)32 b(analysis)f(of)g(sto)r(c)n(hasticit)n(y)i(in)e(deterministic) 601 4551 y(climate)23 b(mo)r(dels.)f(Rep)r(ort)i(450,)g(Univ)n (ersit\177)-35 b(at)24 b(Bremen,)e(Institut)j(f)r(\177)-37 b(ur)23 b(Dynamisc)n(he)h(Systeme,)f(2000.)491 4634 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 4717 y(in)d(the)i(atmosphere)e(and)i(o)r (cean.)f Fq(SIAM)i(J.)g(Math.)f(A)n(nal.)p Ft(,)f(25:1023{1068,)i (1994.)491 4800 y([4])35 b(J.)16 b(R.)g(Brannan,)i(J.)e(Duan,)i(and)f (T.)f(W)-6 b(anner.)17 b(Dissipativ)n(e)f(quasigeostrophic)i(dynamics)e (under)h(random)601 4883 y(forcing.)23 b Fq(J.)i(Math.)h(A)n(nal.)g (Appl.)p Ft(,)e(228:221{233,)i(1998.)491 4967 y([5])35 b(C.)29 b(Castaing)h(and)g(M.)f(V)-6 b(aladier.)29 b Fq(Convex)i(A)n(nalysis)g(and)h(Me)l(asur)l(able)h(Multifunctions)p Ft(.)c(LNM)g(580.)601 5050 y(Springer{V)-6 b(erlag,)23 b(Berlin{Heidelb)r(erg{New)h(Y)-6 b(ork,)23 b(1977.)491 5133 y([6])35 b(B.)28 b(Desjardins)f(and)j(E.)e(Grenier.)g(Deriv)l (ation)h(of)f(quasigeostrophic)i(p)r(oten)n(tial)g(v)n(orticit)n(y)f (equations,)601 5216 y Fq(A)l(dv.)d(Di\013.)e(Eqns.)g Fr(3)f Ft(\(1998\),)i(715-752.)g(T)-6 b(o)24 b(app)r(ear.)p eop %%Page: 20 20 20 19 bop 456 238 a Fp(20)705 b(JINQIA)n(O)29 b(DUAN)g(AND)g(BJ)2028 223 y(\177)2019 238 y(ORN)g(SCHMALFUSS)491 425 y Ft([7])35 b(The)16 b(dissipativ)n(e)h(quasigeostrophic)g(equations.)h(The)e (dissipativ)n(e)h(quasigeostrophic)g(equations.)g Fq(Math-)601 508 y(ematika)p Ft(,)23 b(18:265{288,)i(1981.)491 591 y([8])35 b(J.)24 b(Duan.)g(A)g(remark)f(on)i(three)g(dimensional)f (baro)r(clinic)g(quasigeostrophic)i(dynamics.)d Fq(Appl.)k(Math.)601 674 y(and)f(Comput.)p Ft(,)e(106:285{289,)i(1999.)491 757 y([9])35 b(Y)-6 b(u.)20 b(V.)g(Egoro)n(v)i(and)f(M.A.Sh)n(ubin.)f Fq(Partial)k(Di\013er)l(ential)e(Equations)p Ft(,)g(v)n(olume)e(I)h(of) g Fq(Encyclop)l(ae)l(dia)k(of)601 840 y(Mathematic)l(al)h(Scienc)l(es)p Ft(.)d(Springer,)g(New)h(Y)-6 b(ork,)23 b(1991.)456 923 y([10])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(y)g(w)n(a)n(v)n(es)h (for)e(geoph)n(ysical)i(\015o)n(ws)f(with)601 1006 y(arbitrary)d(p)r (oten)n(tial)i(v)n(orticit)n(y)-6 b(.)24 b Fq(Comm.)i(PDEs)p Ft(,)e(21\(619{658\):619{658,)k(1996.)456 1089 y([11])35 b(F.)20 b(Flandoli)g(and)h(B.)f(Sc)n(hmalfu\031.)f(Random)h(attractors) i(for)e(the)h(sto)r(c)n(hastic)h(3-D)e(Na)n(vier{Stok)n(es)i(equa-)601 1172 y(tion)i(with)f(m)n(ultiplicativ)n(e)g(white)i(noise.)e Fq(Sto)l(chatics)j(and)h(Sto)l(chastics)f(R)l(ep)l(orts)p Ft(,)e(59:21{45,)h(1996.)456 1255 y([12])35 b(K.)23 b(Hasselmann.)f (Sto)r(c)n(hastic)k(climate)d(mo)r(dels,)f(Part)i(I.)f Fq(T)-5 b(el)t(lus)p Ft(,)23 b(28:473{485,)i(1976.)456 1338 y([13])35 b(G.)27 b(Hollo)n(w)n(a)n(y)-6 b(.)27 b(Ocean)i(circulation:)38 b(Flo)n(w)28 b(in)f(probabilit)n(y)g(under)h (statistical)g(dynamical)f(forcing,)h(in)601 1421 y Fq(Sto)l(chastic)e (Mo)l(dels)h(in)e(Ge)l(osystems)p Ft(,)f(S.)f(Molc)n(hano)n(v)i(and)f (W.)g(W)-6 b(o)n(yczynski)25 b(\(eds.