%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: fluc.dvi %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips -t a4 fluc %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1998.12.30:2046 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 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Fo(u)2991 2823 y Fp(H)8 b Fw(\()p Fp(u)p Fw(\))17 b(\010)3306 2782 y Fm(0)3333 2823 y Fw(\()p Fp(\032)p Fw(\))g Fp(@)3533 2838 y Fo(u)3584 2823 y Fp(\025)3642 2838 y Fo(s)3685 2823 y Fw(\()p Fp(u)p Fw(\))27 b Fp(:)0 3108 y Fw(Notice)33 b(that)g(in)g(this)g(form)m(ula)g(\010)1277 3072 y Fm(0)1305 3108 y Fw(\()p Fp(\032)p Fw(\()p Fp(u)p Fw(\)\))26 b(=)i(\010)1773 3072 y Fm(0)1801 3108 y Fw(\()p Fp(\032)1892 3123 y Fk(+)1958 3108 y Fw(\))33 b(\(resp.)44 b(\010)2390 3072 y Fm(0)2418 3108 y Fw(\()p Fp(\032)2509 3123 y Fm(\000)2576 3108 y Fw(\)\))33 b(for)g Fp(u)28 b(>)g Fw(0)33 b(\(resp.)45 b Fp(u)28 b(<)g Fw(0\).)44 b(Hence,)34 b Fp(\025)0 3227 y Fw(is)f(the)h(solution)f(of)g(the)h (linear)f(second)h(order)g(equation)f(with)g(b)s(oundary)g(conditions)h (at)f(the)g(origin:)1154 3366 y Fg(8)1154 3456 y(>)1154 3486 y(>)1154 3515 y(>)1154 3545 y(>)1154 3575 y(<)1154 3755 y(>)1154 3784 y(>)1154 3814 y(>)1154 3844 y(>)1154 3874 y(:)1259 3451 y Fp(@)1312 3466 y Fo(t)1347 3451 y Fp(\025)56 b Fw(=)f(\(1)p Fp(=)p Fw(2\)\010)1893 3410 y Fm(0)1920 3451 y Fw(\()p Fp(\032)p Fw(\)\001)p Fp(\025)1259 3610 y(p)p Fw(\010)1381 3569 y Fm(0)1409 3610 y Fw(\()p Fp(\032)1500 3625 y Fk(+)1566 3610 y Fw(\))17 b Fp(\025)p Fw(\()p Fp(t;)g Fw(0+\))53 b(=)j Fp(q)t Fw(\010)2272 3569 y Fm(0)2300 3610 y Fw(\()p Fp(\032)2391 3625 y Fm(\000)2458 3610 y Fw(\))17 b Fp(\025)p Fw(\()p Fp(t;)g Fw(0)p Fl(\000)p Fw(\))1259 3770 y(\010)1331 3728 y Fm(0)1359 3770 y Fw(\()p Fp(\032)1450 3785 y Fk(+)1515 3770 y Fw(\))p Fp(@)1607 3785 y Fo(u)1659 3770 y Fp(\025)p Fw(\()p Fp(t;)g Fw(0+\))55 b(=)g(\010)2262 3728 y Fm(0)2290 3770 y Fw(\()p Fp(\032)2381 3785 y Fm(\000)2448 3770 y Fw(\))17 b Fp(@)2557 3785 y Fo(u)2608 3770 y Fp(\025)p Fw(\()p Fp(t;)g Fw(0)p Fl(\000)p Fw(\))1259 3929 y Fp(\025)p Fw(\(0)p Fp(;)g Fl(\001)p Fw(\))54 b(=)i Fp(\025)1763 3944 y Fk(0)1807 3929 y Fw(\()p Fl(\001)p Fw(\))17 b Fp(:)3929 3690 y Fw(\(1)p Fp(:)p Fw(5\))-4466 b Fo(dif)8 b(e)p Fk(2)0 4158 y Fw(In)28 b(this)f(form)m(ula,)h Fp(\032)f Fw(is)h(the)f(stationary)g(solution)g (\(1.4\))f(of)i(the)g(original)e(equation.)42 b(The)28 b(aim)e(of)i(this)f(article)0 4277 y(is)h(th)m(us)g(to)g(pro)m(v)m(e)g (that)f(the)i(densit)m(y)f(\014eld)g Fp(Y)1693 4241 y Fo(N)1671 4302 y(t)1797 4277 y Fw(con)m(v)m(erges)g(to)g(the)g (solution)f Fp(Y)2949 4292 y Fo(t)3012 4277 y Fw(of)h(\(1.5\))f(with)g (a)h(white)f(noise)0 4397 y(added.)120 4516 y(Denote)39 b(b)m(y)g Fp(C)689 4480 y Fk(2)682 4544 y Fo(K)759 4516 y Fw(\()p Fi(R)5 b Fw(\))45 b(the)39 b(space)h(of)f(con)m(tin)m(uous)i (functions)f(on)f Fi(R)5 b Fw(,)46 b(with)39 b(compact)f(supp)s(ort)i (and)f(that)0 4636 y(are)33 b(t)m(wice)h(con)m(tin)m(uously)g (di\013eren)m(tiable)g(on)f Fi(R)1805 4651 y Fm(\003)1884 4636 y Fw(=)c(\()p Fl(\0001)p Fp(;)17 b Fw(0\))37 b Fl([)i Fw(\(0)p Fp(;)17 b Fw(+)p Fl(1)p Fw(\).)43 b(Let)313 4895 y Fp(C)391 4847 y Fk(2)p Fo(;)p Fm(\003)384 4924 y Fo(K)545 4895 y Fw(=)667 4784 y Fg(n)733 4895 y Fp(H)36 b Fl(2)28 b Fp(C)1024 4853 y Fk(2)1017 4919 y Fo(K)1110 4895 y Fw(;)44 b Fp(q)t(H)1321 4853 y Fm(0)1349 4895 y Fw(\(0+\))g(=)h Fp(pH)1861 4853 y Fm(0)1889 4895 y Fw(\(0)p Fl(\000)p Fw(\))55 b Fp(;)44 b Fw(\010)2293 4853 y Fm(0)2321 4895 y Fw(\()p Fp(\032)2412 4910 y Fk(+)2477 4895 y Fw(\))17 b Fp(H)2624 4853 y Fm(00)2674 4895 y Fw(\(0+\))56 b(=)f(\010)3139 4853 y Fm(0)3167 4895 y Fw(\()p Fp(\032)3258 4910 y Fm(\000)3325 4895 y Fw(\))17 b Fp(H)3472 4853 y Fm(0)o(0)3522 4895 y Fw(\(0)p Fl(\000)p Fw(\))3727 4784 y Fg(o)3793 4895 y Fp(:)0 5171 y Fw(Notice)33 b(that)g(for)g Fp(H)j Fl(2)28 b Fp(C)972 5124 y Fk(2)p Fo(;)p Fm(\003)965 5201 y Fo(K)1081 5171 y Fw(,)384 5338 y Fg(Z)439 5564 y Fh(R)511 5474 y Fp(du)17 b(\025)695 5489 y Fo(t)730 5474 y Fw(\()p Fp(u)p Fw(\))p Fp(H)8 b Fw(\()p Fp(u)p Fw(\))37 b Fl(\000)1245 5338 y Fg(Z)1300 5564 y Fh(R)1372 5474 y Fp(du)17 b(\025)1556 5489 y Fk(0)1600 5474 y Fw(\()p Fp(u)p Fw(\))p Fp(H)8 b Fw(\()p Fp(u)p Fw(\))44 b(=)g(\(1)p Fp(=)p Fw(2\))2371 5338 y Fg(Z)2469 5363 y Fo(t)2425 5564 y Fk(0)2521 5474 y Fp(ds)2653 5338 y Fg(Z)2708 5564 y Fh(R)2780 5474 y Fp(du)17 b(@)2965 5433 y Fk(2)2959 5498 y Fo(u)3011 5474 y Fp(H)8 b Fw(\()p Fp(u)p Fw(\))17 b(\010)3326 5433 y Fm(0)3353 5474 y Fw(\()p Fp(\032)p Fw(\))g Fp(\025)3558 5489 y Fo(s)3599 5474 y Fw(\()p Fp(u)p Fw(\))2005 5758 y(=)44 b(\(1)p Fp(=)p Fw(2\))2371 5623 y Fg(Z)2469 5647 y Fo(t)2425 5849 y Fk(0)2521 5758 y Fp(ds)2653 5623 y Fg(Z)2708 5849 y Fh(R)2780 5758 y Fp(du)17 b(H)8 b Fw(\()p Fp(u)p Fw(\))17 b(\010)3221 5717 y Fm(0)3248 5758 y Fw(\()p Fp(\032)p Fw(\))g Fp(@)3454 5717 y Fk(2)3448 5783 y Fo(u)3515 5758 y Fp(\025)3573 5773 y Fo(s)3616 5758 y Fw(\()p Fp(u)p Fw(\))0 6037 y(b)s(ecause)35 b(of)e(the)h(b)s(oundary)f(conditions)h(of)f Fp(\025)h Fw(and)f Fp(H)41 b Fw(at)33 b(the)h(origin.)p eop %%Page: 6 6 6 5 bop 0 50 a Fu(6)1123 b(CL)1290 30 y(\023)1279 50 y(AUDIO)38 b(LANDIM,)e(S)2071 30 y(\023)2063 50 y(ER)n(GIO)i(B.)e(V)n (OLCHAN)120 289 y Fw(On)e Fp(C)364 241 y Fk(2)p Fo(;)p Fm(\003)357 318 y Fo(K)506 289 y Fw(de\014ne)h(the)f(op)s(erator)f Fl(A)f Fw(:)985 579 y Fl(A)44 b Fw(=)g(\(1)p Fp(=)p Fw(2\)\010)1530 537 y Fm(0)1557 579 y Fw(\()p Fp(\032)p Fw(\))17 b(\001)54 b(=)1974 438 y Fg(\032)2082 504 y Fw(\(1)p Fp(=)p Fw(2\)\010)2382 468 y Fm(0)2409 504 y Fw(\()p Fp(\032)2500 519 y Fk(+)2565 504 y Fw(\)\001)101 b(in)55 b Fi(R)2992 519 y Fk(+)3064 504 y Fp(;)2082 653 y Fw(\(1)p Fp(=)p Fw(2\)\010)2382 617 y Fm(0)2409 653 y Fw(\()p Fp(\032)2500 668 y Fm(\000)2566 653 y Fw(\)\001)100 b(in)55 b Fi(R)2992 668 y Fm(\000)3093 653 y Fp(:)3929 579 y Fw(\(1)p Fp(:)p Fw(6\))-4466 b Fo(op)0 858 y Fw(De\014ne)35 b Fp(m)396 873 y Fk(+)495 858 y Fw(and)f Fp(m)776 873 y Fm(\000)877 858 y Fw(b)m(y)1228 978 y Fp(m)1315 993 y Fm(\000)1438 978 y Fw(=)56 b Fp(q)20 b Fw(\010)1707 937 y Fm(0)1735 978 y Fw(\()p Fp(\032)1826 993 y Fm(\000)1893 978 y Fw(\))55 b(and)h Fp(m)2290 993 y Fk(+)2412 978 y Fw(=)f Fp(p)17 b Fw(\010)2683 937 y Fm(0)2711 978 y Fw(\()p Fp(\032)2802 993 y Fk(+)2867 978 y Fw(\))0 1159 y(and)32 b(let)g Fp(m)p Fw(\()p Fp(u)p Fw(\))c(=)g Fp(m)777 1174 y Fm(\000)845 1159 y Fy(1)902 1177 y Fm(f)p Fo(u<)p Fk(0)p Fm(g)1157 1159 y Fw(+)20 b Fp(m)1341 1174 y Fk(+)1408 1159 y Fy(1)1465 1177 y Fm(f)p Fo(u>)p Fk(0)p Fm(g)1701 1159 y Fw(.)43 b(It)32 b(is)g(not)g(di\016cult)g(to)f(c)m(hec)m(k)i(that)f Fl(\000A)g Fw(is)g(a)g(symmetric)d(and)0 1285 y(non-negativ)m(e)35 b(op)s(erator)f(on)h Fp(L)1199 1249 y Fk(2)1243 1285 y Fw(\()p Fp(m)p Fw(\()p Fp(u)p Fw(\))p Fp(du)p Fw(\).)47 b(Denote)35 b(b)m(y)g Fp(D)s Fw(\()p Fl(A)p Fw(\))f(its)g(domain)g(in)g Fp(L)3168 1249 y Fk(2)3213 1285 y Fw(\()p Fp(m)p Fw(\()p Fp(u)p Fw(\))p Fp(du)p Fw(\).)47 b(In)35 b(view)f(of)0 1405 y(the)j(Hille-Y)-8 b(osida)35 b(theorem,)i(to)f(sho)m(w)h(that)f Fl(A)g Fw(is)h(the)g(generator)f(of)h(a)g(Mark)m(o)m(v)f(semigroup)g Fp(T)3698 1420 y Fo(t)3733 1405 y Fw(,)h(w)m(e)g(only)0 1524 y(need)d(to)f(c)m(hec)m(k)i(that)e(the)g(P)m(oisson)h(equation) 1723 1745 y Fp(\015)21 b(f)61 b Fl(\000)50 b(A)p Fp(f)65 b Fw(=)56 b Fp(g)1521 b Fw(\(1)p Fp(:)p Fw(7\))-4466 b Fo(P)11 b(oisson)0 1965 y Fw(has)34 b(a)f(solution)g Fp(f)39 b Fl(2)29 b Fp(D)s Fw(\()p Fl(A)p Fw(\))j(for)i(ev)m(ery)f Fp(g)f Fl(2)c Fp(L)1754 1929 y Fk(2)1799 1965 y Fw(\()p Fp(m)p Fw(\()p Fp(u)p Fw(\))p Fp(du)p Fw(\))33 b(and)h(ev)m(ery)f Fp(\015)g(>)c Fw(0.)44 b(Fix)33 b(suc)m(h)i(a)e(function)i Fp(g)h Fw(and)0 2085 y Fp(\015)d(>)28 b Fw(0.)44 b(Let)33 b Fp(f)44 b Fw(b)s(e)34 b(de\014ned)h(b)m(y)726 2510 y Fp(f)11 b Fw(\()p Fp(u)p Fw(\))54 b(=)1108 2216 y Fg(8)1108 2306 y(>)1108 2336 y(>)1108 2366 y(>)1108 2396 y(<)1108 2575 y(>)1108 2605 y(>)1108 2635 y(>)1108 2665 y(:)1213 2371 y Fp(a)1266 2386 y Fm(\000)1333 2371 y Fp(e)1379 2330 y Fo(b)1414 2340 y Fe(\000)1474 2330 y Fo(u)1565 2371 y Fw(+)1681 2235 y Fg(Z)1780 2260 y Fk(0)1736 2462 y Fm(\0001)2023 2304 y Fw(1)p 1911 2348 274 4 v 1911 2439 a(2)p Fp(b)2004 2454 y Fm(\000)2071 2439 y Fp(\022)2118 2454 y Fm(\000)2197 2371 y Fp(e)2243 2330 y Fm(\000)p Fo(b)2340 2340 y Fe(\000)2400 2330 y Fm(j)p Fo(u)p Fm(\000)p Fo(v)s Fm(j)2620 2371 y Fp(g)t Fw(\()p Fp(v)t Fw(\))p Fp(dv)t(;)70 b(u)28 b(<)g Fw(0)1213 2664 y Fp(a)1266 2679 y Fk(+)1332 2664 y Fp(e)1378 2623 y Fm(\000)p Fo(b)1475 2633 y Fj(+)1532 2623 y Fo(u)1623 2664 y Fw(+)1739 2528 y Fg(Z)1839 2553 y Fk(+)p Fm(1)1795 2755 y Fk(0)2124 2597 y Fw(1)p 2013 2641 272 4 v 2013 2732 a(2)p Fp(b)2106 2747 y Fk(+)2172 2732 y Fp(\022)2219 2747 y Fk(+)2297 2664 y Fp(e)2343 2623 y Fm(\000)p Fo(b)2440 2633 y Fj(+)2496 2623 y Fm(j)p Fo(u)p Fm(\000)p Fo(v)s Fm(j)2717 2664 y Fp(g)t Fw(\()p Fp(v)t Fw(\))p Fp(dv)t(;)70 b(u)27 b(>)i Fw(0)e Fp(:)0 2960 y Fw(Here)i Fp(\022)278 2975 y Fm(\006)373 2960 y Fw(=)f(\(1)p Fp(=)p Fw(2\)\010)778 2924 y Fm(0)805 2960 y Fw(\()p Fp(\032)896 2975 y Fm(\006)962 2960 y Fw(\),)h Fp(b)1101 2975 y Fm(\006)1196 2960 y Fw(=)1301 2874 y Fg(p)p 1401 2874 221 4 v 86 x Fp(\015)6 b(=\022)1556 2975 y Fm(\006)1650 2960 y Fw(and)30 b Fp(a)1893 2975 y Fk(+)1958 2960 y Fw(,)g Fp(a)2069 2975 y Fm(\000)2164 2960 y Fw(are)f(c)m(hosen)h(so)f(that)g Fp(f)11 b Fw(\(0+\))27 b(=)h Fp(f)11 b Fw(\(0)p Fl(\000)p Fw(\),)28 b Fp(q)t(f)3796 2924 y Fm(0)3824 2960 y Fw(\(0+\))f(=)0 3079 y Fp(pf)110 3043 y Fm(0)138 3079 y Fw(\(0)p Fl(\000)p Fw(\).)43 b(It)33 b(is)f(straigh)m(tforw)m(ard)h(to)g(c)m(hec)m(k)g(that)g Fp(f)11 b Fw(\()p Fl(\001)p Fw(\))31 b(is)i(a)f(solution)h(of)g (\(1.7\).)42 b(Denote)34 b(b)m(y)e Fp(T)3609 3094 y Fo(t)3677 3079 y Fw(\(resp.)45 b Fp(W)4060 3094 y Fo(t)4095 3079 y Fw(\))0 3199 y(the)34 b(semigroup)e(\(resp.)45 b(di\013usion)34 b(pro)s(cess\))g(asso)s(ciated)f(to)g(the)h(generator)f Fl(A)p Fw(.)120 3318 y(T)-8 b(o)38 b(state)h(the)h(main)e(theorems)g (of)i(this)f(pap)s(er,)i(w)m(e)e(need)i(some)d(more)h(notation)f(and)i (concepts.)63 b(Let)0 3438 y Fc(L)37 b Fw(b)s(e)h(the)f(linear)g(op)s (erator)g Fc(L)d Fw(=)h(\()p Fp(u)1413 3402 y Fk(2)1457 3438 y Fp(=)p Fw(2\010)1629 3402 y Fm(0)1657 3438 y Fw(\()p Fp(\032)p Fw(\)\))23 b Fl(\000)j(A)p Fw(.)55 b(Denote)37 b(b)m(y)g Fp(<)e Fl(\001)p Fp(;)17 b Fl(\001)33 b Fp(>)k Fw(\(resp.)56 b Fp(<)35 b Fl(\001)p Fp(;)17 b Fl(\001)33 b Fp(>)3585 3453 y Fo(m)3661 3438 y Fw(\))k(the)g(inner)0 3557 y(pro)s(duct)42 b(on)f Fp(L)596 3521 y Fk(2)640 3557 y Fw(\()p Fp(du)p Fw(\))f(\(resp.)68 b Fp(L)1247 3521 y Fk(2)1291 3557 y Fw(\()p Fp(m)p Fw(\()p Fp(u)p Fw(\))p Fp(du)p Fw(\)\).)e(F)-8 b(or)41 b Fp(k)i Fl(\025)e Fw(0)g(let)f Fc(H)2547 3572 y Fo(k)2637 3557 y Fw(b)s(e)h(the)g(Sob)s (olev)g(space)g(obtained)h(b)m(y)0 3677 y(completion)32 b(with)h(resp)s(ect)h(to)f(the)h(inner)g(pro)s(duct)1393 3897 y Fp(<)29 b(f)e(;)33 b(g)d(>)1791 3912 y Fo(m;k)1963 3897 y Fw(=)e Fp(<)g(f)f(;)33 b Fc(L)2376 3856 y Fo(k)2426 3897 y Fp(g)d(>)2581 3912 y Fo(m)2713 3897 y Fp(:)0 4118 y Fw(It)j(is)g(easy)g(to)g(c)m(hec)m(k)h(that,)777 4363 y Fc(H)849 4378 y Fo(k)954 4363 y Fw(=)1086 4252 y Fg(n)1153 4363 y Fp(f)55 b Fl(2)44 b Fp(L)1435 4322 y Fk(2)1480 4363 y Fw(\()p Fp(m)p Fw(\()p Fp(u)p Fw(\))p Fp(du)p Fw(\);)1949 4268 y Fg(X)1952 4482 y Fo(j)t 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Fm(\000)p Fo(k)1702 4964 y Fj(0)1744 4939 y Fw(\))27 b(b)s(e)g(the)g(space)h(of)f Fc(H)2543 4954 y Fm(\000)p Fo(k)2647 4964 y Fj(0)2691 4939 y Fw(-v)-6 b(alued)28 b(functions)g(whic)m(h)g(are)f(righ)m(t-)0 5059 y(con)m(tin)m(uous)37 b(with)f(left)g(limits.)50 b(Denote)37 b(b)m(y)f Fp(P)1782 5074 y Fo(N)1893 5059 y Fw(the)g(probabilit)m(y)g(measure)g(on)g Fp(D)s Fw(\([0)p Fp(;)17 b(T)d Fw(])p Fp(;)j Fc(H)3576 5074 y Fm(\000)p Fo(k)3680 5084 y Fj(0)3722 5059 y Fw(\))35 b(induced)0 5178 y(b)m(y)40 b(the)g(pro)s(cess)i Fp(Y)6 b(:)770 5142 y Fo(N)884 5178 y Fw(and)41 b(the)f(pro)s(duct)h(measure)f Fp(\027)2088 5193 y Fo(')2185 5178 y Fw(de\014ned)i(in)e(\(1.8\))-4466 b Fo(nu)4370 b Fw(.)63 b(Let)40 b Fi(P)3199 5193 y Fo(\027)3239 5203 y Ff(')3337 5178 y Fw(b)s(e)g(the)h(probabilit)m(y)0 5307 y(measure)k(on)h Fp(D)s Fw(\([0)p Fp(;)17 b(T)d Fw(])p Fp(;)j Fi(N)1006 5271 y Fh(Z)1064 5307 y Fw(\))45 b(induced)i(b)m(y)e(the)h(probabilit)m(y)f(measure)g Fp(\027)2836 5322 y Fo(')2939 5307 y Fw(and)h(the)f(pro)s(cess)i Fp(\021)3737 5322 y Fo(t)3817 5307 y Fw(and)f(let)0 5427 y Fp(E)74 5442 y Fo(P)125 5452 y Ff(N)193 5427 y Fw([)p Fl(\001)p Fw(])32 b(and)i Fi(E)564 5442 y Fo(\027)604 5452 y Ff(')665 5427 y Fw([)p Fl(\001)p Fw(])e(denote)i(the)g(resp)s (ectiv)m(e)g(exp)s(ectations.)