Abou Hydrodynamc Lm of Some Excluson Processes va Funconal Inegraon Guy Fayolle Cyrl Furlehner Absrac Ths arcle consders some classes of models dealng wh he dynamcs of dscree curves subjeced o sochasc deformaons. I urns ou ha he problems of neres can be se n erms of neracng excluson processes, he ulmae goal beng o derve hydrodynamc lms afer proper scalngs. A seemngly new mehod s proposed, whch reles on he analyss of specfc paral dfferenal operaors, nvolvng varaonal calculus and funconal negraon: ndeed, he varables are he values of some funcons a gven pons, he number of whch ends o become nfne, whch requres he consrucon of generalzed measures. Sarng from a dealed analyss of he asep sysem on he orus Z/Z, we clam ha he argumens a pror work n hgher dmensons (abc, mul-ype excluson processes, ec, leadng o syems of coupled paral dfferenal equaons of Burgers ype. Keywords Cauchy problem, excluson process, funconal negraon, hydrodynamc lm, marngale, weak soluon. 1 Prelmnares Inerplay beween dscree and connuous descrpon s a recurren queson n sascal physcs, whch n some cases can be answered que rgorously va probablsc mehods. In he conex of reacon-dffuson sysems, hs s anamoun o sudyng flud or hydrodynamcs lms. umber of approaches have been proposed, n parcular n he framework of excluson processes, see e.g. [15,[4 [18, [14 and references heren. As far as flud or hydrodynamc lms are a sake, mos of hese mehods have n common o IRIA - Domane de Voluceau, Rocquencour BP 15-78153 Le Chesnay Cedex - France. Conac: Guy.Fayolle@nra.fr 1
2 2 Model defnon be lmed o sysems for whch he saonary saes are gven n closed produc forms, or a leas for whch he nvaran measure for fne (he sze of he sysem s explcly known. For nsance, asep wh open boundary can be descrbed n erms of marx produc form (a sor of a non-commuave produc form and he connuous lms can be undersood by means of brownan brdges (see [5. We propose o address hs queson from he followng dfferen pon of vew: sarng from dscree sample pahs subjeced o sochasc deformaons, he ulmae goal s o undersand he naure of he lm curves when ncreases o nfny. How do hese curves evolve wh me, and whch lmng process do hey represen as goes o nfny (equlbrum curves? Followng [1, 11, 12, we plan o gve some paral answers o hese problems n a seres of forhcomng papers. The mehod proposed n he presen sudy s appled n deal o he asep model. The mahemacal kernel reles on he analyss of specfc paral dfferenal equaons nvolvng varaonal calculus. A usual sequence of emprcal measures s shown o converge n probably o a deermnsc measure, whch s he unque weak soluon of a Cauchy problem. Here varables are n fac he values of some funcon a gven pons, and her number becomes nfne. In our opnon, he approach presens some new feaures, and very lkely exend o hgher dmensons, namely mul-ype excluson processes. A fuure concern wll be o esablsh a complee herarchy of sysems of hydrodynamc equaons, he sudy of whch should allow us o descrbe non-gbbs saes. All hese quesons form also he maer of ongong works. 2 Model defnon 2.1 A general sochasc clock model Consder an orened sample pah of a planar random walk n R 2, conssng of seps (or lnks of equal sze. Each sep can have n dscree possble orenaons, drawn from he se of angles wh some gven orgn {θ k = 2kπ n, k =,..., n 1}. The sochasc dynamcs n force consss n dsplacng one sngle pon a a me whou breakng he pah, whle keepng all lnks whn he se of admssble orenaons. In hs operaon, wo lnks are smulaneously dsplaced, wha consrans que srongly he possble dynamcal rules 2.1.1 Consrucng a relaed connuous-me Markov chan Jumps are produced by ndependen exponenal evens.