\),)f(Springer,)f(1996.)456 1504 y([14])35 b(D.)16 b(D.)h(Holm.)e(Hamiltonian)i(form)n(ulation)e (of)i(the)i(baro)r(clinic)e(quasigeostrophic)h(\015uid)g(equations.)g Fq(Phys.)601 1588 y(Fluids)p Ft(,)24 b(29:7{8,)g(1986.)456 1671 y([15])35 b(P)-6 b(.)30 b(E.)f(Klo)r(eden,)j(H.)e(Keller,)h(and)f (B.)g(Sc)n(hmalfu\031.)f(T)-6 b(o)n(w)n(ards)31 b(a)f(theory)h(of)f(n)n (umerical)f(dynamics.)h(In)601 1754 y(H.)23 b(Crauel)g(and)h(M.)f (Gundlac)n(h,)i(editors,)e Fq(Sto)l(chastic)j(Dynamics)p Ft(.)d(Springer)h(V)-6 b(erlag,)23 b(1999.)456 1837 y([16])35 b(M.)22 b(Leroux.)i Fq(Dynamic)i(A)n(nalysis)g(of)f(We)l(ather)i(and)f (Climate)p Ft(.)e(John)g(Wiley)g(&)f(Sons,)h(1998.)456 1920 y([17])35 b(P)-6 b(.)21 b(M)r(\177)-37 b(uller.)19 b(Sto)r(c)n(hastic)24 b(forcing)d(of)g(quasi-geostrophic)h(eddies.)g (In)g(P)-6 b(.)21 b(M)r(\177)-37 b(uller)20 b(R.)h(J.)g(Adler)g(and)h (B.)f(Ro-)601 2003 y(zo)n(vskii,)j(editors,)h Fq(Sto)l(chastic)i(Mo)l (del)t(ling)g(in)f(Physic)l(al)i(Oc)l(e)l(ano)l(gr)l(aphy)p Ft(,)f(pages)e(381{396.)h(Birkh\177)-35 b(auser,)601 2086 y(Basel,)23 b(1996.)456 2169 y([18])35 b(C.)23 b(Nicolis)f(and)j (G.)e(Nicolis,)f(Is)i(there)h(a)e(climate)g(attractor?)34 b Fq(Natur)l(e)24 b Fr(311)p Ft(,)e(Octob)r(er)i(11,)g(1984.)456 2252 y([19])35 b(J.)23 b(P)n(edlosky)-6 b(.)24 b Fq(Ge)l(ophysic)l(al)k (Fluid)e(Dynamics)p Ft(.)e(Springer)f(V)-6 b(erlag,)23 b(New-Y)-6 b(ork,)23 b(Berlin,)f(1987.)456 2335 y([20])35 b(J.)23 b(P)-6 b(.)23 b(P)n(eixoto)i(and)f(A.)f(H.)g(Oort.)g Fq(Physics)j(of)g(Climate)p Ft(.)e(Springer,)f(New)g(Y)-6 b(ork,)23 b(1992.)456 2418 y([21])35 b(G.)24 b(Da)h(Prato)g(and)h(J.)e (Zab)r(czyk.)i Fq(Sto)l(chastic)h(Equations)h(in)e(In\014nite)h (Dimension)p Ft(.)e(Univ)n(ersit)n(y)f(Press,)601 2501 y(Cam)n(bridge,)e(Cam)n(bridge,)g(1992.)456 2584 y([22])35 b(G.)c(Da)h(Prato)h(and)f(J.)g(Zab)r(czyk.)h(Ev)n(olution)g(equations)g (with)f(white)h(noise)f(b)r(oundary)h(conditions.)601 2667 y Fq(Sto)l(chastics)26 b(and)h(Sto)l(chastics)f(R)l(ep)l(orts)p Ft(,)e(42:167{182,)i(1993.)456 2750 y([23])35 b(G.)25 b(Da)g(Prato)g(and)h(J.)f(Zab)r(czyk.)h Fq(Er)l(go)l(dicity)i(for)f (In\014nite)g(Dimensional)h(Systems)p Ft(.)c(Cam)n(bridge)g(Uni-)601 