-331 5602 y Fo(mt)p Fk(1)190 b Fy(Theorem)43 b(1.1)i(.)57 b Fd(Fix)40 b(a)h(p)-5 b(ositive)38 b(inte)-5 b(ger)40 b Fp(k)1796 5617 y Fk(0)1877 5602 y Fl(\025)c Fw(3)p Fd(.)60 b(L)-5 b(et)40 b Fp(P)55 b Fd(b)-5 b(e)40 b(the)g(pr)-5 b(ob)g(ability)39 b(me)-5 b(asur)g(e)41 b(c)-5 b(onc)g(entr)g(ate)g(d)0 5721 y(on)37 b Fp(C)7 b Fw(\([0)p Fp(;)17 b(T)d Fw(])p Fp(;)j Fc(H)601 5736 y Fm(\000)p Fo(k)705 5746 y Fj(0)748 5721 y Fw(\))37 b Fd(c)-5 b(orr)g(esp)g(onding)35 b(to)j(the)f(gener)-5 b(alize)g(d)36 b(Ornstein-Uhlenb)-5 b(e)g(ck)37 b(pr)-5 b(o)g(c)g(ess)36 b Fp(Y)3527 5736 y Fo(t)3599 5721 y Fd(with)h(me)-5 b(an)38 b Fw(0)0 5841 y Fd(and)d(c)-5 b(ovarianc)g(e)997 5986 y Fp(E)1071 6001 y Fo(P)1137 5876 y Fg(h)1184 5986 y Fp(Y)1242 6001 y Fo(t)1277 5986 y Fw(\()p Fp(H)8 b Fw(\))p Fp(Y)1504 6001 y Fo(s)1546 5986 y Fw(\()p Fp(G)p Fw(\))1702 5876 y Fg(i)1793 5986 y Fw(=)1915 5851 y Fg(Z)1970 6077 y Fh(R)2058 5986 y Fp(\037)p Fw(\()p Fp(u)p Fw(\)\()p Fp(T)2352 6004 y Fm(j)p Fo(t)p Fm(\000)p Fo(s)p Fm(j)2534 5986 y Fp(H)g Fw(\)\()p Fp(u)p Fw(\))p Fp(G)p Fw(\()p Fp(u)p Fw(\))17 b Fp(du)p eop %%Page: 7 7 7 6 bop 1429 50 a Fu(EQUILIBRIUM)38 b(FLUCTUA)-7 b(TIONS)1353 b(7)0 289 y Fd(for)49 b(al)5 b(l)50 b Fp(H)s(;)17 b(G)52 b Fl(2)i Fc(H)782 304 y Fo(k)824 314 y Fj(0)867 289 y Fd(.)88 b(Her)-5 b(e)49 b Fp(\037)p Fw(\()p Fl(\001)p Fw(\))g Fd(is)g(the)g(varianc)-5 b(e)48 b(of)h Fp(\021)t Fw(\()p Fl(\001)p Fw(\))p Fd(:)73 b Fp(\037)p Fw(\()p Fp(u)p Fw(\))52 b(=)i Fp(\037)3003 304 y Fk(+)3069 289 y Fy(1)3126 307 y Fm(f)p Fo(u>)p Fk(0)p Fm(g)3394 289 y Fw(+)33 b Fp(\037)3566 304 y Fm(\000)3633 289 y Fy(1)3690 307 y Fm(f)p Fo(u<)p Fk(0)p Fm(g)3976 289 y Fd(and)0 415 y Fp(\037)62 430 y Fm(\006)157 415 y Fw(=)28 b Fp(E)336 430 y Fo(\027)376 440 y Ff(')432 415 y Fw([)p Fp(\021)t Fw(\()p Fl(\006)p Fw(1\))718 379 y Fk(2)762 415 y Fw(])22 b Fl(\000)g Fp(\032)963 379 y Fk(2)963 440 y Fm(\006)1030 415 y Fd(.)45 b(Then)36 b(the)g(se)-5 b(quenc)g(e)34 b Fp(P)2000 430 y Fo(N)2112 415 y Fd(c)-5 b(onver)g(ges)34 b(we)-5 b(akly)36 b(to)g(the)f(pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(e)36 b Fp(P)14 b Fd(.)120 624 y Fw(Fix)32 b(0)c Fp(<)g(\013)f(<)i Fw(1)k(and)g(denote)h(b)m(y)g Fp(\026)1474 639 y Fo(\013)1564 624 y Fw(the)f(pro)s(duct)h(measure)f (on)h Fl(f)p Fw(0)p Fp(;)17 b Fw(1)p Fl(g)2881 588 y Fh(Z)2932 598 y Fe(\003)3006 624 y Fw(with)33 b(marginals)e(giv)m(en)i (b)m(y)1029 948 y Fp(\026)1089 963 y Fo(\013)1146 948 y Fl(f)p Fp(\030)5 b Fw(;)17 b Fp(\030)5 b Fw(\()p Fp(x)p Fw(\))24 b(=)k(1)p Fl(g)55 b Fw(=)1891 807 y Fg(\032)1999 873 y Fp(\013)746 b(x)28 b(<)g Fw(0)1999 1022 y(1)22 b Fl(\000)h Fw(\()p Fp(q)t(=p)p Fw(\)\(1)e Fl(\000)i Fp(\013)p Fw(\))99 b Fp(x)28 b(>)g Fw(0)p Fp(:)0 1266 y Fw(It)38 b(is)f(easy)h(to)g(c)m(hec)m(k)h(that)e Fp(\026)1121 1281 y Fo(\013)1216 1266 y Fw(is)h(a)f(rev)m(ersible)i(in)m(v)-6 b(arian)m(t)38 b(measure)f(for)i(the)f(en)m(vironmen)m(t)g(as)g(seen)h (from)0 1385 y(the)d(tagged)g(particle.)53 b(Moreo)m(v)m(er,)37 b Fp(\026)1435 1400 y Fo(\013)1528 1385 y Fw(is)f(sen)m(t)h(to)e Fp(\027)2015 1400 y Fo(')2109 1385 y Fw(with)h Fp(')c Fw(=)h Fp(p)2596 1349 y Fm(\000)p Fk(1)2704 1385 y Fw(\(1)23 b Fl(\000)i Fp(\013)p Fw(\))36 b(b)m(y)g(the)g(transformation)f Fl(T)0 1505 y Fw(de\014ned)g(in)e(the)g(b)s(eginning)g(of)g(this)g (section.)45 b(F)-8 b(or)32 b(this)h(\014xed)g(v)-6 b(alue)33 b(of)g Fp(')28 b Fw(=)g Fp(')p Fw(\()p Fp(\013)p Fw(\),)33 b(recall)f(that)h Fp(W)3788 1520 y Fo(t)3856 1505 y Fw(stands)0 1625 y(for)g(the)h(di\013usion)g(pro)s(cess)g(with)f(generator)g Fl(A)g Fw(\(that)g(dep)s(ends)i(on)e Fp(')28 b Fw(=)g Fp(')p Fw(\()p Fp(\013)p Fw(\)\).)-331 1824 y Fo(mt)p Fk(2)190 b Fy(Theorem)29 b(1.2)h(.)57 b Fd(L)-5 b(et)30 b Fp(\016)953 1839 y Fk(0)1005 1824 y Fl(\002)8 b Fp(\026)1150 1839 y Fo(\013)1236 1824 y Fd(b)-5 b(e)28 b(the)h(initial)f(me)-5 b(asur)g(e)29 b(of)g(the)f(pr)-5 b(o)g(c)g(ess)28 b Fw(\()p Fp(X)2897 1839 y Fo(t)2931 1824 y Fp(;)17 b(\030)3020 1839 y Fo(t)3054 1824 y Fw(\))p Fd(.)43 b(The)29 b(\014nite)f (dimensional)0 1947 y(distributions)43 b(of)h Fp(N)800 1911 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)987 1947 y Fp(X)1070 1965 y Fo(tN)1170 1945 y Fj(2)1259 1947 y Fd(c)-5 b(onver)g(ge)43 b(to)h(the)g(\014nite)g(dimensional)g(distributions)f(of)h(the)g(me)-5 b(an)44 b(zer)-5 b(o)0 2067 y(Gaussian)35 b(pr)-5 b(o)g(c)g(ess)34 b Fp(Z)841 2082 y Fo(t)912 2067 y Fd(with)i(c)-5 b(ovarianc)g(e)34 b(given)g(by)220 2376 y Fp(E)6 b Fw([)p Fp(Z)396 2391 y Fo(t)446 2376 y Fp(Z)514 2391 y Fo(s)557 2376 y Fw(])55 b(=)773 2241 y Fg(Z)872 2266 y Fm(1)828 2467 y Fk(0)973 2376 y Fp(P)1037 2391 y Fo(u)1089 2376 y Fw([)p Fp(W)1211 2391 y Fo(t)1274 2376 y Fl(\024)28 b Fw(0])17 b Fp(du)49 b Fw(+)1759 2241 y Fg(Z)1859 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Fw(\))49 b Fl(\000)2405 3808 y Fg(Z)2504 3833 y Fo(t)2460 4035 y Fk(0)2556 3944 y Fp(Y)2614 3959 y Fo(s)2656 3944 y Fw(\()p Fl(A)p Fp(H)8 b Fw(\))17 b Fp(ds)26 b(;)1529 4170 y(K)1621 4122 y Fm(A)p Fo(;H)1614 4195 y(t)1828 4170 y Fw(=)45 b(\()p Fp(M)2097 4122 y Fm(A)p Fo(;H)2086 4195 y(t)2260 4170 y Fw(\))2299 4129 y Fk(2)2393 4170 y Fl(\000)50 b(jjB)s Fp(H)8 b Fl(jj)2791 4129 y Fk(2)2791 4195 y(2)2852 4170 y Fp(t)3929 4022 y Fw(\(2)p Fp(:)p Fw(1\))-4466 b Fo(mp)p Fk(1)0 4429 y Fw(are)33 b(\()p Fp(P)s(;)17 b Fl(F)389 4444 y Fo(t)423 4429 y Fw(\)-martingales)31 b(and)j Fp(Y)1284 4444 y Fk(0)1362 4429 y Fw(is)f(a)g(mean)g(zero)g(Gaussian)h(\014eld)g(with)f (co)m(v)-6 b(ariance)34 b(giv)m(en)f(b)m(y)1147 4739 y Fp(E)1221 4754 y Fo(P)1287 4628 y Fg(h)1334 4739 y Fp(Y)1392 4754 y Fk(0)1437 4739 y Fw(\()p Fp(H)8 b Fw(\))p Fp(Y)1664 4754 y Fk(0)1708 4739 y Fw(\()p Fp(G)p Fw(\))1864 4628 y Fg(i)1966 4739 y Fw(=)2099 4603 y Fg(Z)2154 4830 y Fh(R)2226 4739 y Fp(\037)p Fw(\()p Fp(u)p Fw(\))p Fp(H)g Fw(\()p Fp(u)p Fw(\))p Fp(G)p Fw(\()p Fp(u)p Fw(\))17 b Fp(du)941 b Fw(\(2)p Fp(:)p Fw(2\))-4466 b Fo(b)p Fk(2)0 5072 y Fw(for)37 b Fp(H)8 b Fw(,)36 b Fp(G)h Fw(in)g Fc(H)618 5087 y Fo(k)660 5097 y Fj(0)704 5072 y Fw(.)53 b(Here)37 b Fl(A)f Fw(is)g(the)h(di\013eren)m(tial)g(op)s(erator)f (de\014ned)i(in)e(\(1.6\),)g Fl(B)s Fp(H)41 b Fw(=)3363 4987 y Fg(p)p 3462 4987 301 4 v 3462 5072 a Fw(\010\()p Fp(\032)p Fw(\))p Fp(=)p Fw(2)o Fl(r)p Fp(H)j Fw(and)0 5194 y Fl(k)p Fp(G)p Fl(k)178 5158 y Fk(2)178 5219 y(2)253 5194 y Fw(=)p Fp(<)31 b(G;)17 b(G)30 b(>)p Fw(.)48 b(W)-8 b(e)35 b(conclude)h(the)f(pro)s(of)g(of)g(Theorem)g(1.1)f(in)m(v)m (oking)f(Holley{Stro)s(o)s(c)m(k)g([HS])h(theory)0 5313 y(on)f(generalized)h(Ornstein{Uhlen)m(b)s(ec)m(k)h(pro)s(cesses)g(that) e(states)g(that)g(for)g(eac)m(h)h(measure)f Fp(q)k Fw(on)c Fc(H)3765 5328 y Fo(k)3807 5338 y Fj(0)3851 5313 y Fw(,)g(there)0 5433 y(is)d(a)g(unique)h(probabilit)m(y)e(measure)h(on)h Fp(C)7 b Fw(\([0)p Fp(;)17 b(T)d Fw(])p Fp(;)j Fc(H)1982 5448 y Fm(\000)p Fo(k)2086 5458 y Fj(0)2128 5433 y Fw(\))30 b(that)g(solv)m(es)g(the)g(martingale)f(problem)g(\(2.1\))g(and)0 5553 y(whose)34 b(marginal)d(at)i(time)f(0)h(is)g Fp(q)t Fw(.)120 5679 y(W)-8 b(e)28 b(p)s(ostp)s(one)g(the)g(pro)s(of)g(of)g (the)g(tigh)m(tness)g(to)f(section)h(4.)42 b(W)-8 b(e)28 b(pro)m(v)m(e)h(here)f(that)f(all)g(limit)f(p)s(oin)m(ts)i(of)f(the)0 5798 y(sequence)32 b Fp(P)474 5813 y Fo(N)580 5798 y Fw(solv)m(e)e(the)g(martingale)f(problem)g(\(2.1\),)h(\(2.2\).)42 b(Fix)29 b(a)h(limit)e(p)s(oin)m(t)i Fp(P)44 b Fw(and)31 b(assume)f(without)0 5918 y(loss)40 b(of)f(generalit)m(y)h(that)f Fp(P)1068 5933 y Fo(N)1183 5918 y Fw(con)m(v)m(erges)i(to)e Fp(P)14 b Fw(.)63 b(Recall)39 b(the)h(de\014nition)g(of)g(the)g (\014ltration)f Fl(F)3658 5933 y Fo(t)3692 5918 y Fw(.)64 b(W)-8 b(e)39 b(\014rst)0 6037 y(pro)m(v)m(e)34 b(that)f(the)g (restriction)g(of)h Fp(P)47 b Fw(to)32 b Fl(F)1551 6052 y Fk(0)1629 6037 y Fw(is)h(a)g(Gaussian)h(\014eld)g(with)f(co)m(v)-6 b(ariance)34 b(matrix)d(giv)m(en)i(b)m(y)g(\(2.2\).)p eop %%Page: 8 8 8 7 bop 0 50 a Fu(8)1123 b(CL)1290 30 y(\023)1279 50 y(AUDIO)38 b(LANDIM,)e(S)2071 30 y(\023)2063 50 y(ER)n(GIO)i(B.)e(V)n (OLCHAN)-331 289 y Fo(lemaux)62 b Fy(Lemma)38 b(2.1)h(.)57 b Fd(F)-8 b(or)36 b(every)f(c)-5 b(ontinuous)36 b(function)f Fp(H)44 b Fd(with)35 b(c)-5 b(omp)g(act)35 b(supp)-5 b(ort)36 b(and)g(every)f Fp(t)28 b(>)g Fw(0)p Fd(,)983 561 y Fw(lim)938 625 y Fo(N)7 b Fm(!1)1184 561 y Fw(log)17 b Fi(E)1390 576 y Fo(\027)1430 586 y Ff(')1491 450 y Fg(h)1555 561 y Fw(exp)p Fl(f)p Fp(iY)1849 576 y Fo(t)1884 561 y Fw(\()p Fp(H)8 b Fw(\))p Fl(g)2103 450 y Fg(i)2194 561 y Fw(=)44 b Fl(\000)p Fw(\(1)p Fp(=)p Fw(2\))27 b Fp(<)i(\037H)s(;)17 b(H)34 b(>)28 b(:)0 910 y Fd(Pr)-5 b(o)g(of.)49 b Fw(Recall)37 b(that)g(the)g(initial)f(state)h(is)g(the)h (pro)s(duct)f(in)m(v)-6 b(arian)m(t)37 b(measure)g Fp(\027)3072 925 y Fo(')3130 910 y Fw(.)55 b(Hence,)39 b(for)e(eac)m(h)h(\014xed)0 1030 y(con)m(tin)m(uous)d(function)f(with)f(compact)f(supp)s(ort,)448 1301 y(log)18 b Fi(E)654 1316 y Fo(\027)694 1326 y Ff(')756 1191 y Fg(h)819 1301 y Fw(exp)988 1191 y Fg(n)1054 1301 y Fp(iY)1146 1316 y Fo(t)1182 1301 y Fw(\()p Fp(H)8 b Fw(\))1351 1191 y Fg(oi)1508 1301 y Fw(=)1630 1207 y Fg(X)1630 1421 y Fo(x)p Fm(2)p Fh(Z)1792 1301 y Fw(log)17 b Fp(E)2011 1316 y Fo(\027)2051 1326 y Ff(')2106 1191 y Fg(h)2170 1301 y Fw(exp)2339 1191 y Fg(n)2405 1301 y Fp(iN)2530 1260 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)2718 1301 y Fp(H)8 b Fw(\()p Fp(x=)-6 b(N)11 b Fw(\)[)p Fp(\021)t Fw(\()p Fp(x)p Fw(\))20 b Fl(\000)j Fp(\032)3467 1316 y Fo(x)3517 1301 y Fw(])3545 1191 y Fg(oi)3658 1301 y Fp(:)0 1643 y Fw(By)32 