2.2 Examples 3 Perodc boundary condons wll be assumed, hs pon beng no a crucal resrcon. Dynamcal rules are gven by a se of reacons beween consecuve lnks, an equvalen formulaon beng possble n erms of random grammar. Wh each lnk s assocaed a ype,.e. a leer of an alphabe. Hence, for any n, we can defne whch laer on wll be somemes referred o as a local exchange process. For [1, and k [1, n, le X k represen a lnk of ype k a se. Then we can defne he followng se of reacons. X k X l λ kl +1 XX l +1, k k = 1,..., n, l k + n λ lk 2, (2.1 X k X k+n/2 γ k +1 X k+n/2+1 +1, k = 1,..., n. δ k+1 X k+1 The red equaons does exs only for even n, because of he exsence of folds [wo consecuve lnks wh oppose drecons, whch yeld a rcher dynamcs. X k can also be vewed as a bnary random varable descrbng he occupaon of se by a leer of ype k. Hence, he sae space of he sysem s represened by he array η def = {X k, = 1,..., ; k = 1,..., n}. 2.2 Examples (1 The smple excluson process The frs elemenary and mos suded example s he smple excluson process: hs model, afer mappng parcles ono lnks, corresponds o a onedmensonal flucuang nerface. Here we ake a bnary alphabe and, leng X 1 = τ and X 2 = τ, he se of reacons smply rewres τ τ λ λ + ττ, where λ ± are he ranson raes for he jump of a parcle o he rgh or o he lef. (2 The rangular lace and he ABC model Here he evoluon of he random walk s resrced o he rangular lace. Each lnk (or sep of he walk s eher 1, e 2π/3 or e 4π/3, and que naurally
4 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP wll be sad o be of ype A, B or C, respecvely. Ths corresponds o he so-called ABC model, snce here s a codng by means of a 3-leer alphabe. The se of ransons (or reacons s gven by AB p BA, BC q CB, CA r AC, (2.2 p + q + r + where here s a pror no symmery, bu we wll mpose perodc boundary condons on he sample pahs. Ths model was frs nroduced n [8 n he conex of parcles wh excluson, and for some cases correspondng o he reversbly of he process, a Gbbs form for he nvaran measure was gven n [9 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP As menoned above, we am a obanng hydrodynamc equaons for a class of excluson models. The mehod, alhough relyng on classcal powerful ools (marngales, relave compacness of measures, funconal analyss, has some new feaures whch should hopefully prove fruful n oher conexs. The essence of he approach s n fac conaned n he analyss of he popular asep model, presened below. We noe he dffculy o fnd n he exsng leraure a complee sudy encompassng varous specal cases (symmery, oal or weak asymmery, ec. Consder ses labelled from 1 o, formng a dscree closed curve n he plane, so ha he numberng of ses s mplcly aken modulo,.e. on ( def he dscree orus G = Z/Z. In hgher dmenson, say on he lace Z k, he relaed se of ses would be drawn on he orus (Z/Z k. We gaher below some noaonal maeral vald hroughou hs paper. R (resp. R + sands for he real (resp. posve real lne. C k [, 1 s he collecon of all real-valued, k-connuously dfferenable funcons defned on he nerval [, 1, and M s he space of all fne posve measures on he orus G def = [, 1. For S an arbrary merc space, P(S s he se of probably measures on S, and D S [, T s he space of rgh connuous funcons z : [, S wh lef lms and z. C (K s he space of nfnely dfferenable funcons wh compac suppor ncluded n K S, and we shall wre C[T o denoe he subse of funcons φ(x, C ([, 1 [, T vanshng a = T.
5 For = 1,...,, le A ( ( and B ( ( be bnary random varables represenng respecvely a parcule or a hole a se, so ha, owng o he excluson consran, A ( ( + B ( ( = 1, for all 1. Thus { A ( ( def = ( A ( (,..., A ( (, } s a Markov process. Ω ( F ( wll denoe he generaor of he Markov process A ( (, and = σ ( A ( (s, s s he assocaed naural flraon. Our purpose s o analyze he sequence of emprcal random measures µ ( = 1 A ( G ( (δ, (3.1 when, afer a convenen scalng of he parameers of he generaor Ω (. The probably dsrbuon assocaed wh he pah of he Markov process µ (, [, T, for some fxed T, wll be smply denoed by Q (. As usual, one can embed G ( n G, so ha a pon G ( corresponds