2833 y(v)n(ersit)n(y)g(Press,)e(Cam)n(bridge,)h(1996.)456 2916 y([24])35 b(R.)23 b(Salmon.)f Fq(L)l(e)l(ctur)l(es)k(on)g(Ge)l (ophysic)l(al)i(Fluid)e(Dynamics)p Ft(.)d(Oxford)h(Univ.)f(Press,)f (Oxford,)h(1998.)456 2999 y([25])35 b(R.)24 b(M.)g(Samelson.)h(Sto)r(c) n(hastically)i(forced)e(curren)n(t)g(\015uctuations)j(in)c(v)n(ertical) i(shear)f(and)h(o)n(v)n(er)f(top)r(og-)601 3082 y(raph)n(y)-6 b(,)23 b Fq(J.)j(Ge)l(ophys.)h(R)l(es.)d Fr(94)e Ft(\(1989\))k (8207-8215.)456 3165 y([26])35 b(B.)23 b(Saltzman.)g(Climatic)f (systems)h(analysis.)g Fq(A)l(dv.)j(in)f(Ge)l(ophysics)p Ft(,)g(25:173{233,)g(1983.)456 3248 y([27])35 b(B.)18 b(Sc)n(hmalfu\031.)g(A)h(random)f(\014xed)i(p)r(oin)n(t)g(theorem)f (and)g(the)h(random)e(graph)i(transformation.)e Fq(J.)j(Math.)601 3331 y(A)n(nal.)26 b(Appl.)p Ft(,)e(225\(1\):91{113,)i(1998.)456 3414 y([28])35 b(R.)25 b(T)-6 b(emam.)23 b Fq(Navier{Stokes)28 b(Equation{The)l(ory)h(and)g(Numeric)l(al)f(A)n(nalysis)p Ft(.)e(North-Holland,)f(Ams-)601 3497 y(terdam,)d(1979.)456 3580 y([29])35 b(R.)e(T)-6 b(emam.)33 b Fq(Navier{Stokes)i(e)l(quation) h(and)h(Nonline)l(ar)g(F)-5 b(unctional)36 b(A)n(nalysis)p Ft(.)e(CBMS{NSF)h(Re-)601 3663 y(gional)23 b(Conference)i(Series)f(in)f (Applied)g(Mathematics.)h(SIAM,)f(Philadelphia,)g(1983.)456 3746 y([30])35 b(R.)26 b(T)-6 b(emam.)25 b Fq(In\014nite{Dimensional)30 b(Dynamic)l(al)f(Systems)f(in)h(Me)l(chanics)g(and)h(Physics)p Ft(.)c(Springer{)601 3829 y(V)-6 b(erlag,)23 b(Berlin{Heidelb)r (erg{New)h(Y)-6 b(ork,)23 b(second)i(edition,)e(1997.)456 3912 y([31])35 b(S.)29 b(W)-6 b(ang.)30 b(A)n(ttractors)g(for)e(the)i (3D)f(baro)r(clinic)g(quasi-geostrophic)i(equations)f(of)f(large-scale) g(atmo-)601 3995 y(sphere.)23 b Fq(J.)j(Math.)g(A)n(nal.)g(Appl.)p Ft(,)e(165:266{282,)i(1992.)456 4151 y(\(Jinqiao)e(Duan\))h Fs(Corresponding)h(a)o(uthor:)32 b(Dep)l(ar)l(tment)24 b(of)h(Applied)h(Ma)l(thema)l(tics,)f(Illinois)h(In-)456 4234 y(stitute)20 b(of)f(Technology,)i(Chica)o(go,)f(IL)g(60616,)f (USA.,)i(Tel:)31 b(\(312\)567-5335)75 b(F)-8 b(ax:)30 b(\(312\)567-3135)456 4317 y Fq(E-mail)c(addr)l(ess)p Ft(,)f(Jinqiao)f(Duan:)31 b Fa(duan@iit.edu)456 4458 y Ft(\(Bj\177)-35 b(orn)28 b(Sc)n(hmalfu\031\))h Fs(Dep)l(ar)l(tment)g (of)h(Applied)h(Sciences,)h(University)e(of)g(Technology)g(and)g(Ap-) 456 4541 y(plied)25 b(Sciences,)h(Geusaer)f(Strasse,)g(D{06217)f (Mersebur)o(g,)g(Germany,)456 4624 y Fq(E-mail)i(addr)l(ess)p Ft(,)f(Bj\177)-35 b(orn)23 b(Sc)n(hmalfu\031:)30 b Fa (schmalfuss@in.fh-merse)q(burg)q(.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0012221124945--