b(T)-8 b(a)m(ylor)32 b(expansion,)i(the)f(last)g (exp)s(ectation)g(is)g(equal)g(to)1075 1894 y(1)38 b Fl(\000)i Fw(\(1)p Fp(=)p Fw(2\))p Fp(N)1599 1853 y Fm(\000)p Fk(1)1704 1894 y Fp(\037)p Fw(\()p Fp(x=)-6 b(N)11 b Fw(\))p Fp(H)d Fw(\()p Fp(x=)-6 b(N)11 b Fw(\))2397 1853 y Fk(2)2480 1894 y Fw(+)40 b Fp(O)s Fw(\()p Fp(N)2806 1853 y Fm(\000)p Fk(3)p Fo(=)p Fk(2)2993 1894 y Fw(\))p Fp(:)0 2141 y Fw(Therefore,)34 b(summing)e(o)m(v)m(er)h Fp(x)h Fw(and)f(using)h(that)f(log)q(\(1)21 b(+)i Fp(x)p Fw(\))k(=)h Fp(x)22 b Fw(+)h Fp(O)s Fw(\()p Fp(x)2801 2105 y Fk(2)2845 2141 y Fw(\))33 b(w)m(e)h(obtain)f(that)497 2418 y(log)18 b Fi(E)703 2433 y Fo(\027)743 2443 y Ff(')805 2308 y Fg(h)868 2418 y Fw(exp)q Fl(f)p Fp(iY)1163 2433 y Fo(t)1198 2418 y Fw(\()p Fp(H)8 b Fw(\))p Fl(g)1417 2308 y Fg(i)1507 2418 y Fw(=)45 b Fl(\000)p Fw(\(1)p Fp(=)p Fw(2\))p Fp(N)2025 2377 y Fm(\000)p Fk(1)2148 2324 y Fg(X)2148 2538 y Fo(x)p Fm(2)p Fh(Z)2310 2418 y Fp(\037)p Fw(\()p Fp(x=)-6 b(N)11 b Fw(\))p Fp(H)d Fw(\()p Fp(x=)-6 b(N)11 b Fw(\))3003 2377 y Fk(2)3086 2418 y Fw(+)39 b Fp(O)s Fw(\()p Fp(N)3411 2377 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)3598 2418 y Fw(\))0 2766 y(whic)m(h)34 b(con)m(v)m(erges,)g(as)f Fp(N)39 b Fl(")27 b(1)p Fw(,)33 b(to)g Fl(\000)p Fw(\(1)p Fp(=)p Fw(2\))27 b Fp(<)h(\037H)s(;)17 b(H)35 b(>)p Fw(.)99 b Fb(\003)120 2953 y Fw(F)-8 b(rom)32 b(this)h(lemma)f(and)i(the)f(linearit)m(y)g(of)h Fp(Y)1829 2968 y Fo(t)1897 2953 y Fw(it)f(is)g(not)h(di\016cult)f(\(cf.)46 b([KL],)33 b(Chap.)45 b(11\))33 b(to)g(pro)m(v)m(e)h(that)0 3073 y Fp(P)47 b Fw(restricted)34 b(to)f Fl(F)749 3088 y Fk(0)826 3073 y Fw(is)g(a)g(Gaussian)h(\014eld:)-331 3261 y Fo(colo)192 b Fy(Corollary)39 b(2.2)f(.)57 b Fd(R)-5 b(estricte)g(d)34 b(to)i Fl(F)1441 3276 y Fk(0)1485 3261 y Fd(,)g Fp(P)49 b Fd(is)35 b(a)g(Gaussian)h(\014eld)f(with)h(c)-5 b(ovarianc)g(e)34 b(given)h(by)1348 3532 y Fp(E)1422 3547 y Fo(P)1488 3422 y Fg(h)1535 3532 y Fp(Y)1593 3547 y Fk(0)1638 3532 y Fw(\()p Fp(G)p Fw(\))p Fp(Y)1852 3547 y Fk(0)1896 3532 y Fw(\()p Fp(H)8 b Fw(\))2065 3422 y Fg(i)2156 3532 y Fw(=)17 b Fp(<)28 b(\037H)s(;)17 b(G)27 b(>)h(:)120 3831 y Fw(T)-8 b(o)24 b(pro)m(v)m(e)h(that)f Fp(P)38 b Fw(solv)m(es)25 b(the)g(martingale)e(problem)h(\(2.1\),)h (for)g(eac)m(h)g Fp(H)33 b Fw(in)24 b Fp(C)3053 3783 y Fk(2)p Fo(;)p Fm(\003)3046 3860 y Fo(K)3187 3831 y Fw(de\014ne)i(the)f(martingales)0 3965 y Fp(M)108 3917 y Fo(N)s(;H)97 3990 y(t)273 3965 y Fw(,)33 b Fp(K)426 3917 y Fo(N)s(;H)419 3990 y(t)624 3965 y Fw(b)m(y)496 4282 y Fp(M)604 4234 y Fo(N)s(;H)593 4306 y(t)825 4282 y Fw(=)56 b Fp(Y)1038 4240 y Fo(N)1016 4306 y(t)1113 4282 y Fw(\()p Fp(H)8 b Fw(\))49 b Fl(\000)i Fp(Y)1539 4240 y Fo(N)1517 4306 y Fk(0)1614 4282 y Fw(\()p Fp(H)8 b Fw(\))49 b Fl(\000)1960 4146 y Fg(Z)2059 4171 y Fo(t)2015 4372 y Fk(0)2128 4282 y Fp(N)2219 4240 y Fk(2)2263 4282 y Fl(L)p Fp(Y)2412 4240 y Fo(N)2390 4306 y(s)2487 4282 y Fw(\()p Fp(H)8 b Fw(\))17 b Fp(ds)27 b(;)496 4566 y(K)588 4518 y Fo(N)s(;H)581 4591 y(t)809 4566 y Fw(=)942 4455 y Fg(\020)1001 4566 y Fp(M)1109 4518 y Fo(N)s(;H)1098 4591 y(t)1274 4455 y Fg(\021)1334 4476 y Fk(2)1428 4566 y Fl(\000)51 b Fp(N)1647 4525 y Fk(2)1708 4430 y Fg(Z)1807 4455 y Fo(t)1763 4657 y Fk(0)1859 4455 y Fg(n)1926 4566 y Fl(L)p Fw(\()p Fp(Y)2113 4525 y Fo(N)2092 4591 y(s)2189 4566 y Fw(\()p Fp(H)8 b Fw(\)\))2397 4525 y Fk(2)2490 4566 y Fl(\000)51 b Fw(2)p Fp(Y)2747 4525 y Fo(N)2726 4591 y(s)2823 4566 y Fw(\()p Fp(H)8 b Fw(\))17 b Fl(L)p Fp(Y)3157 4525 y Fo(N)3136 4591 y(s)3232 4566 y Fw(\()p Fp(H)8 b Fw(\))3401 4455 y Fg(o)3484 4566 y Fp(ds)27 b(:)0 4869 y Fw(Denote)h(b)m(y)f Fl(r)551 4884 y Fo(N)654 4869 y Fw(and)h(\001)925 4884 y Fo(N)1028 4869 y Fw(the)g(discrete)g (deriv)-6 b(ativ)m(e)27 b(and)h(the)f(discrete)h(Laplacian:)41 b(\()p Fl(r)3331 4884 y Fo(N)3407 4869 y Fp(f)11 b Fw(\)\()p Fp(x)p Fw(\))26 b(=)i Fp(N)11 b Fw([)p Fp(f)g Fw(\()p Fp(x)f Fw(+)0 4989 y(1\))q Fl(\000)q Fp(f)h Fw(\()p Fp(x)p Fw(\)])22 b(and)i(\(\001)719 5004 y Fo(N)794 4989 y Fp(f)11 b Fw(\)\()p Fp(x)p Fw(\))26 b(=)j Fp(N)1251 4953 y Fk(2)1295 4989 y Fw([)p Fp(f)11 b Fw(\()p Fp(x)q Fw(+)q(1\))q(+)q Fp(f)g Fw(\()p Fp(x)q Fl(\000)q Fw(1\))q Fl(\000)q Fw(2)p Fp(f)g Fw(\()p Fp(x)p Fw(\)].)42 b(Note)22 b(that)h Fp(N)2999 4953 y Fk(2)3043 4989 y Fl(L)p Fp(\021)t Fw(\()p Fp(x)p Fw(\))k(=)h(\(1)p Fp(=)p Fw(2\)\001)3743 5004 y Fo(N)3818 4989 y Fp(g)t Fw(\()p Fp(\021)t Fw(\()p Fp(x)p Fw(\)\))0 5109 y(for)46 b Fp(x)j Fl(6)p Fw(=)h Fl(\000)p Fw(1,)f(0.)82 b(A)46 b(simple)f(computation)g(taking)g(in)m(to)h(accoun)m(t)h(the)f (b)s(oundary)h(terms)e(sho)m(ws)i(that)0 5228 y Fp(N)91 5192 y Fk(2)136 5228 y Fl(L)p Fp(Y)284 5192 y Fo(N)360 5228 y Fw(\()p Fp(H)8 b Fw(\))32 b(is)i(equal)f(to)467 5495 y(\(1)p Fp(=)p Fw(2\))p Fp(N)786 5454 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)1053 5401 y Fg(X)989 5615 y Fo(x)p Fm(6)p Fk(=)p Fm(\000)p Fk(1)p Fo(;)p Fk(0)1277 5495 y Fp(H)8 b Fw(\()p Fp(x=)-6 b(N)11 b Fw(\))17 b(\001)1738 5510 y Fo(N)1813 5495 y Fp(g)t Fw(\()p Fp(\021)t Fw(\()p Fp(x)p Fw(\)\))689 5793 y(+)50 b Fp(N)907 5752 y Fk(3)p Fo(=)p Fk(2)1032 5793 y Fp(H)8 b Fw(\()p Fl(\000)p Fw(1)p Fp(=)-6 b(N)11 b Fw(\))1480 5683 y Fg(n)1546 5793 y Fw(\(1)p Fp(=)p Fw(2\))p Fp(g)t Fw(\()p Fp(\021)t Fw(\()p Fl(\000)p Fw(2)n(\)\))21 b(+)i Fp(pg)t Fw(\()p Fp(\021)t Fw(\(0\)\))d Fl(\000)j Fw([\(1)p Fp(=)p Fw(2\))e(+)h Fp(q)t Fw(])p Fp(g)t Fw(\()p Fp(\021)t Fw(\()p Fl(\000)p Fw(1)o(\)\))3601 5683 y Fg(o)888 6012 y Fw(+)39 b Fp(N)1095 5971 y Fk(3)p Fo(=)p Fk(2)1220 6012 y Fp(H)8 b Fw(\(0\))1439 5902 y Fg(n)1505 6012 y Fw(\(1)p Fp(=)p Fw(2\))p Fp(g)t Fw(\()p Fp(\021)t Fw(\(1\)\))19 b(+)j Fp(q)t(g)t Fw(\()p Fp(\021)t Fw(\()p Fl(\000)p Fw(1)o(\)\))g Fl(\000)h Fw([\(1)p Fp(=)p Fw(2\))d(+)j Fp(p)p Fw(])p Fp(g)t Fw(\()p Fp(\021)t Fw(\(0\)\))3484 5902 y Fg(o)3576 6012 y Fp(:)p eop %%Page: 9 9 9 8 bop 1429 50 a Fu(EQUILIBRIUM)38 b(FLUCTUA)-7 b(TIONS)1353 b(9)0 289 y Fw(Let)39 b Fp(\014)240 304 y Fo(x)366 289 y Fw(=)76 b Fp(E)593 304 y Fo(\027)633 314 y Ff(')688 289 y Fw([)p Fp(g)t Fw(\()p Fp(\021)t Fw(\()p Fp(x)p Fw(\)\)])37 b(and)i(notice)h(that)f(\001)p Fp(\014)1959 304 y Fo(x)2084 289 y Fw(=)76 b(0)39 b(for)g Fp(x)f Fl(6)p Fw(=)g Fl(\000)p Fw(1)p Fp(;)17 b Fw(0.)61 b(W)-8 b(e)40 b(ma)m(y)e(therefore)i(replace)0 408 y(\001)83 423 y Fo(N)159 408 y Fp(g)t Fw(\()p Fp(\021)t Fw(\()p Fp(x)p Fw(\)\))35 b(b)m(y)j(\001)738 423 y Fo(N)813 408 y Fw([)p Fp(g)t Fw(\()p Fp(\021)t Fw(\()p Fp(x)p Fw(\)\))23 b Fl(\000)i Fp(\014)1340 423 y Fo(x)1391 408 y Fw(])37 b(in)g(the)h(\014rst)g(expression.)57 b(A)37 b(double)i(summation)c(b)m (y)j(parts)f(in)h(this)0 528 y(term)32 b(giv)m(es)h(that)g Fp(N)789 492 y Fk(2)834 528 y Fl(L)p Fp(Y)983 492 y Fo(N)1058 528 y Fw(\()p Fp(H)8 b Fw(\))33 b(is)g(equal)g(to)52 777 y(\(1)p Fp(=)p Fw(2\))p Fp(N)371 736 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)637 682 y Fg(X)574 897 y Fo(x)p Fm(6)p Fk(=)p Fm(\000)p Fk(1)p Fo(;)p Fk(0)845 777 y Fw(\(\001)967 792 y Fo(N)1043 777 y Fp(H)8 b Fw(\)\()p Fp(x=)-6 b(N)11 b Fw(\)\()p Fp(g)1530 792 y Fo(x)1600 777 y Fl(\000)23 b Fp(\014)1756 792 y Fo(x)1807 777 y Fw(\))49 b(+)h(\(1)p Fp(=)p Fw(2\))p Fp(N)2341 736 y Fk(3)p Fo(=)p Fk(2)2465 777 y Fw(\()p Fp(g)2552 792 y Fm(\000)p Fk(1)2681 777 y Fl(\000)22 b Fp(\014)2836 792 y Fm(\000)p Fk(1)2944 777 y Fw(\))2983 696 y Fg(\010)3041 777 y Fp(H)8 b Fw(\()p 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Fw(=)d Fp(\014)2925 1349 y Fm(\000)p Fk(2)3032 1334 y Fw(,)j Fp(\014)3152 1349 y Fk(0)3230 1334 y Fw(=)d Fp(\014)3396 1349 y Fk(1)3441 1334 y Fw(.)52 b(Recall)36 b(that)g Fp(H)0 1462 y Fw(b)s(elongs)i(to)f Fp(C)566 1414 y Fk(2)p Fo(;)p Fm(\003)559 1491 y Fo(K)713 1462 y Fw(so)g(that)g Fp(pH)1200 1425 y Fm(0)1228 1462 y Fw(\(0)p Fl(\000)p Fw(\))e(=)g Fp(q)t(H)1719 1425 y Fm(0)1747 1462 y Fw(\(0+\).)57 b(On)38 b(the)g(other)f(hand,)j(an)d (elemen)m(tary)g(computation)0 1581 y(sho)m(ws)g(that)e Fp(q)t(\014)611 1596 y Fm(\000)p Fk(1)750 1581 y Fw(=)d Fp(q)t(E)981 1596 y Fo(\027)1021 1606 y Ff(')1076 1581 y Fw([)p Fp(g)t Fw(\()p Fp(\021)t Fw(\()p Fl(\000)p Fw(1\)\)])e(=)i Fp(q)t(p')k Fw(whic)m(h)h(is)e(also)h(equal)f(to)g Fp(p\014)2946 1596 y Fk(0)2991 1581 y Fw(.)52 b(In)36 b(particular,)f(expanding)0 1701 y(the)f(last)e(three)i(terms)f(in)g(the)h(previous)f(form)m(ula,)f (w)m(e)i(obtain)f(that)g(they)g(are)g(equal)g(to)919 1848 y Fl(p)p 1002 1848 91 4 v 89 x Fp(N)11 b(p)1143 1896 y Fm(\000)p Fk(1)1250 1937 y Fw([)p Fp(q)26 b Fw(+)d(\(1)p Fp(=)p Fw(2\)])p Fp(H)1795 1896 y Fm(0)1821 1937 y Fw(\(0+\)\()p Fp(pg)2163 1952 y Fk(0)2229 1937 y Fl(\000)g Fp(q)t(g)2425 1952 y Fm(\000)p Fk(1)2531 1937 y Fw(\))39 b(+)g Fp(O)s Fw(\()p Fp(N)2934 1896 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)3121 1937 y Fw(\))27 b Fp(:)120 2166 y Fw(F)-8 b(or)28 b Fp(x)h Fw(in)g Fi(Z)d Fw(denote)k(b)m(y)f Fp(W)1132 2181 y Fo(x;x)p Fk(+1)1381 2166 y Fw(the)h(instan)m(taneous)g(curren)m(t)g(o)m(v)m(er)f (the)g(b)s(ond)h Fl(f)p Fp(x;)17 b(x)d Fw(+)g(1)p Fl(g)p Fw(,)28 b(i.e.,)h(the)g(rate)0 2285 y(at)f(whic)m(h)i(a)e(particle)h (jumps)g(from)f Fp(x)h Fw(to)f Fp(x)13 b Fw(+)g(1)29 b(min)m(us)g(the)g(rate)f(at)g(whic)m(h)i(a)e(particle)h(jumps)g(from)f Fp(x)13 b Fw(+)g(1)29 b(to)0 2405 y Fp(x)p Fw(.)41 b(In)26 b(particular,)g(for)f(eac)m(h)h Fp(x)h Fl(2)i Fi(Z)p Fw(,)23 b Fl(L)p Fp(\021)t Fw(\()p Fp(x)p Fw(\))k(=)h Fp(W)1866 2420 y Fo(x)p Fm(\000)p Fk(1)p Fo(;x)2093 2405 y Fl(\000)6 b Fp(W)2270 2420 y Fo(x;x)p Fk(+1)2490 2405 y Fw(,)27 b Fp(W)2639 2420 y Fo(x;x)p Fk(+1)2886 2405 y Fw(=)i(\(1)p Fp(=)p Fw(2\)[)p Fp(g)t Fw(\()p Fp(\021)t Fw(\()p Fp(x)p Fw(\)\))6 b Fl(\000)g Fp(g)t Fw(\()p Fp(\021)t Fw(\()p Fp(x)g Fw(+)g(1\)\)])0 2524 y(for)33 b Fp(x)28 b Fl(6)p Fw(=)g Fl(\000)p Fw(1)34 b(and)f Fp(W)790 2539 y Fm(\000)p Fk(1)p Fo(;)p Fk(0)989 2524 y Fw(=)28 b Fp(q)t(g)t Fw(\()p Fp(\021)t Fw(\()p Fl(\000)p Fw(1\)\))20 b Fl(\000)j Fp(pg)t Fw(\()p Fp(\021)t Fw(\(0\)\).)42 b(Therefore)35 b(w)m(e)e(end)h(up)g(with)127 2826 y Fp(M)235 2779 y Fo(N)s(;H)224 2851 y(t)444 2826 y Fw(=)45 b Fp(Y)646 2785 y Fo(N)624 2851 y(t)721 2826 y Fw(\()p Fp(H)8 b Fw(\))38 b Fl(\000)i Fp(Y)1125 2785 y Fo(N)1103 2851 y Fk(0)1200 2826 y Fw(\()p Fp(H)8 b Fw(\))38 b Fl(\000)1524 2691 y Fg(Z)1623 2715 y Fo(t)1579 2917 y Fk(0)1658 2826 y Fw(\(1)p Fp(=)p Fw(2\))p Fp(N)1977 2785 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)2244 2732 y Fg(X)2180 2946 y Fo(x)p Fm(6)p Fk(=)p Fm(\000)p Fk(1)p Fo(;)p Fk(0)2452 2826 y Fw(\(\001)2574 2841 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Fp(\014)3895 3601 y Fo(x)3946 3586 y Fw(,)41 b(for)0 3705 y(all)34 b(sites)g Fp(x)c Fl(6)p Fw(=)g Fl(\000)p Fw(1)p Fp(;)17 b Fw(0,)34 b(and)h(a)g(b)s(oundary)g(term)e Fp(W)1904 3720 y Fm(\000)p Fk(1)p Fo(;)p Fk(0)2109 3705 y Fw(in)m(v)m(olving)g (only)h(the)h(sites)g Fp(x)30 b Fw(=)g(0)p Fp(;)17 b Fl(\000)p Fw(1.)47 b(In)35 b(the)g(next)0 3825 y(lemma)30 b(w)m(e)i(sho)m(w)h(that)f(the)g(b)s(oundary)g(term)f(con)m(v)m(erges)j (to)d(zero)i(and)f(in)g(the)g(prop)s(osition)g(whic)m(h)h(follo)m(ws)0 3944 y(w)m(e)h(state)g(the)g(Boltzmann{Gibbs)g(principle)g(that)g(allo) m(ws)f(to)g(express)i(the)g(homogeneous)f(part)g(in)f(terms)0 4069 y(of)28 b(the)h(densit)m(y)g(\014eld)f Fp(Y)904 4033 y Fo(N)882 4094 y(s)980 4069 y Fw(\()p Fp(H)8 b Fw(\),)28 b(closing)g(the)h(equation)e(of)i(the)f(martingale)f Fp(M)2972 4021 y Fo(N)s(;H)2961 4094 y(t)3165 4069 y Fw(in)h(terms)f(of)i(the)f(densit)m(y)0 4189 y(\014eld.)