o he pon / n G. Hence, n vew of (3.1, s que naural o le he sequence Q ( be defned on a unque space D M [, T, whch becomes a polsh space (.e. complee and separable va he usual Skorokod opology, as soon as M s self Polsh (see e.g. [7, chaper 3. Whou furher commen, M s assumed o be endowed wh he vague produc opology, as a consequence of he famous Banach-Alaoglo and Tychonoff heorems (see e.g. [17, 13. Choose wo arbrary funcons φ a, φ b C[T and defne he followng realvalued posve measure [ 1 ( ( Z ( [φ a, φ b def = exp, A ( ( + φ b, B ( (, (3.2 G ( φ a vewed as a funconal of φ a, φ b. Snce A ( ( + B ( ( = 1, for 1, he ransform Z ( s essenally a funconal of he sole funcon φ a φ b, up o a consan unformly bounded n. everheless, wll appear laer ha we need 2 ndependen funcons. For he sake of brevy, he explc dependence on, or φ, of quanes lke A ( (, B ( (, Z ( [φ a, φ b, wll frequenly be omed, wherever he meanng remans clear from he conex: for nsance, we smply wre A, B or Z (. Also Z ( sands for he process {Z (, }. A sandard powerful mehod o prove he convergence (n a sense o be specfed laer of he sequence of probably measures nroduced n (3.1 consss frs n showng s relave compacness, and hen n verfyng he concdence of all possble lm pons (see e.g. [14. Moreover here, by he
6 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP choce of he funcons φ a, φ b, suffces o prove hese wo properes for he sequence of projeced measures defned on D R [, T and correspondng o he processes {Z ( [φ a, φ b, }. Le us now nroduce quanes whch, as far as scalng s concerned, are crucal n order o oban meanngful hydrodynamc equaons. λ( def = λab ( + λ ba (, 2 (3.3 µ( def = λ ab ( λ ba (, where he dependence of he raes on s explcly menoned. Theorem 3.1. Le he sysem (3.3 have a gven asympoc expanson of he form, for large, λ( def = λ 2 + o( 2, (3.4 µ( def = µ + o(, where λ and µ are fxed consans. [As for he scalng assumpon (3.4, he random measure log Z ( s a funconal of he underlyng Markov process, n whch he me has been speeded up by a facor 2 and he space shrunk by 1. Assume also he sequence of nal emprcal measures log Z (, aken a me =, converges n probably o some deermnsc measure wh a gven densy ρ(x,, so ha, n probably, ( lm log Z = 1 for any par of funcons φ a, φ b C[T. [ρ(x, φ a (x, + (1 ρ(x, φ b (x, dx, (3.5 Then, for every >, he sequence of random measures µ ( converges n probably, as, o a deermnsc measure havng a densy ρ(x, wh respec o he Lebesgue measure, whch s he unque weak soluon of he Cauchy problem T 1 [ ( φ(x, ρ(x, + λ 2 φ(x, x 2 µρ(x, ( 1 ρ(x, φ(x, x where (3.6 holds for any funcon φ C[T. + 1 ρ(x, φ(x, dx =, dxd (3.6
3.1 Exsence of lm pons: sequenal compacness 7 If, moreover, one assumes he exsence of 2 ρ(x,, for ρ(x, gven, hen x 2 (3.6 reduces o a classcal Burgers equaon ρ(x, = λ 2 ρ(x, ρ(x, x 2 + µ[1 2ρ(x,. x Proof. The proof s conaned n he nex hree subsecons. 3.1 Exsence of lm pons: sequenal compacness As usual n problems dealng wh convergence of sequences of probably measures, our very sarng pon wll be o esablsh he weak relave compacness of he se {log Z (, 1}. Some of he probablsc argumens employed n hs paragraph are classcal and can be found n he leraure, e.g. [18, 14, alhough for slghly dfferen or smpler models. Leng φ a, φ b be wo arbrary funcons n C[T, we refer o equaon (3.2. Usng he exponenal form of Z ( and Lemma [A1-5.1 n [14 (see also chaper 3 n [7 for relaed calculus, one can easly check ha he wo followng random processes U ( V ( def = Z ( Z ( def = (U ( 2 ( ( Ω ( [Z ( s Ω ( [(Z ( s are bounded {F ( }-marngales, where [ φa (, θ ( Seng now def = 1 ψ xy ψ xy (, we have where λ ( xy (, G ( A ( def = φ x φ y = ψ yx, + θ ( s Z s ( ds, (3.7 2 2Z ( Ω ( [Z ( ds (3.8 ( + φ ( b, ( def + 1 ( = ψ xy, ψ xy,, ( 1 def = λ xy ( L ( def = [ exp Ω ( [Z ( λ( G ( ψ xy = L ( ab (, A B +1 + s B ( s (. (3.9 (, 1, xy = ab or ba, Z (, (3.1 λ ( ba (, B A +1. (3.11
8 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP On he oher hand, a sraghforward calculaon n equaon (3.11 allows o rewre (3.8 n he form V ( where he process R ( R ( = G ( = (U ( 2 (Z ( s s srcly posve and gven by [ λ ( ab (, 2 A B +1 + λ ab ( 2 R s ( ds, (3.12 ( [ λ ba (, 2 λ ba ( B A +1. The negral erm n (3.12 s nohng else bu he ncreasng process assocaed wh Doob s decomposon of he submarngale (U ( 2. The folllowng esmaes are crucal. Lemma 3.2. L ( = O(1, (3.13 R ( = O ( 1. (3.14 Proof. We wll derve (3.13 by esmang he rgh-hand sde member of equaon (3.11. From now on, for he sake of shorness, he frs and second paral dervaves of ψ(z, wh respec o z wll be denoed respecvely by ψ (z, and ψ (z,. ( Clearly, ψ xy, = 1 xy( ψ, + O ( 1. Then, akng a second order 2 expanson of he exponenal funcon and usng equaons (3.3 and (3.4, we can rewre (3.11 as L ( = µ( + λ( G ( [ A + A +1 2 G ( (A A +1 ψ ab A A +1 ψ ab (, (, ( 1 + O. (3.15 The frs sum on he rgh n (3.15 s unformly bounded by a consan dependng on ψ. Indeed A 1, and ψ s of bounded varaon snce ψ C[T. As for he second sum comng n (3.15, we have ( (A A +1 ψ ab, = [ ( + 1 ( A +1 ψ ab, ψ ab,. G ( G ( Then he dscree Laplacan ψ ab ( + 1, ψ ab (, ψ ab ( + 2, 2ψ ab ( + 1, +ψ ab (,
3.1 Exsence of lm pons: sequenal compacness 9 adms of he smple expanson ( + 1 ( ψ ab, ψ ab, = 1 ( ( 1 2 ψ ab, + O 2. (3.16 By (3.4, λ( = λ 2 + o( 2, so ha (3.16 mples λ( G ( (A A +1 ψ ab (, = G ( λa ( +1 ( 1 ψ ab, + o = O(1, (3.17 whch concludes he proof of (3.13. The compuaon of R ( (3.14 can be obaned va smlar argumens, remarkng ha leadng o R ( [φ a, φ b = L ( [2φ a, 2φ b 2L ( [φ a, φ b. To show he relave compacness of he famly Z (, whch from he separably and he compleeness of he underlyng spaces s here equvalen o ghness, we proceed as n [14 by means of he followng useful creron. Proposon 3.3 (Aldous s ghness creron, see [1. A sequence {X ( } of random elemens of D R [, T s gh (.e. he dsrbuons of he {X ( } are gh f he wo followng condons hold: ( lm a where X ( def = sup X (. T lm sup P [ X ( a =, (3.18 ( For each ɛ, η, here posve numbers δ and, such ha, f δ δ and, and f τ s an arbrary soppng me wh τ + δ T, hen P [ X ( τ+δ X τ ( ɛ η. (3.19 oe ha condon (3.18 s always necessary for ghness. We shall now apply Lemma 3.2 o equaons (3.7 and (3.12, he role of X ( n Proposon 3.3 beng played by Z (. The random varables Z ( and θ ( are clearly unformly bounded, so ha condon (3.18 s mmedae. To check condon (3.19, rewre (3.7 as +δ Z ( +δ Z ( = U ( +δ U ( + (L ( s + θ s ( Z s ( ds. (3.2
1 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP For [, T, he negral erm n (3.2 s bounded n modulus by Kδ (K beng a consan unformly bounded n and ψ, and sasfes (3.19, whenever s replaced by an arbrary soppng me. We are lef wh he analyss of U (. Bu, from (3.12, (3.14 and Doob s nequaly for submarngales, we have E [ (U ( +δ U ( 2 [ +δ = E (Z s ( P [ sup T U ( ɛ 4 ɛ 2 E [ T (Z ( s 2 R ( s 2 R ( s ds ds Cδ, 4CT ɛ 2, (3.21 where C s a posve consan dependng only on ψ. Thus U ( n probably, as. Ths las propery, ogeher wh (3.7, (3.2 and assumpon (3.5, yeld (3.19 and he announced (weak relave compacness of he sequence Z (. Hence, he sequence of probably measures Q (, defned on D M [, T and correspondng o he process µ (, s also relavely compac: hs s a consequence of classcal projecon heorems (see for nsance Theorem 16.27 n [13. We are now n a poson o esablsh a furher mporan propery. Le Q he lm pon of some arbrary subsequence Q (k, as k, def and Z = lm k Z ( k. Then he suppor of Q s a se of sample pahs absoluely connuous wh respec o he Lebesgue measure. Indeed, he applcaon µ sup T log Z s connuous and we have he mmedae bound sup log Z T 1 [ φ a (x, + φ b (x, dx, whch holds for all ψ a, ψ b C 2 [, 1. Hence, by weak convergence, any lm pon Z has he form [ 1 Z [φ a, φ b = exp [ρ(x, φ a (x, + (1 ρ(x, φ b (x, dx, (3.22 where ρ(x, denoes he lm densy [whch a pror s a random quany of he sequence of emprcal measures µ ( k nroduced n (3.1. 3.2 A funconal negral operaor o characerze lm pons Ths s somehow he Gordan kno of he problem. Relyng on he above weak compacness propery, our nex resul shows ha any arbrary lm pon Q s concenraed on a se of rajecores whch are weak soluons of an negral equaon.