-331 4366 y Fo(lem)p Fk(2)p Fo(:)p Fk(1)93 b Fy(Lemma)44 b(2.3)h(.)57 b Fd(L)-5 b(et)41 b Fl(f)p Fp(\013)983 4381 y Fo(N)1058 4366 y Fp(;)17 b(N)46 b Fl(\025)37 b Fw(1)p Fl(g)j Fd(b)-5 b(e)40 b(a)g(se)-5 b(quenc)g(e)40 b(of)g(numb)-5 b(ers)40 b(such)h(that)f Fw(lim)17 b(sup)3378 4390 y Fo(N)7 b Fm(!1)3629 4366 y Fp(\013)3693 4381 y Fo(N)3768 4366 y Fp(=)-6 b(N)48 b Fw(=)36 b(0)p Fd(.)0 4485 y(Then,)f(for)g(e)-5 b(ach)35 b Fp(t)28 b Fl(\025)g Fw(0)1108 4669 y(lim)17 b(sup)1146 4752 y Fo(N)7 b Fm(!1)1457 4669 y Fi(E)1518 4684 y Fo(\027)1558 4694 y Ff(')1619 4558 y Fg(h\020)1743 4533 y(Z)1842 4558 y Fo(t)1798 4759 y Fk(0)1922 4669 y Fp(\013)1986 4684 y Fo(N)2061 4669 y Fp(W)2155 4684 y Fm(\000)p Fk(1)p Fo(;)p Fk(0)2325 4669 y Fw(\()p Fp(s)p Fw(\))17 b Fp(ds)2583 4558 y Fg(\021)2641 4579 y Fk(2)2685 4558 y Fg(i)2788 4669 y Fw(=)56 b(0)27 b Fp(:)0 4970 y Fd(Pr)-5 b(o)g(of.)49 b Fw(Let)39 b Fp(V)22 b Fw(\()p Fp(\021)665 4985 y Fo(s)708 4970 y Fw(\))35 b(=)h Fp(\013)959 4985 y Fo(N)1034 4970 y Fp(W)1128 4985 y Fm(\000)p Fk(1)p Fo(;)p Fk(0)1299 4970 y Fw(\()p Fp(s)p Fw(\).)57 b(Since)39 b Fp(\027)1823 4985 y Fo(')1919 4970 y Fw(is)f(rev)m(ersible,)h(b)m(y)f (Prop)s(osition)f(A.1.6.1)f(in)i([KL],)g(there)0 5089 y(exists)33 b(a)g(\014nite)h(constan)m(t)g Fp(C)1086 5104 y Fk(0)1164 5089 y Fw(suc)m(h)g(that)863 5378 y Fi(E)923 5393 y Fo(\027)963 5403 y Ff(')1025 5267 y Fg(h\020)1148 5242 y(Z)1248 5267 y Fo(t)1203 5468 y Fk(0)1299 5378 y Fp(V)23 b Fw(\()p Fp(\021)1468 5393 y Fo(s)1510 5378 y Fw(\))17 b Fp(ds)1665 5267 y Fg(\021)1724 5288 y Fk(2)1769 5267 y Fg(i)1871 5378 y Fl(\024)56 b Fp(C)2075 5393 y Fk(0)2120 5378 y Fp(t)27 b(<)i(V)6 b(;)17 b Fw(\()p Fl(\000)p Fp(N)2605 5336 y Fk(2)2648 5378 y Fl(L)p Fw(\))2756 5336 y Fm(\000)p Fk(1)2890 5378 y Fp(V)50 b(>)1289 5623 y Fw(:=)56 b Fp(C)1521 5638 y Fk(0)1566 5623 y Fp(t)44 b Fw(sup)1697 5707 y Fo(h)1812 5512 y Fg(n)1879 5623 y Fw(2)27 b Fp(<)h(V)6 b(;)17 b(h)27 b(>)56 b Fl(\000)28 b Fp(N)2583 5582 y Fk(2)2656 5623 y Fp(<)g(h;)17 b Fl(L)27 b Fp(h)h(>)3149 5512 y Fg(o)3243 5623 y Fp(:)0 5918 y Fw(In)45 b(this)f(form)m(ula)g Fp(<)j Fl(\001)p Fp(;)17 b Fl(\001)46 b Fp(>)f Fw(stands)g(for)g(the)g(inner)g(pro)s(duct)g(in)g Fp(L)2627 5882 y Fk(2)2671 5918 y Fw(\()p Fp(\027)2759 5933 y Fo(')2817 5918 y Fw(\))f(and)h(the)g(suprem)m(um)f(is)h(tak)m (en)0 6037 y(o)m(v)m(er)34 b(all)f(b)s(ounded)i(cylinder)f(functions)h Fp(h)p Fw(.)46 b(F)-8 b(or)34 b Fp(x)28 b Fl(2)h Fi(Z)p Fw(,)i(denote)k(b)m(y)f Fc(d)2700 6052 y Fo(x)2783 6037 y Fw(the)g(con\014guration)g(with)g(just)g(one)p eop %%Page: 10 10 10 9 bop 0 50 a Fu(10)1081 b(CL)1290 30 y(\023)1279 50 y(AUDIO)38 b(LANDIM,)e(S)2071 30 y(\023)2063 50 y(ER)n(GIO)i(B.)e(V)n (OLCHAN)0 289 y Fw(particle)i(at)g Fp(x)p Fw(.)59 b(Also,)38 b(the)h(summation)d(of)j(t)m(w)m(o)f(con\014gurations)h(is)f(to)g(b)s (e)h(understo)s(o)s(d)g(comp)s(onen)m(t)m(wise.)0 408 y(Fix)32 b(a)g(b)s(ounded)i(cylinder)f(function)h Fp(h)p Fw(.)44 b(Making)32 b(the)h(c)m(hange)h(of)f(v)-6 b(ariables)39 b(~)-57 b Fp(\021)32 b Fw(=)c Fp(\021)d Fl(\000)c Fc(d)3356 423 y Fk(0)3401 408 y Fp(;)c(\021)3499 372 y Fm(0)3554 408 y Fw(=)28 b Fp(\021)d Fl(\000)c Fc(d)3881 423 y Fm(\000)p Fk(1)4020 408 y Fw(w)m(e)0 528 y(obtain)33 b(that)802 563 y Fg(Z)918 699 y Fp(W)1012 714 y Fm(\000)p Fk(1)p Fo(;)p Fk(0)1182 699 y Fp(h)28 b(d\027)1368 714 y Fo(')1481 699 y Fw(=)56 b Fp(pq)t(')1822 563 y Fg(Z)1921 699 y Fw([)p Fp(h)p Fw(\()p Fp(\021)43 b Fw(+)c Fc(d)2303 714 y Fm(\000)p Fk(1)2410 699 y Fw(\))49 b Fl(\000)i Fp(h)p Fw(\()p Fp(\021)43 b Fw(+)c Fc(d)2980 714 y Fk(0)3024 699 y Fw(\)])27 b Fp(d\027)3219 714 y Fo(')3304 699 y Fp(:)597 b Fw(\(2)p Fp(:)p Fw(4\))-4466 b Fo(ibp)0 958 y Fw(The)34 b(same)e(c)m(hange)j(of)e(v)-6 b(ariables)33 b(giv)m(es)g(that)900 1268 y Fp(<)28 b(h;)17 b Fl(\000L)p Fp(h)28 b(>)g Fl(\025)56 b Fp(pq)t(')1756 1132 y Fg(Z)1872 1157 y(h)1919 1268 y Fp(h)p Fw(\()p Fp(\021)27 b Fw(+)22 b Fc(d)2240 1283 y Fm(\000)p Fk(1)2347 1268 y Fw(\))g Fl(\000)h Fp(h)p Fw(\()p Fp(\021)j Fw(+)c Fc(d)2828 1283 y Fk(0)2873 1268 y Fw(\))2912 1157 y Fg(i)2959 1178 y Fk(2)3020 1268 y Fp(d\027)3121 1283 y Fo(')3206 1268 y Fp(:)0 1578 y Fw(Therefore,)342 1838 y(2)p Fp(\013)456 1853 y Fo(N)575 1838 y Fp(<)28 b(W)774 1853 y Fm(\000)p Fk(1)p Fo(;)p Fk(0)945 1838 y Fp(;)17 b(h)27 b(>)56 b Fl(\000)28 b Fp(N)1403 1796 y Fk(2)1475 1838 y Fp(<)h(h;)17 b Fl(L)p Fp(h)27 b(>)369 2048 y Fl(\024)56 b Fp(q)21 b(p)c(')716 1912 y Fg(Z)832 1937 y(n)898 2048 y Fw(2)p Fp(\013)1012 2063 y Fo(N)1087 2048 y Fw([)p Fp(h)p Fw(\()p Fp(\021)26 b Fw(+)d Fc(d)1436 2063 y Fk(0)1480 2048 y Fw(\))f Fl(\000)h Fp(h)p Fw(\()p Fp(\021)j Fw(+)d Fc(d)1962 2063 y Fm(\000)p Fk(1)2068 2048 y Fw(\)])49 b Fl(\000)i Fp(N)2403 2007 y Fk(2)2447 2048 y Fw([)p Fp(h)p Fw(\()p Fp(\021)26 b Fw(+)d Fc(d)2796 2063 y Fm(\000)p Fk(1)2903 2048 y Fw(\))f Fl(\000)g Fp(h)p Fw(\()p Fp(\021)k Fw(+)d Fc(d)3384 2063 y Fk(0)3428 2048 y Fw(\)])3495 2007 y Fk(2)3540 1937 y Fg(o)3634 2048 y Fp(d\027)3735 2063 y Fo(')369 2322 y Fl(\024)56 b Fp(q)21 b(p)c(')727 2255 y(\013)791 2219 y Fk(2)791 2282 y Fo(N)p 727 2299 140 4 v 729 2391 a Fp(N)820 2362 y Fk(2)906 2322 y Fp(:)0 2614 y Fw(This)33 b(concludes)i(the)f(pro)s(of)f(of)h(the)f(lemma.)97 b Fb(\003)120 2824 y Fw(W)-8 b(e)35 b(no)m(w)g(close)g(the)g(equation)g (for)g(the)g(martingale)e Fp(M)2259 2776 y Fo(N)s(;H)2248 2849 y(t)2459 2824 y Fw(in)i(terms)f(of)h(the)g(densit)m(y)h(\014eld)f Fp(Y)3787 2788 y Fo(N)3765 2849 y(t)3863 2824 y Fw(.)49 b(This)0 2944 y(is)42 b(the)g(con)m(ten)m(t)i(of)e(the)g (Boltzmann-Gibbs)g(Principle)g(stated)g(b)s(elo)m(w.)71 b(The)43 b(pro)s(of)f(presen)m(ted)i(in)e([KL],)0 3063 y(Chapter)29 b(11)g(is)g(easily)f(adapted)i(to)e(our)h(con)m(text.)43 b(This)29 b(result)g(w)m(as)h(\014rst)f(pro)m(v)m(ed)h(b)m(y)f(Bro)m(x) f(and)h(Rost)h([BR])0 3183 y(in)j(the)h(con)m(text)f(of)h(homogeneous)f (zero-range)h(pro)s(cesses.)-331 3387 y Fo(pr)r(op)p Fk(2)p Fo(:)p Fk(2)67 b Fy(Prop)s(osition)30 b(2.4)g(.)57 b Fd(F)-8 b(or)30 b(al)5 b(l)29 b(c)-5 b(ontinuous)29 b(functions)f Fp(G)p Fw(\()p Fl(\001)p Fw(\))h Fd(on)g Fi(R)2526 3402 y Fm(\003)2607 3387 y Fd(of)g(c)-5 b(omp)g(act)28 b(supp)-5 b(ort)29 b(and)g(every)g Fp(t)e(>)h Fw(0)p Fd(,)217 3720 y Fw(lim)172 3785 y Fo(N)7 b Fm(!1)434 3720 y Fi(E)495 3735 y Fo(\027)535 3745 y Ff(')597 3610 y Fg(h\020)720 3585 y(Z)819 3609 y Fo(t)775 3811 y Fk(0)871 3720 y Fp(N)962 3679 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)1230 3626 y Fg(X)1166 3840 y Fo(x)p Fm(6)p Fk(=0)p Fo(;)p Fm(\000)p Fk(1)1454 3720 y Fp(G)p Fw(\()p Fp(x=)-6 b(N)11 b Fw(\))1802 3610 y Fg(n)1869 3720 y Fp(g)t Fw(\()p Fp(\021)2009 3735 y Fo(s)2051 3720 y Fw(\()p Fp(x)p Fw(\)\))21 b Fl(\000)h Fp(\014)2401 3735 y Fo(x)2474 3720 y Fl(\000)h Fw(\010)2646 3679 y Fm(0)2674 3720 y Fw(\()p Fp(\032)2765 3735 y Fo(x)2814 3720 y Fw(\)[)p Fp(\021)2930 3735 y Fo(s)2973 3720 y Fw(\()p Fp(x)p Fw(\))e Fl(\000)i Fp(\032)3281 3735 y Fo(x)3330 3720 y Fw(])3358 3610 y Fg(o)3441 3720 y Fp(ds)3540 3610 y Fg(\021)3599 3631 y Fk(2)3644 3610 y Fg(i)3735 3720 y Fw(=)45 b(0)27 b Fp(:)120 4161 y Fw(It)k(follo)m(ws)h(from)f (this)g(prop)s(osition)h(applied)g(to)f Fp(G)d Fw(=)g(\001)2262 4176 y Fo(N)2338 4161 y Fp(H)40 b Fw(and)32 b(Lemma)e(2.3)h(that)g Fp(M)3488 4113 y Fo(N)s(;H)3477 4185 y(t)3685 4161 y Fw(is)h(equal)f(to)182 4494 y Fp(Y)262 4452 y Fo(N)240 4518 y(t)337 4494 y Fw(\()p Fp(H)8 b Fw(\))38 b Fl(\000)i Fp(Y)741 4452 y Fo(N)719 4518 y Fk(0)816 4494 y Fw(\()p Fp(H)8 b Fw(\))38 b Fl(\000)1140 4358 y Fg(Z)1239 4383 y Fo(t)1195 4584 y Fk(0)1307 4494 y Fp(ds)17 b Fw(\(1)p Fp(=)p Fw(2\))p Fp(N)1742 4452 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)2008 4399 y Fg(X)1945 4613 y Fo(x)p Fm(6)p Fk(=)p Fm(\000)p Fk(1)p Fo(;)p Fk(0)2216 4494 y Fw(\(\001)2338 4509 y Fo(N)2414 4494 y Fp(H)8 b Fw(\)\()p Fp(x=)-6 b(N)11 b Fw(\)\010)2886 4452 y Fm(0)2913 4494 y Fw(\()p Fp(\032)3004 4509 y Fo(x)3053 4494 y Fw(\)[)p Fp(\021)3169 4509 y Fo(s)3212 4494 y Fw(\()p Fp(x)p Fw(\))21 b Fl(\000)i Fp(\032)3520 4509 y Fo(x)3569 4494 y Fw(])50 b(+)g Fp(A)3849 4509 y Fo(N)3925 4494 y Fp(;)0 4884 y Fw(where)30 b Fp(A)359 4899 y Fo(N)463 4884 y Fw(is)f(a)g(remainder)f(term)g(that)h(con)m(v)m (erges)h(to)f(0)f(in)h Fp(L)2368 4848 y Fk(2)2413 4884 y Fw(\()p Fi(P)2511 4899 y Fo(\027)2551 4909 y Ff(')2608 4884 y Fw(\).)42 b(Since)30 b Fp(H)37 b Fw(b)s(elongs)29 b(to)g Fp(C)3642 4836 y Fk(2)p Fo(;)p Fm(\003)3635 4914 y Fo(K)3751 4884 y Fw(,)h(w)m(e)f(ma)m(y)0 5004 y(replace)g(the)h (discrete)f(Laplacian)g(b)m(y)g(the)g(con)m(tin)m(uous)i(one.)43 b(In)29 b(this)g(case,)h(since)f(\010)3185 4968 y Fm(0)3213 5004 y Fw(\()p Fp(\032)p Fw(\)\001)p Fp(H)36 b Fw(is)29 b(con)m(tin)m(uous,)0 5123 y(w)m(e)34 b(ma)m(y)e(replace)h(summation)f (o)m(v)m(er)h Fp(x)28 b Fl(6)p Fw(=)g Fl(\000)p Fw(1)p Fp(;)17 b Fw(0)32 b(b)m(y)i(a)f(summation)e(o)m(v)m(er)j Fp(x)27 b Fl(2)h Fi(Z)j Fw(to)i(obtain)g(that)831 5451 y Fp(M)939 5403 y Fo(N)s(;H)928 5476 y(t)1159 5451 y Fw(=)56 b Fp(Y)1372 5410 y Fo(N)1350 5476 y(t)1448 5451 y Fw(\()p Fp(H)8 b Fw(\))38 b Fl(\000)h Fp(Y)1851 5410 y Fo(N)1829 5476 y Fk(0)1927 5451 y Fw(\()p Fp(H)8 b Fw(\))38 b Fl(\000)2250 5316 y Fg(Z)2350 5340 y Fo(t)2305 5542 y Fk(0)2401 5451 y Fp(Y)2481 5410 y Fo(N)2459 5476 y(s)2557 5451 y Fw(\()p Fl(A)p Fp(H)8 b Fw(\))17 b Fp(ds)48 b Fw(+)i Fp(A)3172 5466 y Fo(N)3275 5451 y Fp(:)626 b Fw(\(2)p Fp(:)p Fw(5\))-4466 b Fo(bolz)0 5778 y Fw(The)36 b(expression)f(of)h Fp(A)879 5793 y Fo(N)989 5778 y Fw(c)m(hanged)g (but)g(still)e(con)m(v)m(erges)i(to)e(0)h(in)g Fp(L)2598 5742 y Fk(2)2643 5778 y Fw(\()p Fi(P)2741 5793 y Fo(\027)2781 5803 y Ff(')2838 5778 y Fw(\).)49 b(It)35 b(is)g(not)f(di\016cult)h(to) g(deduce)0 5918 y(from)28 b(the)i(previous)f(iden)m(tit)m(y)g(that)f Fp(M)1474 5870 y Fm(A)p Fo(;H)1463 5942 y(t)1666 5918 y Fw(de\014ned)j(in)e(\(2.1\))f(is)h(a)g(\()p Fp(P)s(;)17 b Fl(F)2749 5933 y Fo(t)2782 5918 y Fw(\)-martingale)27 b(for)i(eac)m(h)h Fp(H)37 b Fw(in)29 b Fc(H)4020 5933 y Fo(k)4062 5943 y Fj(0)4106 5918 y Fw(.)0 6037 y(This)k(is)h(the)f (con)m(ten)m(t)h(of)g(the)g(next)f(prop)s(osition.)p eop %%Page: 11 11 11 10 bop 1429 50 a Fu(EQUILIBRIUM)38 b(FLUCTUA)-7 b(TIONS)1310 b(11)-331 289 y Fo(pr)r(op)p Fk(2)p Fo(:)p Fk(4)67 b Fy(Prop)s(osition)38 b(2.5)h(.)57 b Fd(F)-8 b(or)36 b(e)-5 b(ach)35 b Fp(H)44 b Fd(in)35 b Fc(H)1624 304 y Fo(k)1666 314 y Fj(0)1710 289 y Fd(,)g Fp(M)1884 241 y Fm(A)p Fo(;H)1873 313 y(t)2082 289 y Fd(is)g(a)h Fw(\()p Fp(P)s(;)17 b Fl(F)2499 304 y Fo(t)2532 289 y Fw(\))p Fd(-martingale.)0 470 y(Pr)-5 b(o)g(of.)49 b Fw(Fix)25 b(\014rst)h Fp(H)34 b Fw(in)26 b Fp(C)986 422 y Fk(2)p Fo(;)p Fm(\003)979 499 y Fo(K)1103 470 y Fl(\\)7 b Fc(H)1248 485 y Fo(k)1290 495 y Fj(0)1334 470 y Fw(.)42 b(By)25 b(de\014nition,)i Fp(M)2125 422 y Fm(A)p