3.2 A funconal negral operaor o characerze lm pons 11 ( The man dea s o consder for a whle he 2 quanes φ a, (, φ b,, 1, as ordnary free varables, whch for he sake of shorness wll be denoed respecvely by x ( and y (. Wh hs approach, he problem of he hydrodynamc lm wll appear o be mosly of an analycal naure. Le [ ( x ( α xy ( (, = λ ab ( exp [ ( y ( α yx ( (, = λ ba ( exp +1 x( +1 y( + y ( + x ( y ( +1 x ( +1 1, 1. Then, usng (3.7, (3.9, (3.1, (3.11 and he defnon of Z (, we oban mmedaely he followng funconal paral dfferenal equaon (FPDE where L ( d(z ( U ( d s he operaor L ( [h def = 2 α ( G ( xy (, = L ( [Z ( 2 h x ( y ( +1 + θ ( Z ( + α yx ( (, More precsely, nroducng he famly of cylnder ses wh, (3.23 2 h y ( x ( +1 (p def V = [ Φ, Φ p, p = 1, 2..., (3.24 Φ def = sup ( φa (z,, φ b (z,, (3.25 (z, [,1 [,T we see mmedaely ha L ( acs on a subspace of C (V (2, snce Z ( s analyc wh respec o he coordnaes {φ a (.,, φ b (., }, for each fne (hngs wll be made more precse n secon 3.2.3. eedless o say ha L ( s no of parabolc ype, as he quadrac form assocaed wh he second order dervave erms s clearly non defne (see e.g. [6. In addon, equaon (3.23 s a well defned sochasc FPDE, as all underlyng probably spaces emanae from famles of neracng Posson processes. ow, mgh be worh accounng for he use of he word funconal above, and for he reason of solang he hrd erm on he rgh n (3.23. Indeed, by (3.9, θ ( s a funconal nvolvng also a paral dervave of Z ( wh respec o, as we can wre. θ ( Z ( = 1 G ( [ Z ( x ( x ( + Z ( y ( y ( = ( Z. (3.26
12 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP The las equaly n (3.26 mgh look somewha formal, bu wll come ou more clearly n Secon 3.2.3. oe also, by he same argumen whch led o (3.22, ha we have lm k θ( k = 1 [ ρ(x, φ a (x, + (1 ρ(x, φ b (x, dx. (3.27 Our essenal agendum s o prove ha any lm pon of he sequence of random measures µ = lm weak k µ( k sasfes an negral equaon correspondng o a weak soluon (or dsrbuonal n Schwarz sense of a Cauchy ype operaor. To overcome he chef dffculy, namely he behavour of he lm sum n (3.23, we propose a seemngly new approach, whch s anamoun o analyzng he famly of second order lnear paral dfferenal operaors L ( along he sequence k. As brefly emphaszed n he remark a he end of hs secon, a brue force analyss of (3.23 would lead o a dead end. Indeed an mporan prelmnary sep consss n exracng he juce of he esmaes obaned n Lemma 3.2, o rewre he operaor L ( n erms of only prncpal varables, up o quanes of order O( 1. Ths s summarzed n he nex lemma. Lemma 3.4. The followng FPDE holds. where A ( d(z ( U ( d def = A ( [Z ( + θ ( Z ( ( 1 + O, (3.28 s vewed as an operaor wh doman C (V ( such ha A ( [g def = ( [ µψ ab, 1 2 G ( + λ ψ ab G ( (, g x ( +1 ( g x (, + g x ( +1 2 g x ( x ( +1 (3.29 he erm O ( 1 beng n modulus unformly bounded by C, where C denoes a consan dependng only on he quany sup {ψ(x,, ψ (x,, ψ (x, }. (x, [,1 [,T Proof. The resul follows by elemenary algebrac manpulaons from equaons (3.4, (3.7, (3.1, (3.15, (3.17, and deals wll be omed.