Fo(;H)2114 494 y(t)2314 470 y Fw(is)e(measurable)h(with)f(resp)s(ect)i(to)f(the)g(\014ltration)0 604 y Fl(F)72 619 y Fo(t)107 604 y Fw(.)44 b(In)33 b(order)h(to)f(pro)m (v)m(e)g(that)g Fp(M)1278 556 y Fm(A)p Fo(;H)1267 629 y(t)1474 604 y Fw(is)g(a)g(martingale)f(it)g(su\016ces)j(to)e(c)m(hec)m (k)h(that)1400 830 y Fp(E)1474 845 y Fo(P)1540 719 y Fg(h)1587 830 y Fp(M)1695 782 y Fm(A)p Fo(;H)1684 854 y(t)1858 830 y Fp(U)1937 719 y Fg(i)2028 830 y Fw(=)45 b Fp(E)2224 845 y Fo(P)2290 719 y Fg(h)2337 830 y Fp(M)2445 789 y Fm(A)p Fo(;H)2434 854 y(s)2608 830 y Fp(U)2687 719 y Fg(i)0 1050 y Fw(for)36 b(all)g(0)c Fl(\024)h Fp(s)f Fl(\024)h Fp(t)f Fl(\024)h Fp(T)51 b Fw(and)36 b(all)f Fp(U)44 b Fw(=)33 b Fy(1)1581 1068 y Fm(f)p Fo(Y)1669 1078 y Ff(s)1702 1094 y(i)1739 1068 y Fk(\()p Fo(H)1835 1078 y Ff(i)1867 1068 y Fk(\))p Fm(2)p Fo(A)2012 1078 y Ff(i)2043 1068 y Fo(;)13 b Fk(1)p Fm(\024)p Fo(i)p Fm(\024)p Fo(n)p Fm(g)2368 1050 y Fw(,)36 b(where)h Fp(n)f Fw(is)g(a)g(p)s(ositiv)m(e)f(in)m(teger,)i(0)c Fl(\024)g Fp(s)3980 1065 y Fk(1)4056 1050 y Fl(\024)0 1197 y(\001)17 b(\001)g(\001)29 b(\024)j Fp(s)303 1212 y Fo(n)388 1197 y Fl(\024)g Fp(s)j Fw(and,)h(for)g(eac)m(h)g(1)31 b Fl(\024)h Fp(i)g Fl(\024)f Fp(n)p Fw(,)36 b Fp(H)1755 1212 y Fo(i)1823 1197 y Fw(b)s(elongs)g(to)f Fp(C)2385 1149 y Fk(2)p Fo(;)p Fm(\003)2378 1226 y Fo(K)2529 1197 y Fw(and)h 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Ff(')864 1746 y Fg(h)911 1856 y Fp(M)1019 1808 y Fo(N)s(;H)1008 1881 y(t)1201 1856 y Fp(U)1280 1746 y Fg(i)1382 1856 y Fw(=)101 b(lim)1515 1921 y Fo(N)7 b Fm(!1)1761 1856 y Fp(E)1835 1871 y Fo(P)1886 1881 y Ff(N)1954 1746 y Fg(hn)2067 1856 y Fp(Y)2125 1871 y Fo(t)2160 1856 y Fw(\()p Fp(H)h Fw(\))22 b Fl(\000)h Fp(Y)2509 1871 y Fk(0)2553 1856 y Fw(\()p Fp(H)8 b Fw(\))22 b Fl(\000)2843 1721 y Fg(Z)2943 1745 y Fo(t)2899 1947 y Fk(0)2995 1856 y Fp(ds)17 b(Y)3169 1871 y Fo(s)3210 1856 y Fw(\()p Fl(A)p Fp(H)8 b Fw(\))3459 1746 y Fg(o)3524 1856 y Fp(U)3603 1746 y Fg(i)3650 1856 y Fp(:)0 2103 y Fw(A)36 b(simple)f(computation)g(sho)m(ws)i(that)f(the)h(term)e (inside)i(braces)g(is)f(square)g(in)m(tegrable,)h(uniformly)e(in)h Fp(N)11 b Fw(.)0 2222 y(In)45 b(particular,)i(since)e Fp(U)56 b Fw(is)44 b(b)s(ounded)j(and)e Fp(P)1816 2237 y Fo(N)1936 2222 y Fw(con)m(v)m(erges)h(w)m(eakly)d(to)h Fp(P)59 b Fw(and)45 b(since,)j(b)m(y)d(Prop)s(osition)0 2347 y(4.1)37 b(b)s(elo)m(w,)i Fp(P)51 b Fw(is)38 b(concen)m(trated)h (on)f(con)m(tin)m(uous)h(paths,)h Fp(E)2321 2362 y Fo(P)2372 2372 y Ff(N)2440 2347 y Fw([)p Fp(M)2576 2299 y Fo(N)s(;H)2565 2372 y(t)2740 2347 y Fp(U)11 b Fw(])38 b(con)m(v)m(erges)h(to)e Fp(E)3536 2362 y Fo(P)3602 2347 y Fw([)p Fp(M)3738 2299 y Fm(A)p Fo(;H)3727 2372 y(t)3900 2347 y Fp(U)11 b Fw(])38 b(as)0 2478 y Fp(N)h Fl(")27 b(1)p Fw(.)43 b(Since)31 b(this)f(statemen)m(t)f(remains)h(in)g(force)g(with)g Fp(s)g Fw(in)g(place)g(of)g Fp(t)p Fw(,)h Fp(M)2971 2430 y Fm(A)p Fo(;H)2960 2502 y(t)3163 2478 y Fw(is)f(a)g(\()p Fp(P)s(;)17 b Fl(F)3563 2493 y Fo(t)3597 2478 y Fw(\)-martingale)0 2605 y(for)38 b(eac)m(h)h Fp(H)44 b Fl(2)36 b Fp(C)693 2557 y Fk(2)p Fo(;)p Fm(\003)686 2635 y Fo(K)828 2605 y Fl(\\)26 b Fc(H)992 2620 y Fo(k)1034 2630 y Fj(0)1078 2605 y Fw(.)58 b(Fix)38 b(no)m(w)g Fp(H)46 b Fw(in)38 b Fc(H)1882 2620 y Fo(k)1924 2630 y Fj(0)1968 2605 y Fw(.)59 b(With)38 b(a)g(sequence)h Fp(H)2905 2620 y Fo(n)2997 2605 y Fw(of)g(functions)g(in)f Fp(C)3749 2557 y Fk(2)p Fo(;)p Fm(\003)3742 2635 y Fo(K)3884 2605 y Fl(\\)26 b Fc(H)4048 2620 y Fo(k)4090 2630 y Fj(0)0 2740 y Fw(suc)m(h)37 b(that)e Fp(H)528 2755 y Fo(n)582 2740 y Fw(,)g(\010)717 2704 y Fm(0)745 2740 y Fw(\()p Fp(\032)p Fw(\)\001)p Fp(H)1041 2755 y Fo(n)1130 2740 y Fw(con)m(v)m(erges)h(to)f Fp(H)8 b Fw(,)36 b(\010)1929 2704 y Fm(0)1957 2740 y Fw(\()p Fp(\032)p Fw(\)\001)p Fp(H)42 b Fw(in)35 b Fp(L)2481 2704 y Fk(2)2526 2740 y Fw(\()p Fp(P)14 b Fw(\),)35 b(it)f(is)i(easy)f (to)g(sho)m(w)h(that)f Fp(M)3869 2692 y Fm(A)p Fo(;H)3858 2765 y(t)4067 2740 y Fw(is)0 2860 y(a)e(\()p Fp(P)s(;)17 b Fl(F)306 2875 y Fo(t)339 2860 y Fw(\)-martingale.)43 b(This)33 b(concludes)i(the)e(pro)s(of)h(of)f(the)h(lemma.)97 b Fb(\003)120 3040 y Fw(W)-8 b(e)34 b(no)m(w)g(turn)g(to)f(the)h (martingale)e Fp(K)1607 2992 y Fo(N)s(;H)1600 3065 y(t)1772 3040 y Fw(.)45 b(An)34 b(elemen)m(tary)f(computation)f(giv)m(es)i(that) f Fp(N)3652 3004 y Fk(2)3697 3040 y Fl(L)p Fp(Y)3845 3004 y Fo(N)3921 3040 y Fw(\()p Fp(H)8 b Fw(\))4090 3004 y Fk(2)0 3160 y Fl(\000)p Fw(2)p Fp(N)218 3124 y Fk(2)263 3160 y Fp(Y)343 3124 y Fo(N)418 3160 y Fw(\()p Fp(H)g Fw(\))17 b Fl(L)p Fp(Y)752 3124 y Fo(N)828 3160 y Fw(\()p Fp(H)8 b Fw(\))32 b(is)h(equal)g(to)745 3376 y(\(1)p Fp(=)p Fw(2\))p Fp(N)1064 3334 y Fm(\000)p Fk(1)1251 3281 y Fg(X)1187 3495 y Fo(x)p Fm(6)p Fk(=0)p Fo(;)p Fm(\000)p Fk(1)1475 3376 y Fp(g)t Fw(\()p Fp(\021)t Fw(\()p Fp(x)p Fw(\)\)[\()p Fl(r)1943 3391 y Fo(N)2016 3376 y Fp(H)8 b Fw(\)\()p Fp(x=)-6 b(N)11 b Fw(\)])2444 3334 y Fk(2)967 3645 y Fw(+)50 b Fp(N)1185 3604 y Fm(\000)p Fk(1)1292 3645 y Fp(pg)1390 3660 y Fm(\000)p Fk(1)1497 3645 y Fw([\()p Fl(r)1647 3660 y Fo(N)1722 3645 y Fp(H)8 b Fw(\)\()p Fl(\000)p Fw(1)p Fp(=)-6 b(N)11 b Fw(\)])2220 3604 y Fk(2)2313 3645 y Fw(+)51 b Fp(N)2532 3604 y Fm(\000)p Fk(1)2639 3645 y Fp(q)t(g)2735 3660 y Fk(0)2779 3645 y Fw([\()p Fl(r)2929 3660 y Fo(N)3004 3645 y Fp(H)8 b Fw(\)\(0\)])3290 3604 y Fk(2)3361 3645 y Fp(:)-331 3835 y Fo(lem)p Fk(2)p Fo(:)p Fk(2)93 b Fy(Lemma)38 b(2.6)h(.)57 b Fd(F)-8 b(or)36 b(every)f(c)-5 b(ontinuous)36 b(function)f Fp(G)h Fd(on)g Fi(R)2346 3850 y Fm(\003)2433 3835 y Fd(of)f(c)-5 b(omp)g(act)36 b(supp)-5 b(ort)796 4095 y Fw(lim)750 4159 y Fo(N)7 b Fm(!1)1013 4095 y Fi(E)1073 4110 y Fo(\027)1113 4120 y Ff(')1175 3984 y Fg(h\020)1298 3959 y(Z)1398 3984 y Fo(t)1354 4186 y Fk(0)1450 4095 y Fp(N)1541 4054 y Fm(\000)p Fk(1)1728 4000 y Fg(X)1664 4215 y Fo(x)p Fm(6)p Fk(=0)p Fo(;)p Fm(\000)p Fk(1)1952 4095 y Fp(G)p Fw(\()p Fp(x=)-6 b(N)11 b Fw(\)[)p Fp(g)t Fw(\()p Fp(\021)2468 4110 y Fo(s)2510 4095 y Fw(\()p Fp(x)p Fw(\)\))21 b Fl(\000)i Fp(\014)2861 4110 y Fo(x)2911 4095 y Fw(])2939 3984 y Fg(\021)2998 4005 y Fk(2)3043 3984 y Fg(i)3145 4095 y Fw(=)56 b(0)28 b Fp(:)120 4394 y Fw(The)42 b(pro)s(of)g(is)g(elemen)m (tary)g(b)s(ecause)h Fp(\027)1656 4409 y Fo(')1756 4394 y Fw(is)f(a)g(pro)s(duct)g(stationary)f(measure.)71 b(It)42 b(follo)m(ws)f(from)h(this)0 4513 y(result)34 b(applied)f(to)g Fp(G)28 b Fw(=)g Fl(r)1042 4528 y Fo(N)1118 4513 y Fp(H)41 b Fw(and)34 b(from)e(the)i(previous)f(computation)f(that)733 4754 y Fp(K)825 4706 y Fo(N)s(;H)818 4779 y(t)1035 4754 y Fw(=)44 b(\()p Fp(M)1303 4706 y Fo(N)s(;H)1292 4779 y(t)1468 4754 y Fw(\))1507 4713 y Fk(2)1590 4754 y Fl(\000)1771 4687 y Fp(t)p 1719 4731 141 4 v 1719 4822 a Fw(2)p Fp(N)1952 4659 y Fg(X)1888 4874 y Fo(x)p Fm(6)p Fk(=)p Fm(\000)p Fk(1)p Fo(;)p Fk(0)2176 4643 y Fg(h)2223 4754 y Fw(\()p Fl(r)2345 4769 y Fo(N)2420 4754 y Fp(H)8 b Fw(\)\()p Fp(x=)-6 b(N)11 b Fw(\))2820 4643 y Fg(i)2867 4664 y Fk(2)2912 4754 y Fp(\014)2968 4769 y Fo(x)3068 4754 y Fw(+)3221 4729 y(~)3195 4754 y Fp(A)3270 4769 y Fo(N)3373 4754 y Fp(;)0 5075 y Fw(where)318 5050 y(~)292 5075 y Fp(A)367 5090 y Fo(N)479 5075 y Fw(con)m(v)m(erges)39 b(to)d(0)h(in)g Fp(L)1330 5039 y Fk(2)1375 5075 y Fw(\()p Fi(P)1473 5090 y Fo(\027)1513 5100 y Ff(')1570 5075 y Fw(\).)55 b(Here)38 b(again)e(w)m(e)i(ma)m(y)e(replace)h(the)h (discrete)g(gradien)m(t)f(b)m(y)g(the)0 5195 y(con)m(tin)m(uous)45 b(one)f(and)g(extend)g(the)g(summation)e(to)h Fi(Z)p Fw(.)73 b(Rep)s(eating)44 b(the)g(argumen)m(ts)f(presen)m(ted)j(in)d (the)0 5320 y(pro)s(of)33 b(of)h(Prop)s(osition)e(2.5,)g(w)m(e)i(sho)m (w)g(that)f Fp(K)1800 5272 y Fm(A)p Fo(;H)1793 5344 y(t)1996 5320 y Fw(is)g(a)g(\()p Fp(P)s(;)17 b Fl(F)2402 5335 y Fo(t)2436 5320 y Fw(\)-martingale)31 b(for)i(eac)m(h)i Fp(H)g Fl(2)28 b Fc(H)3667 5335 y Fo(k)3709 5345 y Fj(0)3753 5320 y Fw(.)0 5490 y Fd(Pr)-5 b(o)g(of)36 b(of)h(The)-5 b(or)g(em)37 b(1.1.)50 b Fw(By)34 b(Prop)s(osition)h(4.1)f(the)h (sequence)i Fp(P)2528 5505 y Fo(N)2639 5490 y Fw(is)e(tigh)m(t)f(and)i (all)e(its)h(limit)d(p)s(oin)m(ts)k(are)0 5609 y(concen)m(trated)h(on)e (con)m(tin)m(uous)i(paths.)51 b(Fix)35 b(a)g(limit)e(p)s(oin)m(t)j Fp(P)49 b Fw(of)36 b(the)f(sequence)i Fp(P)3200 5624 y Fo(N)3276 5609 y Fw(.)50 b(W)-8 b(e)36 b(pro)m(v)m(ed)g(in)g(this)0 5729 y(section)h(that)e Fp(P)50 b Fw(solv)m(es)37 b(the)f(martingale)f (problem)h(\(2.1\),)f(\(2.2\).)52 b(The)36 b(pro)s(of)h(of)f(Theorem)g (\(1.4\))f(in)h([HS])0 5848 y(can)j(b)s(e)g(adapted)g(to)f(sho)m(w)h (that)f(there)h(exists)f(at)g(most)f(one)i(solution)f(of)h(\(2.1\),)f (\(2.2\).)59 b(This)38 b(concludes)0 5968 y(the)c(pro)s(of)f(of)g(the)h (theorem.)99 b Fb(\003)p eop %%Page: 12 12 12 11 bop 0 50 a Fu(12)1081 b(CL)1290 30 y(\023)1279 50 y(AUDIO)38 b(LANDIM,)e(S)2071 30 y(\023)2063 50 y(ER)n(GIO)i(B.)e(V) n(OLCHAN)0 289 y Fy(3.)51 b(Fluctuation)36 b(of)i(the)f(asymmetric)i (tagged)e(particle.)120 408 y Fw(In)c(this)g(section)g(w)m(e)h(use)f (the)h(results)f(pro)m(v)m(en)h(for)f(the)h(zero-range)g(mo)s(del)d(to) i(deduce)i(a)d(Cen)m(tral)h(Limit)0 528 y(Theorem)27 b(for)h(the)g(asymmetric)e(tagged)h(particle)h(in)g(the)g (nearest-neigh)m(b)s(or)h(symmetric)d(simple)h(exclusion)0 648 y(pro)s(cess.)45 b(W)-8 b(e)32 b(follo)m(w)g(the)h(approac)m(h)g (presen)m(ted)h(b)m(y)e([R)-11 b(V])32 b(for)g(the)h(case)g(of)f(a)g (symmetric)e(tagged)i(particle.)120 767 y(Recall)c(from)g(section)i(2)e (that)g(\()p Fp(X)1386 782 y Fo(t)1421 767 y Fp(;)17 b(\030)1510 782 y Fo(t)1544 767 y Fw(\))28 b(is)h(a)f(one-dimensional)h (nearest)h(neigh)m(b)s(or)f(symmetric)e(exclusion)0 887 y(pro)s(cess)g(with)f(an)h(asymmetric)d(tagged)i(particle)g(that)f (sits)i(at)e(the)i(origin)e(at)h Fp(t)i Fw(=)g(0.)41 b(W)-8 b(e)27 b(de\014ned)h(in)e(section)0 1006 y(1)33 b(a)f(corresp)s(ondence)j(b)s(et)m(w)m(een)f(this)f(exclusion)g(pro)s (cess)h(and)f(a)g(symmetric)d(zero)j(range)g(pro)s(cess)h(with)f(an)0 1126 y(asymmetry)h(at)j(the)g(b)s(ond)h Fl(f\000)p Fw(1)p Fp(;)17 b Fw(0)p Fl(g)p Fw(.)54 b(In)37 b(this)g(corresp)s(ondence)j (the)d(p)s(osition)f(of)h(the)h(tagged)f(particle)f(at)0 1245 y(time)i Fp(t)h Fw(is)g(giv)m(en)h(b)m(y)f(the)h(total)e(n)m(um)m (b)s(er)i(of)f(jumps)h(from)f(0)g(to)g Fl(\000)p Fw(1)g(min)m(us)g(the) h(total)e(n)m(um)m(b)s(er)i(of)f(jumps)0 1365 y(from)33 b Fl(\000)p Fw(1)g(to)g(0)g(in)g(the)h(time)e(in)m(terv)-6 b(al)33 b([0)p Fp(;)17 b(t)p Fw(])31 b(.)120 1484 y(Fix)37 b(0)g Fl(\024)g Fp(\013)f Fl(\024)h Fw(1)i(and)g(recall)f(the)h (de\014nition)g(of)g(the)g(pro)s(duct)g(measure)g Fp(\026)3031 1499 y Fo(\013)3126 1484 y Fw(giv)m(en)f(in)h(section)g(1.)60 b(Let)0 1604 y Fp(')30 b Fw(=)h Fp(p)253 1568 y Fm(\000)p Fk(1)360 1604 y Fw(\(1)23 b Fl(\000)h Fp(\013)p Fw(\).)48 b(Since)36 b Fp(\021)1063 1619 y Fo(t)1129 1604 y Fw(=)30 b Fp(\021)1289 1568 y Fo(N)1285 1629 y(t)1400 1604 y Fw(is)k(the)i(zero)f(range)g(pro)s(cess)h(with)e(generator)h Fp(N)3273 1568 y Fk(2)3318 1604 y Fl(L)p Fw(,)f(it)g(is)h(accelerated)0 1724 y(b)m(y)i Fp(N)233 1687 y Fk(2)278 1724 y Fw(,)g(in)g(con)m(trast) g(with)g(the)g(pro)s(cess)h(\()p Fp(X)1731 1739 y Fo(t)1766 1724 y Fp(;)17 b(\030)1855 1739 y Fo(t)1889 1724 y Fw(\))36 b(whic)m(h)i(is)f(not.)55 b(Denote)37 b(b)m(y)g Fp(S)3138 1687 y Fo(N)3132 1748 y Fk(+)3214 1724 y Fw(\()p Fp(t)p Fw(\))f(\(resp.)56 b Fp(S)3731 1687 y Fo(N)3725 1748 y Fm(\000)3806 1724 y Fw(\()p Fp(t)p Fw(\)\))36 b(the)0 1843 y(total)31 b(n)m(um)m(b)s(er)i(of)g(jumps)g(from)e Fl(\000)p Fw(1)i(to)f(0)g(\(resp.)44 b(0)33 b(to)f Fl(\000)p Fw(1\))g(in)g(the)h(time)e(in)m(terv)-6 b(al)32 b([0)p Fp(;)17 b(t)p Fw(])31 b(for)i(the)g(pro)s(cess)g Fp(\021)t Fw(.)0 1963 y(The)h(p)s(osition)e(of)i(the)g(tagged)f(particle)g(is)g (suc)m(h)h(that)1487 2191 y Fp(X)1570 2209 y Fo(tN)1670 2189 y Fj(2)1770 2191 y Fw(=)55 b Fp(S)1969 2150 y Fo(N)1963 2216 y Fm(\000)2045 2191 y Fw(\()p Fp(t)p Fw(\))49 b Fl(\000)h Fp(S)2402 2150 y Fo(N)2396 2216 y Fk(+)2478 2191 y Fw(\()p Fp(t)p Fw(\))27 b Fp(:)120 2422 y Fw(F)-8 b(or)41 b Fp(n)h Fl(\025)h Fw(1,)h(let)e Fp(G)882 2437 y Fo(n)936 2422 y Fw(\()p Fp(u)p Fw(\))59 b(=)h([1)27 b Fl(\000)i Fw(\()p Fp(u=n)p Fw(\)])1751 2386 y Fk(+)1832 2422 y Fy(1)1889 2440 y Fm(f)p Fo(u)p Fm(\025)p Fk(0)p Fm(g)2125 2422 y Fw(.)71 b(The)42 b(next)g(result)h(giv)m(es)f(a)g (precise)h(meaning)e(to)0 2542 y(iden)m(tit)m(y)33 b(\(1.1\).)-331 2719 y Fo(lem)p Fk(3)p Fo(:)p Fk(1)93 b Fy(Lemma)38 b(3.1)h(.)57 b Fd(Ther)-5 b(e)35 b(exists)g(a)h(\014nite)f(c)-5 b(onstant)35 b Fp(c)h Fd(such)f(that)687 2993 y Fw(sup)675 3077 y Fo(N)7 b Fm(\025)p Fk(1)865 2993 y Fi(E)925 3008 y Fo(\027)965 3018 y Ff(')1027 2882 y Fg(h\020)1145 2925 y Fp(X)1228 2943 y Fo(tN)1328 2923 y Fj(2)p 1145 2970 228 4 v 1172 2990 a Fl(p)p 1255 2990 91 4 v 84 x Fp(N)1423 2993 y Fl(\000)1613 2925 y Fw(1)p 1551 2970 174 4 v 1551 2990 a Fl(p)p 1634 2990 91 4 v 84 x Fp(N)1755 2898 y Fg(X)1754 3112 y Fo(x)p Fm(\025)p Fk(0)1917 2993 y Fp(G)1995 3008 y Fo(n)2050 2993 y Fw(\()p Fp(x=)-6 b(N)11 b Fw(\))17 b([)p Fp(\021)2414 3008 y Fk(0)2458 2993 y Fw(\()p Fp(x)p Fw(\))49 b Fl(\000)h Fp(\021)2818 3008 y Fo(t)2854 2993 y Fw(\()p Fp(x)p Fw(\)])3017 2882 y Fg(\021)3076 2903 y Fk(2)3120 2882 y Fg(i)3223 2993 y Fl(\024)3367 2925 y Fp(ct)p 3367 2970 80 4 v 3377 3061 a(n)0 3328 y Fd(for)35 b(al)5 b(l)37 b Fp(n)27 b Fl(\025)h Fw(1)36 b Fd(and)f Fp(t)28 b Fl(2)g Fw([0)p Fp(;)17 b(T)d Fw(])p Fd(.)0 3505 y(Pr)-5 b(o)g(of.)49 b Fw(First)33 b(observ)m(e)h(that)666 3797 y Fp(S)733 3756 y Fo(N)727 3822 y Fk(+)808 3797 y Fw(\()p Fp(t)p Fw(\))39 b Fl(\000)g Fp(q)t(N)1216 3756 y Fk(2)1277 3662 y Fg(Z)1377 3686 y Fo(t)1332 3888 y Fk(0)1428 3797 y Fp(g)t Fw(\()p Fp(\021)1568 3812 y Fo(s)1610 3797 y Fw(\()p Fl(\000)p Fw(1\)\))17 b Fp(ds)27 b(;)215 b(S)2307 3756 y Fo(N)2301 3822 y Fm(\000)2383 3797 y Fw(\()p Fp(t)p Fw(\))38 b Fl(\000)h Fp(pN)2792 3756 y Fk(2)2853 3662 y Fg(Z)2953 3686 y Fo(t)2909 3888 y Fk(0)3005 3797 y Fp(g)t Fw(\()p Fp(\021)3145 3812 y Fo(s)3187 3797 y Fw(\(0\)\))17 b Fp(ds)0 4105 y Fw(are)48 b Fi(P)240 4120 y Fo(\027)280 4130 y Ff(')337 4105 y Fw(-martingales.)86 b(Hence,)51 b(so)d(is)f Fp(M)1679 4120 y Fo(t)1765 4105 y Fw(=)52 b Fp(X)1977 4123 y Fo(tN)2077 4103 y Fj(2)2153 4105 y Fw(+)32 b Fp(N)2353 4069 y Fk(2)2414 4025 y Fg(R)2481 4050 y Fo(t)2461 4141 y Fk(0)2532 4105 y Fp(W)2626 4120 y Fm(\000)p Fk(1)p Fo(;)p Fk(0)2797 4105 y Fw(\()p Fp(s)p Fw(\))17 b Fp(ds)p Fw(.)85 b(F)-8 b(or)47 b Fp(n)k Fl(\025)h Fw(1,)f(let)c Fp(Y)3951 4069 y Fo(n)3929 4130 y(t)4056 4105 y Fw(=)0 4151 y Fg(P)105 4255 y Fo(x)p Fm(\025)p Fk(0)274 4226 y Fp(G)352 4241 y Fo(n)406 4226 y Fw(\()p Fp(x=)-6 b(N)11 b Fw(\))17 b(\()p Fp(\021)781 4241 y Fo(t)816 4226 y Fw(\()p Fp(x)p Fw(\))22 b Fl(\000)g Fp(\032)1124 4241 y Fo(x)1174 4226 y Fw(\))33 b(and)h Fp(V)1520 4189 y Fo(n)1498 4250 y(t)1602 4226 y Fw(=)28 b Fp(Y)1787 4189 y Fo(n)1765 4250 y Fk(0)1863 4226 y Fl(\000)23 b Fp(Y)2043 4189 y Fo(n)2021 4250 y(t)2097 4226 y Fw(.)44 b(Then)1088 4541 y Fp(M)1196 4500 y Fo(n)1185 4566 y(t)1305 4541 y Fw(=)56 b Fp(V)1518 4500 y Fo(n)1496 4566 y(t)1622 4541 y Fw(+)1749 4405 y Fg(Z)1849 4430 y Fo(t)1805 4632 y Fk(0)1901 4541 y Fp(N)1992 4500 y Fk(2)2055 4446 y Fg(X)2053 4660 y Fo(x)p Fm(\025)p Fk(0)2217 4541 y Fp(G)2295 4556 y Fo(n)2349 4541 y Fw(\()p Fp(x=)-6 b(N)11 b Fw(\))17 b Fl(L)p Fp(\021)2754 4556 y Fo(s)2796 4541 y Fw(\()p Fp(x)p Fw(\))g Fp(ds)0 4871 y Fw(is)26 b(a)g Fi(P)228 4886 y Fo(\027)268 4896 y Ff(')325 4871 y Fw(-martingale.)40 b(A)26 b(straigh)m(tforw)m(ard)g(computation)f(sho)m(ws)i(that)e(the)i (in)m(tegral)e(term)g(of)h(the)g(previous)0 4991 y(expression)34 b(is)f(equal)g(to)1058 5311 y Fp(N)1149 5270 y Fk(2)1210 5175 y Fg(Z)1310 5200 y Fo(t)1265 5402 y Fk(0)1361 5200 y Fg(n)1428 5311 y Fp(W)1522 5326 y Fm(\000)p Fk(1)p Fo(;)p Fk(0)1692 5311 y Fw(\()p Fp(s)p Fw(\))38 b(+)2034 5244 y(1)p 1983 5288 151 4 v 1983 5379 a Fp(nN)2163 5187 y Fo(nN)7 b Fm(\000)p Fk(1)2202 5216 y Fg(X)2200 5427 y Fo(x)p Fk(=0)2401 5311 y Fp(W)2495 5326 y Fo(x;x)p Fk(+1)2715 5311 y Fw(\()p Fp(s)p Fw(\))2840 5200 y Fg(o)2922 5311 y Fp(ds)27 b(:)0 5625 y Fw(Th)m(us,)34 b Fp(X)363 5643 y Fo(tN)463 5623 y Fj(2)529 5625 y Fl(\000)23 b Fp(V)709 5589 y Fo(n)687 5650 y(t)797 5625 y Fw(is)33 b(equal)g(to)1170 5951 y Fp(M)1267 5966 y Fo(t)1351 5951 y Fl(\000)51 b Fp(M)1587 5910 y Fo(n)1576 5976 y(t)1690 5951 y Fw(+)1830 5884 y Fp(N)p 1830 5928 91 4 v 1846 6019 a(n)1949 5815 y Fg(Z)2049 5840 y Fo(t)2004 6042 y Fk(0)2117 5826 y Fo(nN)7 b Fm(\000)p Fk(1)2156 5856 y Fg(X)2155 6067 y Fo(x)p Fk(=0)2355 5951 y Fp(W)2449 5966 y Fo(x;x)p Fk(+1)2669 5951 y Fw(\()p Fp(s)p Fw(\))17 b Fp(ds)26 b(:)p eop %%Page: 13 13 13 12 bop 1429 50 a Fu(EQUILIBRIUM)38 b(FLUCTUA)-7 b(TIONS)1310 b(13)0 289 y Fw(W)-8 b(e)36 b(claim)e(that)i(this)g(expression)h(con)m (v)m(erges)g(to)e(0)h(in)g Fp(L)2187 253 y Fk(2)2231 289 y Fw(\()p Fi(P)2329 304 y Fo(\027)2369 314 y Ff(')2426 289 y Fw(\))g(as)g Fp(n)c Fl(")f(1)p Fw(.)52 b(Notice)35 b(that)h(the)g(expression)0 408 y(app)s(earing)g(in)g(the)g(statemen)m (t)f(of)h(Lemma)f(2.3)g(is)g(just)i(the)f(curren)m(t)h Fp(W)2780 423 y Fm(\000)p Fk(1)p Fo(;)p Fk(0)2951 408 y Fw(.)52 b(Rep)s(eating)36 b(the)g(argumen)m(ts)0 528 y(presen)m(ted)f(there,)f(w)m(e)g(deduce)h(that)1045 811 y Fi(E)1106 826 y Fo(\027)1146 836 y Ff(')1207 700 y Fg(h\020)1330 675 y(Z)1430 700 y Fo(t)1386 902 y Fk(0)1494 744 y Fp(N)p 1494 788 91 4 v 1510 879 a(n)1613 687 y Fo(nN)7 b Fm(\000)p Fk(1)1652 716 y Fg(X)1651 927 y Fo(x)p Fk(=0)1852 811 y Fp(W)1946 826 y Fo(x;x)p Fk(+1)2166 811 y Fw(\()p Fp(s)p Fw(\))17 b Fp(ds)2407 700 y Fg(\021)2464 721 y Fk(2)2509 700 y Fg(i)2573 811 y Fp(ds)55 b Fl(\024)2871 744 y Fp(C)7 b(tN)p 2871 788 206 4 v 2944 879 a(n)0 1088 y Fw(for)33 b(some)g(\014nite)h(constan)m(t)g Fp(C)7 b Fw(.)120 1208 y(F)-8 b(or)42 b(eac)m(h)j Fp(x)e Fw(denote)h(b)m(y)f Fp(J)1187 1160 y Fo(x;x)p Fk(+1)1177 1232 y Fo(t)1450 1208 y Fw(the)h(total)e(n)m(um)m(b)s(er)i(of)g(jumps)f(from)g Fp(x)g Fw(to)g Fp(x)29 b Fw(+)g(1)43 b(min)m(us)g(the)h(total)0 1335 y(n)m(um)m(b)s(er)39 b(of)f(jumps)h(from)e Fp(x)26 b Fw(+)g(1)38 b(to)f Fp(x)h Fw(in)h(the)f(time)f(in)m(terv)-6 b(al)38 b([0)p Fp(;)17 b(t)p Fw(])37 b(for)h(the)g(pro)s(cess)i Fp(\021)t Fw(.)59 b(Th)m(us,)40 b Fp(j)3801 1287 y Fo(x;x)p Fk(+1)3795 1359 y Fo(t)4056 1335 y Fw(=)0 1468 y Fp(J)65 1420 y Fo(x;x)p Fk(+1)55 1493 y Fo(t)308 1468 y Fl(\000)23 b Fp(N)499 1432 y Fk(2)560 1388 y Fg(R)626 1413 y Fo(t)607 1504 y Fk(0)678 1468 y Fp(W)772 1483 y Fo(x;x)p Fk(+1)992 1468 y Fw(\()p Fp(s)p Fw(\))p Fp(ds)33 b Fw(is)h(a)g(martingale)f(for)h (eac)m(h)h Fp(x)g Fw(in)f Fi(Z)e Fw(and)i Fp(M)2913 1483 y Fo(t)2977 1468 y Fw(=)c Fl(\000)p Fp(j)3208 1420 y Fm(\000)p Fk(1)p Fo(;)p Fk(0)3202 1493 y Fo(t)3379 1468 y Fw(,)k Fp(M)3549 1432 y Fo(n)3538 1493 y(t)3632 1468 y Fw(=)c Fl(\000)p Fp(j)3863 1420 y Fm(\000)p Fk(1)p Fo(;)p Fk(0)3857 1493 y Fo(t)4056 1468 y Fw(+)0 1611 y(\()p Fp(nN)11 b Fw(\))229 1575 y Fm(\000)p Fk(1)352 1537 y Fg(P)457 1561 y Fo(nN)c Fm(\000)p Fk(1)457 1641 y Fo(x)p Fk(=0)700 1611 y Fp(j)747 1564 y Fo(x;x)p Fk(+1)741 1636 y Fo(t)967 1611 y Fw(.)44 b(In)33 b(particular,)997 1905 y Fi(E)1058 1920 y Fo(\027)1098 1930 y Ff(')1159 1794 y Fg(h\020)1266 1905 y Fp(M)1374 1864 y Fo(n)1363 1930 y(t)1450 1905 y Fl(\000)22 b Fp(M)1646 1920 y Fo(t)1681 1794 y Fg(\021)1741 1815 y Fk(2)1785 1794 y Fg(i)1888 1905 y Fw(=)56 b Fi(E)2081 1920 y Fo(\027)2121 1930 y Ff(')2183 1794 y Fg(h\020)2352 1838 y Fw(1)p 2301 1882 151 4 v 2301 1973 a Fp(nN)2480 1780 y Fo(nN)7 b Fm(\000)p Fk(1)2519 1810 y Fg(X)2518 2021 y Fo(x)p Fk(=0)2719 1905 y Fp(j)2766 1857 y Fo(x;x)p Fk(+1)2760 1930 y Fo(t)2985 1794 y Fg(\021)3045 1815 y Fk(2)3090 1794 y Fg(i)1224 2253 y Fw(=)1481 2186 y(1)p 1369 2230 273 4 v 1369 2322 a(\()p Fp(nN)k Fw(\))1598 2293 y Fk(2)1670 2129 y Fo(nN)c Fm(\000)p Fk(1)1709 2159 y Fg(X)1708 2370 y Fo(x)p Fk(=0)1909 2253 y Fp(tN)2036 2212 y Fk(2)2080 2253 y Fp(\014)2136 2268 y Fo(x)2242 2253 y Fw(=)2387 2186 y Fp(N)k(\014)2534 2201 y Fk(0)2579 2186 y Fp(t)p 2387 2230 228 4 v 2471 2322 a(n)2627 2253 y Fl(\001)0 2554 y Fw(Here)40 b(w)m(e)h(used)g(the)g (fact)f(that)g(the)g(martingales)f Fl(f)p Fp(j)2055 2506 y Fo(x;x)p Fk(+1)2049 2579 y Fo(t)2274 2554 y Fp(;)33 b(x)39 b Fl(2)g Fi(Z)p Fl(g)f Fw(are)i(m)m(utually)e(orthogonal)h(b)s (ecause)0 2682 y(they)33 b(don't)g(ha)m(v)m(e)h(common)d(jumps)j(and)g (that)f Fi(E)1879 2697 y Fo(\027)1919 2707 y Ff(')1981 2682 y Fw([\()p Fp(j)2095 2634 y Fo(x;x)p Fk(+1)2089 2706 y Fo(t)2314 2682 y Fw(\))2353 2646 y Fk(2)2397 2682 y Fw(])28 b(=)g Fp(tN)2685 2646 y Fk(2)2729 2682 y Fp(\014)2785 2697 y Fo(x)2836 2682 y Fw(.)99 b Fb(\003)120 2859 y Fw(It)35 b(follo)m(ws)g(from)f(this)i(lemma)d(that)i(for)g(eac)m(h)h (\014xed)g Fp(t)f Fw(the)h(sequence)h Fp(Y)2897 2823 y Fo(N)2875 2884 y Fk(0)2972 2859 y Fw(\()p Fp(G)3089 2874 y Fo(n)3144 2859 y Fw(\))23 b Fl(\000)h Fp(Y)3387 2823 y Fo(N)3365 2884 y(t)3463 2859 y Fw(\()p Fp(G)3580 2874 y Fo(n)3634 2859 y Fw(\))35 b(is)g(Cauc)m(h)m(y)0 2979 y(in)k Fp(L)190 2943 y Fk(2)234 2979 y Fw(\()p Fi(P)332 2994 y Fo(\027)372 3004 y Ff(')430 2979 y Fw(\),)g(uniformly)f(o)m(v)m (er)h Fp(N)48 b Fl(\025)38 b Fw(1.)61 b(In)39 b(consequence,)j(the)e (sequence)g Fp(Y)2978 2994 y Fk(0)3023 2979 y Fw(\()p Fp(G)3140 2994 y Fo(n)3194 2979 y Fw(\))26 b Fl(\000)g Fp(Y)3420 2994 y Fo(t)3455 2979 y Fw(\()p Fp(G)3572 2994 y Fo(n)3627 2979 y Fw(\))38 b(is)h(Cauc)m(h)m(y)0 3104 y(in)e Fp(L)188 3068 y Fk(2)233 3104 y Fw(\()p Fp(P)14 b Fw(\))36 b(\(Since)i 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Fw(\()p Fp(u)p Fw(\))1811 4271 y Fg(i)1874 4382 y Fp(du)50 b Fw(+)2160 4246 y Fg(Z)2215 4472 y Fh(R)2287 4382 y Fp(G)2365 4397 y Fo(n)2420 4382 y Fw(\()p Fp(u)p Fw(\))2555 4271 y Fg(h)2601 4382 y Fp(G)2679 4397 y Fo(n)2734 4382 y Fw(\()p Fp(u)p Fw(\))22 b Fl(\000)g Fp(T)3048 4397 y Fo(s)3091 4382 y Fp(G)3169 4397 y Fo(n)3224 4382 y Fw(\()p Fp(u)p Fw(\))3359 4271 y Fg(i)3422 4382 y Fp(du)825 4648 y Fl(\000)952 4512 y Fg(Z)1007 4739 y Fh(R)1079 4648 y Fp(G)1157 4663 y Fo(n)1212 4648 y Fw(\()p Fp(u)p Fw(\))1347 4537 y Fg(h)1393 4648 y Fp(G)1471 4663 y Fo(n)1526 4648 y Fw(\()p Fp(u)p Fw(\))f Fl(\000)i Fp(T)1840 4666 y Fm(j)p Fo(t)p Fm(\000)p Fo(s)p Fm(j)2023 4648 y Fp(G)2101 4663 y