3.2 A funconal negral operaor o characerze lm pons 13 Takng Lemma 3.4 as a sarng pon, we skech ou below n Secons 3.2.1 and 3.2.2 he man lnes of our analycal approach, whch ndeed can be brefly summarzed by means of some keywords. Couplng, aken here n Skohorod s conex. Regularzaon and funconal negraon. Regularzaon refers o he fundamenal mehod used n he heory of dsrbuons o approxmae eher funcons or dsrbuons. We shall apply o he FPDE (3.28, consderng and he values of he funcon φ a (., (aken a pons of he orus as + 1 ordnary varables. Then, passng o he lm as, we nroduce convenen funconal negrals ogeher wh varaonal dervaves. Ths mgh lkely exend o much wder sysems, alhough hs asseron could ceranly be debaed. 3.2.1 Inerm reducon o an almos sure convergence seng Ths can be acheved by means of he exended Skohorod couplng (or ransfer heorem (see Corollary 6.12 n [13, whch n bref says ha, f a sequence of real random varables (ξ k s such ha lm k f k (ξ k = f(ξ converges n dsrbuon, hen here exs a probably space V and a new random sequence ξ k, such ha ξ L k = ξk and lm k f k ( ξ k = f(ξ, almos surely n V, wh ξ = L ξ. Here hs heorem wll be appled o he famly Z ( k, whch hus gves rse o a new sequence, named Y ( k n he sequel. Clearly hs sep s n no way oblgaory, bu raher a maer of ase. Indeed, one could sll keep on workng n a weak convergence conex, wh Alexandrov s pormaneau heorem (see e.g. [7 whenever needed. 3.2.2 Consderng (3.29 as a paral dfferenal operaor wh consan coeffcens For each fne, we consder he quanes ψ ab ( (,, ψ ab,, = 1,...,, as consan parameers, whle he x ( s wll be aken as free varables from a varaonal calculus pon of vew. Ths s clearly feasble, rememberng ha, by defnon, ψ ab = φ a φ b, for all φ a, φ b C[T. Hereafer, wll be vewed as an exogeneous mue varable, no parcpang concreely n he proposed varaonal approach. Then, accordng o he noaon nroduced n secon 3.2.1, we can rewre
14 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP (3.28 n he form d(y ( U ( d def = A ( [Y ( + θ ( Y ( ( 1 + O, (3.3 where θ ( s sll gven by (3.9, keepng n mnd ha all random varables n (3.3 are defned wh respec o hs new (hough unspecfed probably space nroduced n secon 3.2.1 above. In parcular, from he ghness proved n secon 3.1, leng along some subsequence k, we have lm Y ( k [φ a, φ b a.s. Y [φ a, φ b. k 3.2.3 Analyss of he FDPE (3.3 The dea now s o propose a regularzaon procedure, whch consss n carryng ou he convoluon of (3.3 wh a suably chosen es funcon, nong ha Y ( nowhere vanshes and s unformly bounded. As usual, he convoluon f g of wo negrable funcons f, g C (V (, wh V ( gven n (3.24, wll be defned by (f g(u = f(u vg(vdv. V ( Le ω be he funcon of he real varable z defned by exp ( 1 ω(z def z f z <, = f z, where wll be convenen o wre ω (z def = dω(z dz def and ω (z def = d2 ω(z. dz 2 Seng x ( = (x ( 1, x( 2,..., ( x(, wh x( = φ a,, 1, we nroduce he followng famly of posve es funcons χ ( ε C (V (, ( 1 χ ε ( ( x ( = ω (x ( 2 ε, 2 ε. (3.31 =1 oe: I s worh keepng n mnd ha, as ofen as possble, he me varable wll be omed n mos of he mahemacal quanes, e.g. x ( (. Indeed, as menoned before, plays n some sense he role of a parameer. From (3.3, follows mmedaely ha ( d(y ( U ( χ ε ( d ( x ( = ( A ( + ( (θ ( [Y ( Y ( χ ( ( x ( ε ( χ ( ( x ( 1 + O. ε (3.32
3.2 A funconal negral operaor o characerze lm pons 15 ow, by (3.29 and accordng o he saemen made n Secon 3.2.2, he frs erm n he rgh-hand sde member of (3.32 can be negraed by pars, so ha, for any fne and ε suffcenly small, ( (Ã( A ( [Y ( χ ε ( ( x ( = [χ ( ε Y ( ( x (, (3.33 where Ã( s by defnon he adjon operaor of A ( n he Lagrange sense. Here he doman of à ( consss of all funcons h ( of he form [ 1 h ( = exp dσ a ( (xv [φ a (x, + dσ ( (xv [φ b (x,, where V : C[T R + sands for an arbrary analyc funcon; σ ( a and σ ( b are arbrary dscree probably measures on G (. Clearly χ ε ( belongs o he doman of à (. Under he assumpons made n Secon 3.2.2, a drec negraon by pars n (3.29 yelds he formula à ( [h = λ G ( µψ ab G ( ψ ab ( [ (, 1 h 2 x ( (, h x ( +1 where Ã( has been defned n (3.33. Le, for each φ φ a C[T, χ ε (φ def = lm χ( ε, ( 1 ( x ( = ω b + h x ( +1 + 2 h x ( x ( +1 (3.34 φ 2 (x, dx ε 2, (3.35 where he negral n (3.35 s readly obaned as he lm of he Remann sum n (3.31. Lemma 3.5. For each φ(x, C[T, he followng lm holds unformly. lm à ( [χ ε ( ( x ( = 1 [ µψ ab (x, K(φ, x, + λψ ab (x, H(φ, x, dx, (3.36 wh ( 1 H(φ, z, = 2φ(z, ω φ 2 (u, du ε, 2 ( 1 K(φ, z, = H(φ, z, + 4φ 2 (z, ω φ 2 (u, du ε. 2 (3.37
16 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP Proof. For fxed, he funcon Ã( [χ ( ε has paral dervaves of any order wh respec o he coordnaes x (, = 1,...,. In parcular, we ge from (3.31, for each G (, χ ( ε x ( 2 χ ( ε x ( x ( +1 ( ( x ( = 2 x( 1 ω ( x ( = 4 x( x ( +1 ω =1 ( 1 ( x ( =1 2 ε 2 ( x (, 2 ε 2 So we are agan lef wh Remann sums, whch, for any connuous funcon k(x,, yeld a once and lm k G ( (, χ ( ε x ( ( x ( = ( 1 1 2ω φ 2 (x, dx ε 2 k(x, φ(x, dx, lm G ( k ( 2, χ ( ε x ( x ( +1 ( x ( = ( 1 1 4ω φ 2 (x, dx ε 2 k(x, φ 2 (x, dx. Keepng n mnd ha, for all, he vecor x ( (from s very defnon mus be a dscrezaon of some funcon elemen n C[T, he las mporan sep o derve (3.36 requres o gve a precse meanng o he lm lm χ ( ε ( x ( d x (, V ( allowng o carry ou funconal negraon and varaonal dfferenaon. In hs respec, le us emphasze (f necessary a all! ha he usual consrucons of measures and negrals do no apply n general when he doman of negraon s an nfne-dmensonal space of funcons or mappngs, all he more because a complee axomac for funconal negraon does no really exs; ndeed each case requres he consrucon of ad hoc generalzed measures, see promeasures n [2, 3 or quas-measures n [16. These quesons prove o be of mporance n varous problems relaed o heorecal physcs. In our case sudy, negraon over pahs belongng o C[T needs o be properly consruced. The pon s o defne, as, a volume elemen denoed by δ(φ. We shall no do here, bu hs could be acheved by mmckng classcal.
3.2 A funconal negral operaor o characerze lm pons 17 fundamenal approaches, see e.g. [2, 3, 16. The man ool s he mporan F. Resz represenaon heorem, whch o every posve lnear funconal le correspond a unque posve measure. ow from equaon (3.35 s permssble o nroduce he normalzed es funconal χ ε (φ = χ ε(φ D, where D s chosen o ensure C ([ Φ, Φ χ ε (φ δ(φ = 1, and Φ s gven by (3.25, so ha D = C ([ Φ, Φ ω ( 1 φ 2 (x, dx ε 2 δ(φ. From Skohorod s couplng heorem, Y does sasfy an equaon of he form (3.22. Hence, we can wre he followng funconal dervaves (whch are planly of a Radon-ykodym naure Y φ = ρ(., Y, 2 Y φ 2 = ρ2 (., Y. (3.38 ow everyhng s n order o complee he puzzle, accordng o he followng seps. 1. Frs, usng (3.33, rewre (3.32 as ( d(y ( U ( χ ε ( d (Ã( ( x ( = + ( (θ ( [χ ( ε Y ( Y ( ( x ( ( χ ( ( x ( 1 + O. ε (3.39 2. Le n (3.39 and hen replace χ ε (φ by χ ε (φ, rememberng ha by (3.21 U ( = O(1/ unformly. 3. Carry ou wo funconal negraon by pars n equaon (3.36 by makng use of (3.38. 4. Fnally, negrae on [, T, le ε and swch back o he orgnal probably space, where Z ( [φ a, φ b, by secon 3.2.1, converges n dsrbuon o Z [φ a, φ b : hs yelds exacly he announced Cauchy problem (3.6.
18 References Hence, he famly of random measures µ ( converges n dsrbuon o a deermnsc measure µ, whch n hs pecular case mples also convergence n probably. 3.3 Unqueness The problem of unqueness of weak soluons of he Cauchy problem (3.6 for nonlnear equaon s n fac already solved n he leraure. For a wde bblography on he subjec, we refer he reader for nsance o [6. The proof of Theorem 3.1 s concluded 4 Conjecure for he n-speces model We wll sae a conjecure abou hydrodynamc equaons for he n-speces model, brefly nroduced n secon 2.1, n he so-called equdffusve case, precsely descrbed hereafer. Defnon 4.1. The n-speces sysem s sad o be equdffusve whenever here exss a consan λ, such ha, for all pars (k, l, Then, leng λ kl ( lm 2 = λ. [ kl def λ kl α = lm log ( λ lk, ( we asser he followng hydrodynamc sysem holds. ρ k = β 2 ρ k x 2 + ( α lk ρ k ρ l, k = 1,..., n. x l k The dea s o apply he funconal approach presened n hs paper: hs s he subjec maer of an ongong work. References [1 P. Bllngsley. Convergence of Probably Measures. Wley Seres n Probably and Sascs. John Wley & Sons Inc., 2 edon, 1999. [2. Bourbak. Inegraon, Chapre IX. Hermann, Pars, 1969.
References 19 [3 P. Carer and C. DeW-Moree. A rgorous mahemacal foundaon of funconal negraon. In Funconal Inegraon : Bascs and Applcaons, volume 361 of ATO ASI - Seres B: Physcs, pages 1 5. Plenum Press, 1997. [4 A. De Mas and E. Presu. Mahemacal Mehods for Hydrodynamc Lms, volume 151 of Lecure oes n Mahemacs. Sprnger-Verlag, 1991. [5 B. Derrda, M. Evans, V. Hakm, and V. Pasquer. Exac soluon for 1d asymmerc excluson model usng a marx formulaon. J. Phys. A: Mah. Gen., 26:1493 1517, 1993. [6 Y. Egorov and M. Shubn, edors. Paral Dfferenal Equaons, volume I-II-III of Encyclopeda of Mahemacal Scences. Sprnger Verlag, 1992. [7 S. Eher and T. Kurz. Markov Processes, Characerzaon and Convergence. John Wley & Sons, 1986. [8 M. Evans, D. P. Foser, C. Godrèche, and D. Mukamel. Sponaneous symmery breakng n a one dmensonal drven dffusve sysem. Phys. Rev. Le., 74:28 211, 1995. [9 M. Evans, Y. Kafr, M. Koduvely, and D. Mukamel. Phase Separaon and Coarsenng n one-dmensonal Drven Dffusve Sysems. Phys. Rev. E., 58:2764, 1998. [1 G. Fayolle and C. Furlehner. Dynamcal Wndngs of Random Walks and Excluson Models. Par I: Thermodynamc lm n Z 2. Journal of Sascal Physcs, 114(1/2:229 26, January 24. [11 G. Fayolle and C. Furlehner. Sochasc deformaons of sample pahs of random walks and excluson models. In Mahemacs and compuer scence. III, Trends Mah., pages 415 428. Brkhäuser, Basel, 24. [12 G. Fayolle and C. Furlehner. Sochasc Dynamcs of Dscree Curves and Mul-Type Excluson Processes. Journal of Sascal Physcs, 127(5:149 194, 27. [13 O. Kallenberg. Foundaons of Modern Probably. Sprnger, 2 edon, 22. [14 C. Kpns and C. Landm. Scalng lms of Ineracng Parcles Sysems. Sprnger-Verlag, 1999. [15 T. M. Lgge. Sochasc Ineracng Sysems: Conac, Voer and Excluson Processes, volume 324 of Grundlehren der mahemaschen Wssenschafen. Sprnger, 1999.
2 References [16 E. V. Maïkov. τ-smooh funconals. Tans. Moscow Mah. Soc., 2:1 4, 1969. [17 W. Rudn. Funconal Analyss. Inernaonal Seres n Pure and Appled Mahemacs. McGraw-Hll, 2 edon, 1991. [18 H. Spohn. Large Scale Dynamcs of Ineracng Parcles. Sprnger, 1991.