Fo(n)2155 4648 y Fw(\()p Fp(u)p Fw(\))2290 4537 y Fg(i)2353 4648 y Fp(du:)0 4906 y Fw(Since)259 4826 y Fg(R)306 4941 y Fh(R)378 4906 y Fp(G)456 4921 y Fo(n)511 4906 y Fw(\()p Fp(u)p Fw(\)[)p Fp(G)752 4921 y Fo(n)805 4906 y Fw(\()p Fp(u)p Fw(\))d Fl(\000)h Fp(T)1116 4921 y Fo(t)1151 4906 y Fp(G)1229 4921 y Fo(n)1283 4906 y Fw(\()p Fp(u)p Fw(\)])c Fp(du)31 b Fw(con)m(v)m(erges,)i(as)g Fp(n)27 b Fl(")h(1)p Fw(,)j(to)2643 4826 y Fg(R)2709 4850 y Fm(1)2690 4941 y Fk(0)2810 4906 y Fp(P)2874 4921 y Fo(u)2927 4906 y Fw([)p Fp(W)3049 4921 y Fo(t)3111 4906 y Fl(\024)d Fw(0])17 b Fp(du)p Fw(,)31 b(\(1.9\))g(is)h(pro)m(v)m (en.)120 5025 y(W)-8 b(e)33 b(are)g(no)m(w)h(in)f(a)g(p)s(osition)g(to) g(pro)m(v)m(e)g(Theorem)g(1.2.)0 5194 y Fd(Pr)-5 b(o)g(of)31 b(of)h(The)-5 b(or)g(em)32 b(1.2.)50 b Fw(Since)31 b Fp(Y)1338 5158 y Fo(N)1317 5219 y Fk(0)1414 5194 y Fw(\()p Fp(G)1531 5209 y Fo(n)1585 5194 y Fw(\))15 b Fl(\000)g Fp(Y)1811 5158 y Fo(N)1789 5219 y(t)1887 5194 y Fw(\()p Fp(G)2004 5209 y Fo(n)2058 5194 y Fw(\))29 b(con)m(v)m(erges,)i(as)f Fp(n)e Fl(")f(1)p Fw(,)j(to)f Fp(N)3248 5158 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)3435 5194 y Fp(X)3518 5212 y Fo(tN)3618 5192 y Fj(2)3692 5194 y Fw(in)g Fp(L)3872 5158 y Fk(2)3917 5194 y Fw(\()p Fp(P)4020 5209 y Fo(N)4095 5194 y Fw(\))0 5314 y(uniformly)34 b(in)g Fp(N)46 b Fw(and)35 b(since)g Fp(Y)1197 5329 y Fk(0)1242 5314 y Fw(\()p Fp(G)1359 5329 y Fo(n)1413 5314 y Fw(\))23 b Fl(\000)g Fp(Y)1633 5329 y Fo(t)1668 5314 y Fw(\()p Fp(G)1785 5329 y Fo(n)1840 5314 y Fw(\))34 b(con)m(v)m(erges,)i(as)f Fp(n)29 b Fl(")h(1)p Fw(,)k(to)h Fp(Z)3036 5329 y Fo(t)3105 5314 y Fw(in)g Fp(L)3291 5277 y Fk(2)3335 5314 y Fw(\()p Fp(P)14 b Fw(\),)34 b(it)g(is)h(not)f(hard)0 5433 y(to)f(sho)m(w)h(that)1201 5558 y Fi(E)1261 5573 y Fo(\027)1301 5583 y Ff(')1363 5447 y Fg(h)1410 5558 y Fp(f)1470 5447 y Fg(\020)1529 5558 y Fp(N)1620 5517 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)1807 5558 y Fp(X)1890 5575 y Fo(t)1920 5585 y Fj(1)1959 5575 y Fo(N)2029 5555 y Fj(2)2073 5558 y Fp(;)f(:::)17 b(;)g(N)2371 5517 y Fm(\000)p Fk(1)p Fo(=)p Fk(2)2556 5558 y Fp(X)2639 5575 y Fo(t)2669 5586 y Ff(k)2712 5575 y Fo(N)2782 5555 y Fj(2)2827 5447 y Fg(\021i)0 5756 y Fw(con)m(v)m(erges)43 b(to)f Fp(E)660 5771 y Fo(P)725 5756 y Fw([)p Fp(f)11 b Fw(\()p Fp(Z)920 5771 y Fo(t)950 5781 y Fj(1)993 5756 y Fp(;)17 b(:::;)g(Z)1235 5771 y Fo(t)1265 5782 y Ff(k)1312 5756 y Fw(\)])41 b(for)h(ev)m(ery)g(b)s(ounded)h(Lipsc)m(hitz-con)m (tin)m(uous)i(function)e Fp(f)52 b Fw(and)43 b(ev)m(ery)0 5875 y(0)27 b Fl(\024)i Fp(t)219 5890 y Fk(1)291 5875 y Fp(<)f(:::)f(<)h(t)648 5890 y Fo(k)725 5875 y Fl(\024)g Fp(T)14 b Fw(.)99 b Fb(\003)p eop %%Page: 14 14 14 13 bop 0 50 a Fu(14)1081 b(CL)1290 30 y(\023)1279 50 y(AUDIO)38 b(LANDIM,)e(S)2071 30 y(\023)2063 50 y(ER)n(GIO)i(B.)e(V) n(OLCHAN)0 289 y Fy(4.)51 b(Tigh)m(tness.)120 408 y Fw(W)-8 b(e)33 b(pro)m(v)m(e)h(in)f(this)g(section)h(that)f(the)g(sequence)i Fp(P)2067 423 y Fo(N)2176 408 y Fw(de\014ned)g(in)e(section)h(1)f(is)g (tigh)m(t.)-331 585 y Fo(j)t(usta)136 b Fy(Prop)s(osition)52 b(4.1)g(.)57 b Fd(The)47 b(se)-5 b(quenc)g(e)46 b Fl(f)p Fp(P)1665 600 y Fo(N)1740 585 y Fp(;)64 b(N)58 b Fl(\025)48 b Fw(1)p Fl(g)e Fd(is)g(tight.)78 b(Mor)-5 b(e)g(over,)48 b(al)5 b(l)48 b(its)e(limit)g(p)-5 b(oints)46 b(ar)-5 b(e)0 704 y(c)g(onc)g(entr)g(ate)g(d)35 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Fo(j)2173 1599 y Fp(>)2250 1614 y Fo(m)2371 1599 y Fw(\(2)p Fp(j)27 b Fw(+)c(1\))17 b Fp(H)2817 1614 y Fo(j)0 1932 y Fw(for)29 b(all)f Fp(f)40 b Fw(in)28 b Fp(D)s Fw(\()p Fl(A)p Fw(\).)42 b(Recall)29 b(that)g(for)g Fp(k)h Fl(\025)f Fw(1,)g Fc(H)1818 1947 y Fo(k)1896 1932 y Fw(stands)g(for)g(the)h(Hilb)s(ert)e(space)i (obtained)f(b)m(y)g(completion)0 2051 y(of)j Fp(D)s Fw(\()p Fc(L)302 2015 y Fo(k)352 2051 y Fw(\))f(with)h(resp)s(ect)h(to)f(the)g (inner)h(pro)s(duct)f Fp(<)d Fl(\001)p Fp(;)17 b Fl(\001)26 b Fp(>)2215 2066 y Fo(m;k)2391 2051 y Fw(de\014ned)34 b(b)m(y)e Fp(<)c(f)f(;)33 b(g)e(>)3268 2066 y Fo(m;k)3439 2051 y Fw(=)d Fp(<)h(f)e(;)33 b Fc(L)3853 2015 y Fo(k)3902 2051 y Fp(g)e(>)4058 2066 y Fo(m)0 2174 y Fw(for)i Fp(f)5 b(;)17 b(g)36 b Fw(in)d Fp(D)s Fw(\()p Fc(L)641 2138 y Fo(k)691 2174 y Fw(\).)43 b(It)33 b(can)h(b)s(e)g(sho)m(wn)g(that)f Fc(H)1818 2189 y Fo(k)1922 2174 y Fw(=)56 b Fl(f)p Fp(f)e Fl(2)45 b Fp(L)2387 2138 y Fk(2)2448 2174 y Fw(;)2537 2100 y Fg(P)2520 2264 y 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5242 y Fk(0)1313 5227 y Fw(.)44 b(This)34 b(concludes)h(the)e (pro)s(of)h(of)f(the)h(lemma.)97 b Fb(\003)-331 5408 y Fo(col)231 b Fy(Corollary)39 b(4.3)f(.)57 b Fd(F)-8 b(or)36 b Fp(k)31 b Fl(\025)d Fw(3)p Fd(,)764 5673 y Fw(lim)17 b(sup)801 5757 y Fo(N)7 b Fm(!1)1102 5673 y Fi(E)1162 5688 y Fo(\027)1202 5698 y Ff(')1264 5563 y Fg(h)1379 5673 y Fw(sup)1328 5757 y Fk(0)p Fm(\024)p Fo(t)p Fm(\024)p Fo(T)1613 5673 y Fl(k)p Fp(Y)1721 5688 y Fo(t)1756 5673 y Fl(k)1806 5632 y Fk(2)1806 5698 y Fm(\000)p Fo(k)1917 5563 y Fg(i)2036 5673 y Fp(<)56 b Fl(1)p Fp(;)799 5934 y Fw(lim)764 5994 y Fo(n)p Fm(!1)1005 5934 y Fw(lim)17 b(sup)1043 6018 y Fo(N)7 b Fm(!1)1343 5934 y Fi(E)1404 5949 y Fo(\027)1444 5959 y Ff(')1505 5824 y Fg(h)1620 5934 y Fw(sup)1569 6018 y Fk(0)p Fm(\024)p Fo(t)p Fm(\024)p Fo(T)1868 5840 y Fg(X)1865 6054 y Fo(j)t Fm(\025)p Fo(n)2014 5934 y Fw(\(2)p Fp(j)27 b Fw(+)c(1\))2360 5893 y Fm(\000)p Fo(k)2498 5934 y Fp(<)29 b(Y)2662 5949 y Fo(t)2696 5934 y Fp(;)17 b(H)2824 5949 y Fo(j)2893 5934 y Fp(>)2970 5893 y Fk(2)2970 5959 y Fo(m)3074 5824 y Fg(i)3176 5934 y Fw(=)56 b(0)p Fp(:)p eop %%Page: 16 16 16 15 bop 0 50 a Fu(16)1081 b(CL)1290 30 y(\023)1279 50 y(AUDIO)38 b(LANDIM,)e(S)2071 30 y(\023)2063 50 y(ER)n(GIO)i(B.)e(V) n(OLCHAN)0 289 y Fd(Pr)-5 b(o)g(of.)49 b Fw(By)33 b(de\014nition)g(of)h Fl(k)p Fp(Y)1134 304 y Fo(t)1169 289 y Fl(k)1219 253 y Fk(2)1219 317 y Fm(\000)p Fo(k)1330 289 y Fw(,)1197 519 y Fl(k)p Fp(Y)1305 534 y Fo(t)1339 519 y Fl(k)1389 478 y Fk(2)1389 543 y Fm(\000)p Fo(k)1556 519 y Fw(=)1689 424 y Fg(X)1691 638 y Fo(j)t Fm(\025)p Fk(0)1833 519 y Fw(\(2)p Fp(j)27 b Fw(+)c(1\))2179 478 y Fm(\000)p Fo(k)2334 519 y Fp(<)28 b(Y)2497 534 y Fo(t)2532 519 y Fp(;)17 b(H)2660 534 y Fo(j)2729 519 y Fp(>)2806 478 y Fk(2)2806 543 y Fo(m)2909 519 y Fp(:)0 819 y Fw(In)34 b(particular,)e(b)m(y)i(the)f(previous)h(lemma,)252 1042 y(lim)206 1106 y Fo(N)7 b Fm(!1)452 1042 y Fi(E)512 1057 y Fo(\027)552 1067 y Ff(')614 932 y Fg(h)729 1042 y Fw(sup)678 1126 y 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Fo(s)p Fm(\000)p Fo(t)p Fm(j\024)p Fo(\016)1267 3404 y Fk(0)p Fm(\024)p Fo(s;t)p Fm(\024)p Fo(T)1627 3144 y Fg(X)1625 3358 y Fo(j)t Fm(\024)p Fo(n)1801 3239 y Fp(<)28 b(Y)1964 3254 y Fo(t)2021 3239 y Fl(\000)23 b Fp(Y)2179 3254 y Fo(s)2238 3239 y Fp(;)33 b(H)2382 3254 y Fo(j)2451 3239 y Fp(>)2528 3198 y Fk(2)2601 3239 y Fw(\(2)p Fp(j)28 b Fw(+)22 b(1\))2947 3198 y Fm(\000)p Fo(k)3113 3239 y Fp(>)56 b(\017)3286 3128 y Fg(i)3389 3239 y Fw(=)g(0)17 b Fp(:)0 3615 y Fw(W)-8 b(e)39 b(observ)m(e)h(that)e Fp(Y)839 3579 y Fo(N)817 3640 y(t)914 3615 y Fw(\()p Fp(H)1036 3630 y Fo(j)1078 3615 y Fw(\))26 b Fl(\000)g Fp(Y)1326 3579 y Fo(N)1304 3640 y(s)1401 3615 y Fw(\()p Fp(H)1523 3630 y Fo(j)1565 3615 y Fw(\))36 b(=)i Fp(M)1852 3630 y Fo(t)1886 3615 y Fw(\()p Fp(j)6 b Fw(\))25 b Fl(\000)i Fp(M)2237 3630 y Fo(s)2279 3615 y Fw(\()p Fp(j)6 b Fw(\))25 b(+)2532 3535 y Fg(R)2599 3559 y Fo(t)2579 3650 y(s)2650 3615 y Fw(\000)2712 3630 y Fk(1)p Fo(;j)2818 3615 y Fw(\()p Fp(r)s Fw(\))p Fp(dr)s Fw(.)59 b(The)39 b(martingale)e(part)i(is)0 3735 y(handled)j(as)f(in)g([KL],)g(Chap.11.)67 b(F)-8 b(or)40 b(the)h(in)m(tegral)g(part)g(one)g(uses)h(the)f(pro)s(of)g(in)g ([L])g(for)g(nongradien)m(t)0 3855 y(systems.)i(W)-8 b(e)34 b(lea)m(v)m(e)f(the)h(details)f(to)g(the)g(reader.)100 b Fb(\003)120 4025 y Fw(W)-8 b(e)33 b(conclude)i(this)e(section)h(pro)m (ving)f(an)g(estimate)f(used)j(in)e(Lemma)e(4.2.)-331 4195 y Fo(lem)p Fk(5)p Fo(:)p Fk(3)93 b Fy(Lemma)38 b(4.4)h(.)57 b Fd(F)-8 b(or)36 b(al)5 b(l)36 b Fw(0)28 b Fl(\024)g Fp(t)g Fl(\024)g Fp(T)50 b Fd(and)35 b Fp(a)27 b(>)i Fw(0)p Fd(,)437 4457 y Fi(E)498 4472 y Fo(\027)538 4482 y Ff(')599 4347 y Fg(h)663 4457 y Fw(exp)832 4347 y Fg(n)898 4457 y Fp(a)p Fl(j)996 4322 y Fg(Z)1095 4346 y Fo(t)1050 4548 y(s)1146 4457 y Fw(\000)1208 4472 y Fk(1)p Fo(;j)1314 4457 y Fw(\()p Fp(u)p Fw(\))p Fp(du)p Fl(j)1586 4347 y Fg(o)o(i)1754 4457 y Fl(\024)56 b Fw(2)17 b(exp)2122 4347 y Fg(n)2188 4457 y Fp(a)2241 4416 y Fk(2)2286 4457 y Fw(\()p Fp(t)22 b Fl(\000)g Fp(s)p Fw(\))p Fp(N)2659 4416 y Fm(\000)p Fk(1)2783 4363 y Fg(X)2782 4577 y Fo(x)p Fm(2)p Fh(Z)2927 4457 y Fw(\()p Fl(r)3049 4472 y Fo(N)3125 4457 y Fp(H)3208 4472 y Fo(j)3250 4457 y Fw(\()p Fp(x=)-6 b(N)11 b Fw(\)\))3559 4416 y Fk(2)3603 4347 y Fg(o)3669 4457 y Fp(:)0 4820 y Fd(Pr)-5 b(o)g(of.)49 b Fw(Since)38 b Fp(e)623 4784 y Fm(j)p Fo(x)p Fm(j)754 4820 y Fl(\024)33 b Fp(e)910 4784 y Fo(x)985 4820 y Fw(+)25 b Fp(e)1133 4784 y Fo(x)1183 4820 y Fw(,)37 b(it)f(is)g(enough)i(to)e(pro)m(v)m(e)h (that)f(the)h(left)f(hand)i(side)f(without)f(the)h(absolute)0 4939 y(v)-6 b(alue)47 b(is)g(b)s(ounded)h(b)m(y)f(one)g(half)g(of)g (the)g(righ)m(t)g(hand)g(side.)86 b(Since)47 b Fp(\027)2794 4954 y Fo(')2899 4939 y Fw(is)f(an)h(in)m(v)-6 b(arian)m(t)47 b(measure,)j(b)m(y)0 5059 y(F)-8 b(eynman-Kac)33 b(form)m(ula,)298 5321 y Fi(E)358 5336 y Fo(\027)398 5346 y Ff(')460 5211 y Fg(h)523 5321 y Fw(exp)692 5211 y Fg(n)775 5186 y(Z)875 5210 y Fo(s)p Fm(\000)p Fo(t)831 5412 y Fk(0)1026 5321 y Fp(a)p Fw(\000)1141 5336 y Fk(1)p Fo(;j)1247 5321 y Fw(\()p Fp(u)p Fw(\))17 b Fp(du)1508 5211 y Fg(o)o(i)1676 5321 y Fl(\024)55 b Fw(exp)1977 5211 y Fg(n)2044 5321 y Fw(\()p Fp(t)21 b Fl(\000)i Fp(s)p Fw(\))17 b(sup)2394 5406 y Fo(f)2509 5211 y Fg(n)2592 5186 y(Z)2708 5321 y Fp(a)p Fw(\000)2823 5336 y Fk(1)p Fo(;j)2928 5321 y Fp(f)27 b(d\027)3105 5336 y Fo(')3202 5321 y Fl(\000)39 b Fp(N)3409 5280 y Fk(2)3453 5321 y Fp(D)s Fw(\()p Fp(f)11 b Fw(\))3676 5211 y Fg(oo)3808 5321 y Fp(;)0 5601 y Fw(where)26 b(the)g(suprem)m(um)g(is)f(tak)m(en)h(o)m(v)m(er)g(all)f(densities)h (with)f(resp)s(ect)i(to)e Fp(\027)2728 5616 y Fo(')2811 5601 y Fw(and)h Fl(\000)p Fp(D)s Fw(\()p Fp(f)11 b Fw(\))28 b(=)p Fp(<)3507 5526 y Fl(p)p 3590 5526 60 4 v 75 x Fp(f)11 b(;)17 b Fl(L)3764 5526 y(p)p 3846 5526 V 3846 5601 a Fp(f)38 b(>)4010 5616 y Fo(\027)4050 5626 y Ff(')4106 5601 y Fw(.)0 5721 y(Recall)33 b(the)h(de\014nition)g(of)f(the)h (curren)m(t)g Fp(W)1640 5736 y Fo(x;x)p Fk(+1)1893 5721 y Fw(in)m(tro)s(duced)h(in)e(section)h(2.)43 b(W)-8 b(e)34 b(ha)m(v)m(e)g(that)1166 5943 y Fp(a)17 b Fw(\000)1298 5958 y Fk(1)p Fo(;j)1458 5943 y Fw(=)56 b Fp(a)1661 5853 y Fl(p)p 1743 5853 91 4 v 1743 5943 a Fp(N)1851 5848 y Fg(X)1851 6062 y Fo(x)p Fm(2)p Fh(Z)1996 5943 y Fw(\()p Fl(r)2118 5958 y Fo(N)2193 5943 y Fp(H)2276 5958 y Fo(j)2318 5943 y Fw(\)\()p Fp(x=)-6 b(N)11 b Fw(\))p Fp(W)2721 5958 y Fo(x;x)p Fk(+1)2940 5943 y Fp(:)p eop %%Page: 17 17 17 16 bop 1429 50 a Fu(EQUILIBRIUM)38 b(FLUCTUA)-7 b(TIONS)1310 b(17)0 289 y Fw(The)27 b(in)m(tegration)f(b)m(y)h(parts)g(form)m(ula)f (\(2.4\))g(for)h(the)g(curren)m(t)h(sho)m(ws)f(that)g Fp(a)2880 209 y Fg(R)2962 289 y Fw(\000)3024 304 y Fk(1)p Fo(;j)3130 289 y Fp(f)g(d\027)3307 304 y Fo(')3391 289 y Fw(is)g(b)s(ounded)h(ab)s(o)m(v)m(e)0 408 y(b)m(y)1249 525 y(2)p Fp(a)1352 489 y Fk(2)p 1249 570 148 4 v 1277 661 a Fp(N)1426 498 y Fg(X)1425 712 y Fo(x)p Fm(2)p Fh(Z)1587 482 y Fg(h)1634 593 y Fw(\()p Fl(r)1756 608 y Fo(N)1831 593 y Fp(H)1914 608 y Fo(j)1956 593 y Fw(\)\()p Fp(x=)-6 b(N)11 b Fw(\))2265 482 y Fg(i)2311 503 y Fk(2)2395 593 y Fw(+)39 b Fp(N)2602 552 y Fk(2)2647 593 y Fp(D)s Fw(\()p Fp(f)11 b Fw(\))p Fp(:)0 883 y Fw(Therefore,)630 1174 y(sup)682 1258 y Fo(f)797 1063 y Fg(n)880 1038 y(Z)996 1174 y Fp(a)p Fw(\000)1111 1189 y Fk(1)p Fo(;j)1216 1174 y Fp(f)27 b(d\027)1393 1189 y Fo(')1490 1174 y Fl(\000)39 b Fp(N)1697 1133 y Fk(2)1741 1174 y Fp(D)s Fw(\()p Fp(f)11 b Fw(\))1964 1063 y Fg(o)2085 1174 y Fl(\024)56 b Fw(\()p Fp(a)2310 1133 y Fk(2)2354 1174 y Fp(=)-6 b(N)11 b Fw(\))2546 1079 y Fg(X)2545 1293 y Fo(x)p Fm(2)p Fh(Z)2707 1063 y Fg(h)2754 1174 y Fw(\()p Fl(r)2876 1189 y Fo(N)2951 1174 y Fp(H)3034 1189 y Fo(j)3076 1174 y Fw(\)\()p Fp(x=)-6 b(N)11 b Fw(\))3385 1063 y Fg(i)3431 1084 y Fk(2)3476 1174 y Fp(;)0 1513 y Fw(whic)m(h)34 b(concludes)h(the)e(pro)s(of)h(of)f (the)h(lemma.)97 b Fb(\003)0 1699 y Fy(App)s(endix.)120 1821 y Fw(Consider)43 b(the)h(linear)e(op)s(erator)h Fc(L)h Fw(=)h(\()p Fp(u)1753 1785 y Fk(2)1797 1821 y Fp(=)p Fw(2\010)1969 1785 y Fm(0)1997 1821 y Fw(\()p Fp(\032)p Fw(\)\))27 b Fl(\000)i(A)p Fw(.)73 b(In)43 b(this)g(app)s(endix)g(w)m(e)h(construct)g(an)f(or-)0 1940 y(thonormal)32 b(basis)h(of)h Fc(L)p Fw(-eigenfunctions)h Fl(f)p Fp(H)1723 1955 y Fo(j)1765 1940 y Fp(;)17 b(j)33 b Fl(\025)28 b Fw(0)p Fl(g)33 b Fw(for)g Fp(L)2342 1904 y Fk(2)2387 1940 y Fw(\()p Fp(m)p Fw(\()p Fp(u)p Fw(\))p Fp(du)p Fw(\))f(suc)m(h)j(that)1262 2186 y Fp(H)1345 2201 y Fo(j)1415 2186 y Fl(2)28 b Fp(D)s Fw(\()p Fl(A)p Fw(\))82 b(and)h Fc(L)p Fp(H)2226 2201 y Fo(j)2296 2186 y Fw(=)28 b(\(2)p Fp(j)g Fw(+)22 b(1\))p Fp(H)2830 2201 y Fo(j)3929 2186 y Fw(\(5)p Fp(:)p Fw(1\))-4466 b Fo(r)r(eq)0 2432 y Fw(for)33 b(eac)m(h)i Fp(j)e Fl(\025)28 b Fw(0.)120 2553 y(Recall)j(that)h Fp(\022)s Fw(\()p Fp(u)p Fw(\))26 b(=)j Fp(\022)997 2568 y Fm(\000)1064 2553 y Fy(1)1121 2571 y Fm(f)p Fo(u<)p Fk(0)p Fm(g)1376 2553 y Fw(+)20 b Fp(\022)1520 2568 y Fk(+)1585 2553 y Fy(1)1642 2571 y Fm(f)p Fo(u>)p Fk(0)p Fm(g)1878 2553 y Fw(,)32 b(where)h Fp(\022)2272 2568 y Fm(\006)2366 2553 y Fw(=)c(\010)2544 2517 y Fm(0)2572 2553 y Fw(\()p Fp(\032)2663 2568 y Fm(\006)2729 2553 y Fw(\).)43 b(Let)32 b Fp(J)3071 2568 y Fk(0)3116 2553 y Fw(\()p Fp(u)p Fw(\))27 b(=)i(exp\()p Fl(\000)p Fp(u)3709 2517 y Fk(2)3754 2553 y Fp(=)p Fw(2)p Fp(\022)s Fw(\))h(and)0 2673 y(notice)j(that)g(it)g(b)s(elongs)g(to)g(the)h (space)518 2943 y Fp(C)596 2895 y Fm(1)p Fo(;)p Fm(\003)589 2969 y Fk(0)773 2943 y Fw(=)878 2832 y Fg(n)944 2943 y Fp(H)i Fl(2)28 b Fp(C)1235 2902 y Fm(1)1228 2967 y Fk(0)1320 2943 y Fw(\()p Fi(R)t Fw(\);)17 b Fp(q)t(H)1647 2902 y Fm(0)1680 2943 y Fw(\(0+\))28 b(=)g Fp(pH)2159 2902 y Fm(0)2187 2943 y Fw(\(0)p Fl(\000)p Fw(\))17 b Fp(;)32 b(\022)2516 2958 y Fk(+)2582 2943 y Fp(H)2673 2902 y Fm(00)2724 2943 y Fw(\(0+\))27 b(=)h Fp(\022)3108 2958 y Fm(\000)3175 2943 y Fp(H)3266 2902 y Fm(00)3317 2943 y Fw(\(0)p Fl(\000)p Fw(\))3522 2832 y Fg(o)3588 2943 y Fp(;)0 3218 y Fw(where)42 b Fp(C)374 3182 y Fm(1)367 3243 y Fk(0)459 3218 y Fw(\()p Fi(R)5 b Fw(\))47 b(stands)42 b(for)f(the)h(space)h(of)e(smo)s(oth)f(functions)j(on)e Fi(R)2715 3233 y Fm(\003)2808 3218 y Fw(that)g(are)h(rapidly)f(deca)m (ying)g(and)0 3346 y(con)m(tin)m(uous)49 b(on)e Fi(R)5 b Fw(.)93 b(F)-8 b(or)47 b Fp(n)k Fl(\025)h Fw(1,)f(let)c Fp( )1637 3361 y Fk(2)p Fo(n)1730 3346 y Fw(\()p Fp(u)p Fw(\))k(=)h(\()p Fp(u=)2191 3261 y Fl(p)p 2274 3261 50 4 v 85 x Fp(\022)r Fw(\))2362 3310 y Fk(2)p Fo(n)2456 3346 y Fw(,)f Fp( )2600 3361 y Fk(2)p Fo(n)p Fm(\000)p Fk(1)2796 3346 y Fw(\()p Fp(u)p Fw(\))f(=)i Fp(b)p Fw(\()p Fp(u)p Fw(\)\()p Fp(u=)3434 3261 y Fl(p)p 3516 3261 V 3516 3346 a Fp(\022)s Fw(\))3605 3310 y Fk(2)p Fo(n)p Fm(\000)p Fk(1)3800 3346 y Fw(,)f(where)0 3479 y Fp(b)p Fw(\()p Fp(u)p Fw(\))38 b(=)333 3396 y Fg(p)p 433 3396 113 4 v 83 x Fp(\022)480 3494 y Fk(+)546 3479 y Fp(p)p Fy(1)653 3497 y Fm(f)p Fo(u>)p Fk(0)p Fm(g)915 3479 y Fw(+)1020 3396 y Fg(p)p 1119 3396 115 4 v 1119 3479 a Fp(\022)1166 3494 y Fm(\000)1233 3479 y Fp(q)t Fy(1)1338 3497 y Fm(f)p Fo(u<)p Fk(0)p Fm(g)1574 3479 y Fw(.)65 b(Set)41 b Fp(J)1901 3494 y Fo(n)1955 3479 y Fw(\()p Fp(u)p Fw(\))e(=)g Fp( )2310 3494 y Fo(n)2364 3479 y Fw(\()p Fp(u)p Fw(\))p Fp(e)2545 3443 y Fm(\000)p Fo(u)2654 3413 y Fj(2)2694 3443 y Fo(=)p Fk(2)p Fo(\022)2859 3479 y Fw(for)i Fp(n)e Fl(\025)g Fw(1.)65 b(Notice)39 b(that)h Fp(J)3974 3494 y Fo(n)4029 3479 y Fw(\()p Fl(\001)p Fw(\))0 3605 y(b)s(elong)33 b(to)g Fp(C)518 3557 y Fm(1)p Fo(;)p Fm(\003)511 3631 y Fk(0)700 3605 y Fw(for)h Fp(n)27 b Fl(\025)i Fw(1.)120 3726 y(W)-8 b(e)41 b(claim)e(that)i(the)h(set)f Fl(f)p Fp(J)1244 3741 y Fo(n)1298 3726 y Fw(\()p Fl(\001)p Fw(\))p Fp(;)17 b(n)39 b Fl(\025)i Fw(0)p Fl(g)g Fw(is)g(complete)g(in) g Fp(L)2536 3690 y Fk(2)2580 3726 y Fw(\()p Fp(m)p Fw(\()p Fp(u)p Fw(\))p Fp(du)p Fw(\))g(or,)h(equiv)-6 b(alen)m(tly)e(,)42 b(that)e(the)0 3860 y(set)k Fl(f)p Fp( )281 3875 y Fo(n)335 3860 y Fw(\()p Fp(u)p Fw(\))p Fp(;)17 b(n)44 b Fl(\025)j Fw(0)p Fl(g)c 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Fw(\))27 b(=)h Fp(f)11 b Fw(\()p Fp(u)p Fw(\))p Fp(m)p Fw(\()p Fp(u)p Fw(\),)32 b(whic)m(h)i(b)s(elongs)f(to)g Fp(L)1968 4206 y Fk(2)2013 4242 y Fw(\()p Fp(e)2098 4206 y Fm(\000)p Fo(u)2207 4176 y Fj(2)2246 4206 y Fo(=)p Fk(2)p Fo(\022)2372 4242 y Fp(du)p Fw(\).)43 b(F)-8 b(or)33 b(the)h(ev)m(en)f(terms,)g(w)m (e)g(ha)m(v)m(e)1334 4403 y Fg(Z)1389 4629 y Fh(R)1444 4539 y Fw(\()p Fp(u=)1590 4449 y Fl(p)p 1673 4449 50 4 v 90 x Fp(\022)s Fw(\))1762 4498 y Fk(2)p Fo(n)1872 4539 y Fp(f)1921 4554 y Fk(0)1965 4539 y Fw(\()p Fp(u)p Fw(\))17 b Fp(e)2163 4498 y Fm(\000)p Fo(u)2272 4467 y Fj(2)2311 4498 y Fo(=)p Fk(2)p Fo(\022)2453 4539 y Fp(du)56 b Fw(=)f(0)0 4835 y(from)33 b(whic)m(h)h(follo)m(ws,)e(after)i (a)f(c)m(hange)h(of)g(v)-6 b(ariables,)32 b(that)1146 4990 y Fg(Z)1201 5216 y Fh(R)1256 5125 y Fw(\()p Fp(t)1331 5084 y Fk(2)p Fo(n)1425 5125 y Fp(=)p Fw(2)p Fp(n)p Fw(!\))17 b Fp(u)1726 5084 y Fk(2)p Fo(n)1818 5125 y Fp(f)1867 5140 y Fk(0)1912 5125 y Fw(\()p Fp(u)2008 5036 y Fl(p)p 2091 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b(LANDIM,)e(S)2071 30 y(\023)2063 50 y(ER)n(GIO)i(B.)e(V) n(OLCHAN)120 289 y Fw(Rep)s(eating)d(the)h(same)e(argumen)m(t)h(for)h (the)f(o)s(dd)h(terms,)e(w)m(e)i(get)f(that)1443 441 y Fg(Z)1498 667 y Fh(R)1570 577 y Fw(sin\()p Fp(tu)p Fw(\))17 b Fp(b)p Fw(\()p Fp(u)p Fw(\))g Fp(A)2150 592 y Fo(\022)2193 577 y Fw(\()p Fp(u)p Fw(\))g Fp(du)54 b Fw(=)i(0)0 870 y(for)33 b(all)g Fp(t)28 b Fl(2)g Fi(R)5 b Fw(.)120 990 y(Decomp)s(osing)38 b Fp(A)811 1005 y Fo(\022)856 990 y Fw(\()p Fl(\001)p Fw(\))g(in)h(its)g(ev)m(en)g(and)h (o)s(dd)f(parts,)i(denoted)f(resp)s(ectiv)m(ely)f(b)m(y)g Fp(A)3342 1005 y Fo(e)3384 990 y Fw(\()p Fl(\001)p Fw(\))g(and)g Fp(A)3803 1005 y Fo(o)3847 990 y Fw(\()p Fl(\001)p Fw(\),)g(w)m(e)0 1110 y(get)33 b(that)907 1142 y Fg(Z)962 1369 y Fh(R)1034 1278 y Fp(A)1109 1293 y Fo(e)1152 1278 y Fw(\()p Fp(u)p Fw(\))17 b(sin)o(\()p Fp(tu)p Fw(\))g Fp(du)27 b Fw(=)h(0)55 b(and)2191 1142 y Fg(Z)2247 1369 y Fh(R)2319 1278 y Fp(A)2394 1293 y Fo(e)2436 1278 y Fw(\()p Fp(u)p Fw(\))17 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y(ac)m(kno)m(wledge)k(partial)f(supp)s(ort)i(of)f(PR)m(ONEX)g (41.96.0923.00)d(\\F)-8 b(en^)-50 b(omenos)30 b(Cr)-11 b(\023)-39 b(\020ticos)31 b(em)f(Probabilidade)0 3088 y(e)j(Pro)s(cessos)h(Esto)s(c\023)-50 b(asticos".)0 3333 y Fy(References.)-99 3516 y Fw([Ar])49 b(Arratia)33 b(R.;)j(The)f (motion)f(of)h(a)g(tagged)f(particle)h(in)g(the)g(simple)g(symmetric)d (exclusion)k(system)e(in)h Fi(Z)p Fw(,)120 3635 y(Ann.Prob.)44 b Fy(11)33 b Fw(362-373)g(\(1983\).)-130 3756 y([BR])50 b(T.)44 b(Bro)m(x)f(and)i(H.)f(Rost;)50 b(Equilibrium)43 b(\015uctuations)i(of)g(sto)s(c)m(hastic)g(particle)f(systems:)66 b(the)45 b(role)g(of)120 3875 y(conserv)m(ed)34 b(quan)m(tities.)44 b(Ann)34 b(Prob)f Fy(12)h Fw(742-759)e(\(1984\).)-58 3996 y([C])50 b(C.)26 b(C.)g(Chang;)j(Equilibrium)c(\015uctuations)j (of)f(gradien)m(t)g(rev)m(ersible)g(particle)g(systems.)41 b(Prob.)h(Th.)g(Rel.)120 4116 y(Fields)33 b Fy(100)h Fw(269{283)e(\(1994\).)-252 4236 y([DGL1])50 b(D)s(\177)-53 b(urr)41 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b(T.)32 b(E.;)g(Di\013usions)h(with)f(collisions)g(b)s(et)m(w)m(een)i (particles.)44 b(J.)32 b(Appl.)44 b(Prob.)g Fy(2)32 b Fw(323{338)f(\(1965\).)-116 5197 y([HS])50 b(R.)e(A.)h(Holley)e(and)j (D.)e(W.)h(Stro)s(o)s(c)m(k;)56 b(Generalized)50 b(Ornstein-Uhlen)m(b)s (ec)m(k)h(pro)s(cesses)g(and)e(in\014nite)120 5317 y(branc)m(hing)34 b(Bro)m(wnian)f(motions.)43 b(Ky)m(oto)33 b(Univ.)43 b(R.I.M.S.)33 b Fy(14)g Fw(741-814)g(\(1978\).)-125 5437 y([KL])50 b(Kipnis)29 b(C.,)g(Landim)g(C.;)g Fd(Sc)-5 b(aling)32 b(Limit)f(of)h(Inter)-5 b(acting)31 b(Particle)g(Systems.)p Fw(,)e(Springer)h(V)-8 b(erlag,)28 b(Berlin)120 5557 y(1999.)-197 5678 y([LO)m(V])50 b(Landim)42 b(C.,)k(Olla)d(S.)g(and)h (V)-8 b(olc)m(han)43 b(S.;)49 b(Driv)m(en)43 b(T)-8 b(racer)44 b(particle)f(in)g(one)h(dimensional)f(symmetric)120 5797 y(simple)32 b(exclusion.)44 b(Comm)m(un.)f(Math.)h(Ph)m(ys.)g Fy(192)p Fw(,)34 b(287{307)e(\(1998\))-48 5918 y([L])50 b(S.)24 b(L.)h(Lu;)i(Equilibrium)c(\015uctuations)i(of)g(a)f 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b(motion)f(with)i (elastic)f(collisions)g(of)h(an)g(in\014nite)h(particle)e(system;)47 b(J.)42 b(math.)120 1126 y(Mec)m(h.)j Fy(18)34 b Fw(973{989)e (\(1968\).)-41 1245 y([S])50 b(H.)32 b(Sp)s(ohn;)i Fd(L)-5 b(ar)g(ge)36 b(sc)-5 b(ale)35 b(dynamics)g(of)g(inter)-5 b(acting)35 b(p)-5 b(articles)p Fw(.)43 b(Springer-V)-8 b(erlag,)33 b(New)g(Y)-8 b(ork)32 b(\(1991\).)120 1465 y Fv(IMP)-7 b(A-Instituto)30 b(de)f(Ma)-5 b(tem)1231 1459 y(\023)1229 1465 y(atica)31 b(Pura)f(e)f(Aplicad)n(a,)i(Estrad)n (a)f(Dona)g(Castorina)g(110,)h(Jardim)f(Bot)-5 b(an-)0 1564 y(ico,)41 b(CEP)e(22460,)j(Rio)c(de)h(Janeir)n(o,)j(Brasil)c(and)h (CNRS)h(UPRES-A)h(6085,)g(Universit)3322 1558 y(\023)3322 1564 y(e)g(de)d(R)n(ouen,)j(76.128)0 1664 y(Mont)32 b(Saint)g(Aignan)i (Cedex,)e(France.)120 1763 y Fn(E-mail)e(addr)l(ess)6 b Fu(:)41 b Fa(landim@impa.br)120 1983 y Fv(Dep)-5 b(ar)g(t)g(amento)34 b(de)g(Ma)-5 b(tem)1177 1977 y(\023)1175 1983 y(atica,)38 b(PUC-Rio-Pontif)2138 1977 y(\023)2148 1983 y(\020cia)g(Universid)n (ade)f(Ca)-5 b(t)3085 1977 y(\023)3082 1983 y(olica)35 b(do)e(Rio)i(de)f(Janeir)n(o,)0 2082 y(R)n(ua)e(Mar)n(ques)g(de)f(S)779 2076 y(~)777 2082 y(ao)h(Vicente,)h(G)1378 2076 y(\023)1376 2082 y(avea,)g(CEP)f(22453-900)h(,)f(Rio)f(de)g(Janeir)n(o,)i(Brasil.) 120 2182 y Fn(E-mail)d(addr)l(ess)6 b Fu(:)41 b Fa (volchan@saci.mat.puc)q(-ri)q(o.)q(br)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF