Noise prevents collapse of Vlasov Poisson point charges

Transcription

1 Nose prevens collapse of Vlasov Posson pon charges Franços Delarue, Franco Flandol, Daro Vncenz To ce hs verson: Franços Delarue, Franco Flandol, Daro Vncenz. Nose prevens collapse of Vlasov Posson pon charges. Communcaons on Pure and Appled Mahemacs, Wley, 214, 67, pp <hal v2> HAL Id: hal hps://hal.archves-ouveres.fr/hal v2 Submed on 5 Jul 212 HAL s a mul-dscplnary open access archve for he depos and dssemnaon of scenfc research documens, wheher hey are publshed or no. The documens may come from eachng and research nsuons n France or abroad, or from publc or prvae research ceners. L archve ouvere plurdscplnare HAL, es desnée au dépô e à la dffuson de documens scenfques de nveau recherche, publés ou non, émanan des éablssemens d ensegnemen e de recherche franças ou érangers, des laboraores publcs ou prvés.

2 Nose prevens collapse of Vlasov Posson pon charges F. Delarue, F. Flandol, D. Vncenz Absrac We elucdae he effec of nose on he dynamcs of N pon charges n a Vlasov-Posson model drven by a sngular bounded neracon force. A oo smple nose does no mpac he srucure nhered from he deermnsc case and, n parcular, canno preven coalescence. Inspred by he heory of random ranspor n passve scalars, we denfy a class of random felds whch generae random pulses ha are chaoc enough o dsorganze he deermnsc srucure and preven any collapse of he parcles. We oban srong unque solvably of he sochasc model for any nal confguraon of dfferen pon charges. In he case where here are exacly wo parcles, we mplemen he vanshng nose mehod for deermnng he connuaon of he deermnsc model afer collapse. 1 Inroducon I s a well-known fac ha whe nose perurbaons mprove he wellposedness properes of Ordnary Dfferenal Equaons (ODEs, n parcular her unqueness; see for nsance Krylov and Röckner [19]. The nfluence of nose on pahologes of Paral Dfferenal Equaons (PDEs s no as well undersood. A revew of recen resuls n he drecon of unqueness can be found n [12]. On he conrary, wheher nose can preven he emergence of sngulares n PDEs s sll que obscure. A furher challengng queson s wheher nose can selec a naural canddae for he connuaon of soluons afer he sngulares. A well-known sysem n whch sngulares may develop explcly s he Vlasov Posson equaon on he lne. We refer he reader o [26] for several 1

3 examples and for an exensve dscusson of he underlyng sakes, ncludng he connecon wh he 2D Euler equaons. (See also [5], [27] and [34]. The movaon for he presen sudy s o undersand he nfluence of nose on such sngulares. 1.1 Vlasov Posson equaon on he lne Consder he followng sysem n he unknown f : [, R R (, x, v f (, x, v R: f f (, x, v + v (, x, v + E(, x f x v (, x, v =, f(, x, v = f (x, v, ρ (, x = f (, x, v dv, E (, x = F (x y ρ (, y dy, R where F (x s a bounded funcon ha s connuous everywhere excep a x = and has sde lms n + and. If F (x = sgn(x (wh sgn( =, Eq. (1 s he 1D Vlasov Posson model descrbng he evoluon of he phasespace densy f of a sysem of elecrons (n naural uns, where he elemenary charge and he mass of he elecron are se equal o one. Such an equaon s known o develop sngulares n he case of measure-valued soluons, see [26] and he works dscussed heren. For nsance, s possble o desgn examples of so-called elecron-shee srucures ha perss n posve me, bu collapse o one pon a a ceran me (f s an elecron shee f s concenraed on lnes,.e. reads f (x, v = f (x δ(v v (x. A smplfed verson of hs phenomenon s he coalescence of N pon charges n fne me: as shown below, here are examples of nal condons of he form f (x, v = N =1 a δ(x x δ(v v, wh dfferen pars (x 1, v 1,..., (x N, v N R 2, for whch f remans of he same form on some nerval [, bu degeneraes no f (, x, v = δ(x x δ(v v a some me, wh (x, v R 2. The man queson movang our work s he followng: does he presence of nose modfy hese facs? In hs framework, a naural way o conceve a nosy verson of (1 may be he followng: when he elecrc charge s no oally solaed, bu lves n a medum (a sor of elecrc bah, a random exernal force adds o he force generaed by he elecrc feld. Assumng ha he elecrc charge does no affec he exernal random feld, he smples srucure modellng hs suaon s a Sochasc PDE (SPDE of he form f f (, x, v + v (, x, v + x R ( E(, x + ε dw d 2 (1 f (, x, v =, (2 v

4 where W s a Brownan moon and he Sraonovch negral s used (hs s he naural choce when passng o he lm along regular noses. Unforunaely, he nose n Eq. (2 s oo smple o avod he emergence of sngulares such as hose descrbed above. Indeed, he random feld f(, x, v = f(, x + ε W sds, v + εw formally sasfes [ f + v f x + Ẽ f ](, x, v = wh v Ẽ(, x = F (x y f(, y, vdydv, R 2 so ha any concenraon pon z = (x, v of f a some me ranslaes no he random concenraon pon (x ε W sds, v εw of f a he same me. To expec a posve effec of he nose, we mus consder noses of a refned space srucure. We consder a nosy equaon of he form [ d f + v f ( ] f x d + E(, xd + ε σ k (x dw k (, x, v =, (3 v k=1 where ((W k k 1 s a famly of ndependen Brownan moons, and we prove ha, under very general condons on he covarance funcon Q (x, y = k=1 σ k (x σ k (y, he followng resul holds: Theorem 1 Gven f (x, v = N =1 a δ (z z as nal condon, wh he generc noaon z = (x, v and wh dfferen nal pons z R 2, = 1,..., N, here s a unque global soluon o sysem (3 of he form f (, x, v = N =1 a δ (z z (, where ((z ( 1 N s a connuous adaped sochasc process wh values n R 2N whou coalescence n R 2,.e., wh probably one, z ( z j ( for all and 1, j N, j. The precse assumpons of Theorem 1 and he defnons used heren wll be clarfed nex. Here, s worh remarkng ha our sudy does no cover he case of an elecron shee. We noneheless expec our resul o be a frs sep forward n hs drecon snce he number N of parcles s here arbrary for a gven covarance funcon Q(x, y. Indeed, he assumpon ha we shall mpose on Q (x, y guaranees ha any N-uple, wh arbrary N, of dfferen pons (z 1,..., z N n he (x, v-space feels he orgnal nose as random mpulses ha are no oo coordnaed one wh each oher. Such a propagaon may be seen as a sor of mld spaal chaos produced by he nose. The negave example dscussed above does no enjoy a smlar propery snce he nose (εw plugged heren produces he same mpulse a every space pon, hus acng as a random Gallean ransformaon. 3

5 1.2 Non-Markovan connuaon afer a sngulary As menoned above, he random perurbaon nroduced n Eq. (3 may provde some ndcaons abou he naural connuaon of he soluons afer he coalescence of wo pon charges. (More dffcul or generc cases are no clear a hs sage of our undersandng of he problem. Consder he smple example n whch F (x = sgn (x and f (x, v = 1 2 δ (z z δ (z z 2, z 1 = ( 1, v, z 2 = (1, v wh v >. As we shall dscuss below, he Lagrangan dynamcs correspondng o (1 consss of he sysem dx d ( = v (, dv d ( = F (x ( xī(,, (4 for = 1, 2 and ī = 2 f = 1 and vce versa, wh (x (, v ( = (ɛ, ɛ v as nal condon, wh ɛ = 1 f = 2 and ɛ = 1 f = 1. The funcons ( v ( = ɛ ( v, x ( = ɛ 1 v + 2 (5 2 solve he sysem for [, v, and he lms of x 1 ( and x 2 ( as v concde when v 2 = 2. Ths means ha, wh he choce v = 2, he soluons (x (, v (, = 1, 2, converge o he same pon (, as v, so ha he orgn (, s a sngular pon of he Lagrangan dynamcs. On he conrary, Theorem 1 saes ha, for any posve level of nose ε n he nosy formulaon (3, he random soluons ((x ε (, v ε ( =1,2 never mee, wh probably one. I s hen naural o nvesgae he behavor of he sochasc soluon as ε. In Secon 2, we shall prove, under general condons on he covarance funcon Q: Theorem 2 Assume ha v = 2. Then, as ε, he par process ((z ε ( =1,2 converges n dsrbuon on he space C([,, R 2 R 2 oward 1 2 δ( (z1(, z2( δ( (z1 (, z2 (, (6 where (z ( = (x (, v (, for and = 1, 2, and (z1 (, z2 ( maches (z1(, z2( for 2 and (z2(, z1( for > 2. 4

6 Theorem 2 mus be seen as a rule for he connuaon of he soluons o he deermnsc sysem (1 afer sngulary. When he parcles mee, hey spl nsananeously, bu hey can do n wo dfferen ways: ( wh probably 1/2, he rajecores mee a coalescence me and hen spl whou crossng each oher, namely each of he wo rajecores keeps of consan sgn before and afer coalescence; ( wh probably 1/2, he rajecores mee, cross each oher and hen spl for ever, namely he sgn of each of hem changes a coalescence me exacly. Ths sands for a mahemacal descrpon of he repulsve effec of he neracon force F : here s no way for he parcles o merge and hen form a sngle parcle wh a double charge. Ths suaon sounds as a physcal loss of he Markov propery: jus afer coalescence, splng occurs because he sysem keeps memory of wha s sae was before. Precsely, f we model he dynamcs of a sac sngle parcle wh a double charge by a par (z1 (, z2 ( n phase space, subjec o z1 ( = z2 ( and v 1 ( = v 2 ( =, for, we ge a non-markovan famly of soluons o (4. When he rajecores z1 and z2 mee, hey do no resar wh he same dynamcs as z1 and z2 do. We refer he reader o [9] for oher mahemacal examples of non-markovan connuaons. 1.3 Vlasov Posson ype sysem of N parcles n R d The problem descrbed n Secon 1.1 wll be reaed as a parcular case of he followng generalzaon n R d : [ d f + (v x f + E(, x v f d + k=1 ] σ k (x v f dw k (, x, v =, subjec o smlar consrans as n (1, where σ k : R d R d are Lpschzconnuous felds subjec o addonal assumpons, whch wll be specfed laer (see (A.2 3 n Secon 3, and ((W k k N\{} are ndependen onedmensonal Brownan moons. Below, F wll be assumed o be bounded and locally Lpschz connuous on any compac subse of R d \ {}, he Lpschz consan on any rng of he form {x R d : r x 1} growng a mos as 1/r as r ends o. In parcular, F may be dsconnuous a. A relevan case s when F = U where U s a poenal wh a Lpschz pon a zero, namely U s Lpschz connuous on R d and smooh on R d \ {}. Ths framework ncludes he example F (x = x/ x, x R d, and, as a parcular case, he one-dmensonal model dscussed above,.e. F (x = 5

7 sgn(x, x R. On he conrary, he d-dmensonal Posson case, where F (x = ±x x d, x R d, d 2, (7 does no sasfy he aforemenoned assumpons, and herefore falls ousde hs sudy. The sgns + and - descrbe repulsve and aracve neracons respecvely; correspondng models are referred o as elecrosac and gravaonal respecvely. For he elecrosac poenal, our analyss urns ou o be rrelevan n dmenson d 2 snce he deermnsc sysem self s free of coalescence. The Lagrangan moon assocaed wh he SPDE s dx d = V, dv = a j F ( X X j j d + k=1 σ k ( X dw k, (8 for and 1 N. The measure valued process f (, x, v = N a δ ( ( x X δ v V =1 (9 solves he SPDE n weak form, from Iô s formula n Sraonovch form appled o ( 1 N a ϕ(x, V, for a es funcon ϕ : R d R d R. Ths paper s devoed o he analyss of sysem (8. Takng hs sysem as a sarng pon, he problem would be much easer o handle f each parcle were o be forced wh s own Brownan moon, ndependenly of he oher ones. Ths choce of he nose, however, would break he relaon beween he Lagrangan dynamcs and he SPDE nroduced above. The paper s organzed as follows. In Secon 2, we sar wh he proof of Theorem 2, as we feel o have a srkng nerpreaon from he physcal pon of vew. The vanshng nose mehod for selecng soluons o sngular dfferenal equaons goes back o he earler paper by [3], bu he examples nvesgaed heren are one-dmensonal only. (See also [12], ogeher wh [15] and [2]. Here Theorem 2 apples o a four-dmensonal sysem. In [3], he vanshng nose behavor of he sochasc dfferenal equaon a hand s manly of analycal essence. Below, he proof of Theorem 2 reles on a new approach: pahwse, he asympoc dynamcs of (z ε =1,2 are expanded wh respec o he parameer ε ll coalescence occurs; he possble lm regmes hen read on he (random coeffcens of he expanson. The resul 6

8 s obaned under general condons on he covarance marx Q: specfc examples are gven n Secon 3. The remander of he paper s devoed o he proof of Theorem 1. We frs dscuss he srucure of he nose ha wll preven coalescence from emergng. I should be emphaszed ha he effec of he nose on he 2Nsysem (8 s hghly non-rval. Indeed, alhough s doubly sngular, he nose makes he sysem flucuae enough o avod pahologcal phenomena such as hose observed n he deermnsc case. The frs sngulary s due o he fac ha he same Brownan moons ac on all he parcles. If he (σ k k N\{} were consan, he parcles would feel he same mpulses and he nose would no have any real effec; bascally, would jus ac as a random ranslaon of he sysem, as n (2. Thus he pon s o desgn a nose allowng dsplacemens of dfferen parcles n dfferen drecons. To reach he desred effec, we requre he covarance marx (Q(x, x j 1,j N o be srcly posve for any vecor (x 1,..., x N (R d N wh parwse dfferen enres. We gve examples n Secon 3. These examples are nspred from he Krachnan nose used n he heory of random ranspor of passve scalars. (See for example [11], [22] and [29]. However, he model consdered here does no have he same nerpreaon as he Krachnan model, snce he nose here acs ono he veloces. A possble way o relae Eq. (8 o urbulence heory would conss n penalzng he drf of he velocy of he h parcle by V. Ths model would descrbe he moon of neracng heavy parcles n a flud whose velocy s a random feld, see [4]. The second sngulary of he model s nhered from he knec srucure of he deermnsc counerpar: he nose acs as an addonal random force only, namely s plugged no he equaon of he velocy only. In oher words, he coupled sysem for (X, V 1 N s degenerae. We shall show n Subsecon 3.3 ha he ellpcy properes of he nose n R Nd acually lf up o hypoellpcy properes n R 2Nd. Once he se-up for he nose s defned, we are ready o ackle he problem of no-coalescence. We frs esablsh ha he Lagrangan dynamcs are well posed for Lebesgue a.e. nal confguraon of dfferen parcles. Ths does no requre any specal feaures of he nose. Specfyng he form of he nose accordng o he requremens dscussed n Secon 3, we hen prove well-posedness and no-collapse for all nal condons of he parcle sysem wh parwse dfferen enres by akng advanage of he hypoellpcy propery of he whole sysem, see Subsecon 4.3. The man lnes of Secon 4 are conneced wh he sraegy already developed by [13] n order o prove 7

9 ha nose may preven N pon verces, when drven by 2D Euler equaons, from collapsng. However, boh he framework and he resuls are here que dfferen. In [13], he nose s fne dmensonal, he dmenson dependng upon he number of parcles; s gven mplcly only, from a generc exsence resul; moreover, he dynamcs of he parcles are non-degenerae. Here, he srucure of he nose s explc and s ndependen of he number of parcles; morever, he dynamcs of he parcles are degenerae. 1.4 Assumpon For smplcy, we choose a = 1/N for 1 N. We also assume ha (A.1 F s bounded everywhere on R d and locally Lpschz-connuous on any compac subse of R d \ {}. Moreover, [ ] F (x F (y sup sup r < +. x y <r 1 r x, y 1,x y The examples we have n mnd are: for d = 1, F (x = sgn(x, x R \ {}; for d 2, F (x = x/ x, x R d \ {}. (A.2 For each k N\{}, σ k : R d R d s Lpschz-connuous, and he seres k=1 σα k ( xσβ k (ỹ converges unformly w.r.. ( x, ỹ n compac subse of R d R d, for each α, β = 1,..., d. We defne Q ( x, ỹ = σ k ( x σ k (ỹ R d d, (1 k=1 as he covarance funcon of he random feld R d x k=1 σ k ( x W1 k. I s of posve ype, ha s n,j=1 Q( xj, x ṽ, ṽ j R d, for any n 1, x 1,..., x n, ṽ 1,..., ṽ n R d. (A.3 Q( x, ỹ s bounded on he dagonal, ha s sup x R d Q( x, x < +. Moreover, sasfes he Lpschz ype regulary propery Q ( x, x + Q (ỹ, ỹ Q ( x, ỹ Q (ỹ, x < +. (11 x ỹ 2 sup x,ỹ R d x ỹ (A.4 Q ( x, ỹ s srcly posve on Γ x,n = {(x 1,..., x N R Nd : x x j whenever j}, ha s, for all ( x 1,..., x N Γ x,n and v = ( v 1,..., v N R Nd \ {}, N,j=1 Q (xj, x v, v j R d >. 8

10 The regulary assumpons on σ k and Q n (A.2 and (A.3 respecvely are srongly relaed one wh each oher. Specfcally, he Lpschz condon (11 mples a srong Lpschz propery of he felds (σ k k N\{} : ( σ α k ( x σk α (ỹ ( σ β k ( x σβ k (ỹ k=1 = Q αβ ( x, x Q αβ ( x, ỹ Q αβ (ỹ, x + Q αβ (ỹ, ỹ C x ỹ 2. (12 Conversely, Eq. (11 holds f he Lpschz consans of he (σ k k N\{} are square summable. In pracce, he covarance funcon Q s gven frs. Precsely, gven a funcon Q : R 2d ( x, ỹ Q( x, ỹ wh values n he se of symmerc marces of sze d d, sasfyng (A.3 and of posve ype, may be expressed as a covarance funcon as n (1, for some felds (σ k k N\{} sasfyng (A.2. We refer o Theorem n [2]. In hs framework, a suffcen condon o guaranee (11 s: Q s of class C 2 wh bounded mxed dervaves, ha s sup ( x,ỹ R d R 2 x,ỹq( x, ỹ < +. Indeed, Lpschz propery (11 hen d follows from a sraghforward Taylor expanson. By (A.2, Sraonovch negrals n SDE (8 are (formally equal o Iô negrals, so ha (8 wll be nerpreed n he usual Iô form dx d = V, dv = 1 N F ( X X j d + j k=1 σ k ( X dw k, (13, {1,..., N}. Indeed, he local marngale par of (σ k (X s zero, snce (σ k (X s of bounded varaon. We shall no rea more rgorously hs equvalence, and, from now, we shall adop he Iô formulaon. 1.5 Useful noaon In he whole paper, he number N of parcles s fxed, so ha he dependence of consans upon N s no nvesgaed. For any n N \ {}, z R n and r >, B n (z, r s he closed ball of dmenson n, cener z and radus r; Leb n s he Lebesgue measure on R n. The volume of B n (z, r s denoed by V n (r. The confguraons of he N-parcle sysem n he phase space are generally denoed by z or Z. Posons are denoed by x or X and veloces by v or V. Smlarly, he ypcal noaon for a sngle parcle n he phase space s z = ( x, ṽ, x sandng for s poson and ṽ for s velocy. The se of 9

11 pars of dfferen ndces n he parcle sysem s denoed by N = {(, j {1,..., N} 2 : j}. Moreover, we se Γ N = {(z 1,..., z N R 2Nd : (, j N, z z j } and Γ x,n = {(x 1,..., x N R Nd : (, j N, x x j }. We also defne he projecon mappngs: Π x : R 2Nd z = (z j 1 j N = ( (x j, v j 1 j N Π x(z = (x j 1 j N R Nd, π x : R 2d z = ( x, ṽ x R d, π,x : R 2Nd z = (z j 1 j N = ( (x j, v j 1 j N π,x(z = x R d, wh a smlar defnon for Π v, π v and π,v. We hen pu π = (π,x, π,v. Below, Eq. (13 wll be also wren n he compac form dz = F (Z d + A k (Z dw k,, (14 k=1 where Z = (X, V, wh X = (X 1,..., X N and V = (V 1,..., V N, and F : R Nd R Nd ( x = (x 1,..., x N, v ( ( 1 v, F (x x j R Nd R Nd, N 1 N (15 j A k : R Nd R Nd (x, v (, A k (x R Nd R Nd, A k : ( R d N ( x 1,..., x N ( σ k (x 1,..., σ k (x N ( R d N. For any, he 2d-coordnaes of Z wll be denoed by Z = π (Z = (X, V, {1,..., N}. Smlarly, we shall denoe by (F π (F 1 N and (A k π (A k 1 N he 2d-componens of F and A k, for k N \ {}. 2 Connuaon: Proof of Theorem 2 Here, we denfy general condons on he srucure of he nose n (3 under whch Theorem 2 holds. Typcal examples are gven n Proposon 19 n Secon 3. In he whole secon, we hus consder he 4D sysem: dx,ε = V,ε d, dv,ε = sgn ( X,ε Xī,ε + ε k=1 σ k (X,ε dw k, (16 for, = 1, 2 and ī = 2 f = 1 and vce versa. Below, we assume ha (X,ε, V,ε = (ɛ, ɛ 2, wh ɛ = 1 f = 2 and ɛ = 1 f = 1. As a frs general condon (agan, we refer o Proposon 19 for examples, we se: 1

12 Condon 3 For any ε >, Theorem 1 apples and hus (16 has a unque srong soluon, whch sasfes P{, (X 1,ε, V 1,ε (X 2,ε, V 2,ε } = 1. We wll se Z,ε = (X,ε, V,ε, ε >, = 1, 2. When ε =, he curves X, = ɛ ( 1 2 2, V, = ɛ ( 2 +,, solve he sysem (16, bu merge a me = 2. Agan, we wll se Z, = (X,, V,, = 1, 2. Noe ha Z 2, = Z 1, for all. We are o prove ha (Z 1,ε, Z 2,ε converges n dsrbuon on he space C([, +, R 4 oward (1/2δ (Z 1,+,Z 2,+ + (1/2δ (Z 1,,Z 2,, where { (Z 1,+, Z 2,+ = (Z 1,, Z 2,, ; (Z 1,, Z 2, (Z 1,, Z 2,, [, = ], (Z 2,, Z 1,, >. The whole pon s o nvesgae he dfferences: X ε = X2,ε X 1,ε, V ε = V 2,ε V 1,ε,, ε >. ( We wll use he second condon (see as an example Proposon 19: Condon 4 Assume ha (Z 1,ε, Z 2,ε sasfes (16, bu wh an arbrary random nal condon (Z 1,ε, Z 2,ε Γ 2, ndependen of he nose ((W k k 1. Denoe by (F he augmened flraon generaed by he nal condon (Z 1,ε, Z 2,ε and by he nose ((W k k 1. Then, here exss an (F -Brownan moon (B ε such ha, for all, dx ε = V ε d, dv ε = sgn(x ε d + εσ(x ε db ε, (18 where σ s C 2 funcon from R o R, dependng on he (σ k k 1 only (n parcular, σ s ndependen of he nal condon (Z 1,ε, Z 2,ε and of ε, wh bounded dervaves of order 1 and 2, such ha σ( = and σ(1 >. Seng Z ε = (X ε, V ε, for any, we frs nvesgae he soluons of (18 when ε =. We have he obvous Lemma 5 For ε =, all he soluons of (18 wh ε = and (X, V = (1, 2, have he form ( Z = (X, V ( 2 =, for = 2. (

13 We emphasze ha unqueness fals afer coalescence me. Indeed, any (Z, wh (Z as n (19, Z = (, for 1 and Z = ±(( 1 2 /2, 1, for 1, where 1 may be real or nfne, s a soluon o (18 when ε = heren. We clam Proposon 6 Gven τ ε = nf{ : X ε }, for any δ > and M >, lm P{τ ε ( δ, + δ} = 1 ε 2, lm P{τ ε M} = 1 ε 2. (2 Moreover, defnng τ ε 2 = nf{ > τ ε : X ε }, we have, for any M >, lm P{τ 2 ε M} = 1. (21 ε Proposon 6 suggess ha, n he lm regme ε, he rajecores of he wo parcles cross wh probably 1/2 exacly, and, f so, hey jus cross once, a coalescence me. Ths s one sep forward n he proof of Theorem 2. Precsely, we prove below ha Proposon 6 mples Theorem 2. Proof. (Proposon 6 Theorem 2. The famly ((Z 1,ε, Z 2,ε <ε 1 s gh by (A.3. We denoe by µ a weak lm, on he space of connuous funcons C([, +, R 4, of he famly of measures (P (Z 1,ε,Z 2,ε <ε 1 as ε, he canoncal process on C([, +, R 4 beng denoed by (ξ 1 = (χ 1, ν 1, ξ 2 = (χ 2, ν 2. We wll also denoe χ = (χ 2 χ 1 /2 and ν = (ν 2 ν 1 /2,. Under he measure µ, ξ = (χ, ν, = 1, 2, sasfes χ = ν, ν 1,, = 1, 2. We now make use of Proposon 6. Gven M >, we have, on he se {τ ε M}, V,ε = V,ε + ɛ + ε k 1 σ k (X,ε s dw k s, M, where ɛ maches 1 f = 2 and 1 f = 1, so ha, for any η >, lm nf ε P { sup V,ε V,ε ɛ η, = 1, 2 } lm P{τ ε M}. M ε By pore-maneau Theorem, we deduce ha { } { = 1 f M < µ ν = ν, + ɛ, M, = 1, 2 1/2 f M >. (22 12

14 Therefore, under µ, (ξ 1 concdes wh (Z 1, and (ξ 2 concdes wh (Z 2,. In parcular, under µ, ξ 1 = ξ 2 = (,. Smlarly, we also deduce from (22 ha, wh probably greaer han 1/2 under µ, (ξ 1, ξ 2 = (Z 1,+, Z 2,+ for any. By he same argumen, for δ > small and M > + δ, we deduce from Proposon 6 ha, µ { ν = ν +δ ɛ [ ( + δ], + δ M, = 1, 2 } lm ε P{τ ε + δ, τ ε 2 M} = 1 2. Leng δ end oward and M oward +, we deduce ha, wh probably greaer han 1/2 under µ, (ξ 1, ξ 2 = (Z 1,, Z 2, for any. 2.1 Key Lemmas by Inegraon by Pars The proof of Proposon 6 reles on wo key lemmas, each of hem beng proven by negraon by pars. The frs one s Lemma 7 Se N + = [, + 2 \ {(, } and, smlarly, N = (, ] 2 \ {(, }. Consder also he ses of nal condons for (16: Γ ± = {(z 1 = (x 1, v 1, z 2 = (x 2, v 2 : (x 2 x 1, v 2 v 1 N ± }. Then, here exss a consan c > such ha, for any M > and any compac subse K R 4, lm nf P{ [, M], ε ±Xε (z 1,z 2 K Γ ± c 2 (Z 1,ε, Z 2,ε = (z 1, z 2 } = 1. Proof. In he whole proof, he nal condon (z 1, z 2 K Γ + s gven,.e. (Z 1,ε, Z 2,ε = (z 1, z 2 K Γ +. Wrng z = (x, v, = 1, 2, we se x = (x 2 x 1 /2 and v = (v 2 v 1 /2. W.l.o.g., we assume ha x >. Indeed, when x =, v mus be posve, so ha, n very shor me, boh X ε and V ε are posve. By Markov propery (whch holds for he 4D sysem because of srong unqueness, we are hen led back o he case when x and v are posve. By Condon 4, we can wre dv ε = sgn(x ε d + εσ(x ε db ε,, where (B ε s a 1D Browan moon. Takng advanage of he smoohness of σ, we perform he followng negraon by pars: d ( V ε εσ(x ε B ε = ( sgn(x ε εσ (X ε V ε B ε d. 13

15 Keepng n mnd ha τ ε = nf{ : X ε }, we have V ε εσ(x ε B ε ε σ (X ε sv ε sb ε sds, τ ε. On he even A ε 1 = { sup σ (X ε V ε B ε 1 }, we have M 2ε dx ε ( 2 + ( εσ(xε B ε d 2 CεXε B ε d, τ ε M, where C here sands for he Lpschz consan of σ. We deduce d X ε ( (Cε 2 exp Bs ds ε d, wh Xε = X ε exp Cε Bs ds ε, for τ ε M. Therefore, on A ε 1, τ ε mus be greaer han M, so ha he above expresson holds up o me M (a leas. We deduce d X ε (/2d, for M, so ha X ε 2 /4, for M. Inersec now A ε 1 wh A ε 2 = { sup B ε 1 }. Then, on A ε M εm 1 A ε 2, X ε 2 4 exp( C, M. To complee he proof, remans o noce (from a sandard ghness argumen ha P(A ε 1 A ε 2 1 as ε, unformly n (z 1, z 2 K. (The proof when (z 1, z 2 s n Γ s smlar. We now come back o he case when he nal condon of he 4D sysem s ((1, 2, ( 1, 2. The second key lemma consss n expandng he dfference process (X ε, V ε w.r.. ε, up o τ ε = nf{ : X ε }. Lemma 8 There exs a famly of 1D Brownan moons ((B ε ε> and a famly of random connuous processes (g ε : R + R ε>, such ha and he processes sasfy T >, lm sup P { sup g ε > R } =, (23 R + <ε 1 T dv ( = d, dx ( = V ( d, (X (, V ( = (1, 2, dv (1,ε = σ(x ( db ε, dx (1,ε = V (1,ε d, (X (1,ε, V (1,ε = (,, X ε (X ( + εx (1,ε + V ε (V ( + εv (1,ε ε 2 g ε, τ ε. (24 14

16 Proof. By Condon 4, we can wre dv ε = d + εσ(x ε db ε, τ ε, for some 1D Brownan moon (B ε, so ha d [ δx] ε = δv ε d, d [ ] [ δv ε = ε σ(x ε σ(x ( ] db ε, τ ε, + εx (1,ε, δv ε = V ε (V ( wh δx ε = X ε (X ( he same negraon by pars as above, wh δ V ε = δv ε ε ( σ(x ε σ(x ( B ε. + εv (1,ε. We perform Then, we can fnd a famly of random connuous funcons ((v,ε ε>, sasfyng (23, such ha d [ δ V ] ( ε = ε σ (X ε V ε σ (X ( V ( B ε d = ε ( σ (X ε V ε σ (X ( = εσ (X ε B ε δv ε d ε ( σ (X ε σ (X ( + εx (1,ε (V ( + εv (1,ε B ε d + ε 2 v,ε d + εx (1,ε (V ( + εv (1,ε B ε d + ε 2 v,ε Snce σ s Lpschz-connuous, we can fnd wo famles of random funcons ((v 1,ε ε> and ((v 2,ε ε>, sasfyng (23, such ha d [ δ V ] ε = εv 1,ε δx ε d + εv 2,ε δ V ε d + ε 2 v,ε d. (25 In a smlar way, we can fnd wo famles of random funcons ((x,ε ε> and ((x 1,ε ε>, sasfyng (23, such ha d [ ] δx ε = εx 1,ε δx ε d + δ V ε d + ε 2 x,ε d. (26 Boundng he resolven of he lnear sysem (25 26 n erms of he bounds for x 1,ε, v 1,ε and v 2,ε, he resul easly follows. d. 2.2 Proof of Proposon 6 We emphasze ha V ( = 2 + and X ( = (1 / 2 2 and ha V (1,ε = X (1,ε = σ(x ( s s db ε s = σ [( 1 σ [( 1 r 2 2 ] db ε r = s 2 2 ] db ε s, ( rσ [( 1 r 2 ] db ε r. 2 Choosng = 2 and keepng n mnd ha σ(1, we deduce (27 15

17 Lemma 9 The r.v. s (X (1,ε ε> have he same Gaussan law wh zero mean and non-zero varance. In parcular, P{X (1,ε We clam Lemma 1 For any real M > = 2 and any δ >, > } = P{X (1,ε } = 1/2. ( ( ( lm P ( {τ ε M} {X (1,ε ε > δ} =, lm P ( {τ ε > M} {X (1,ε ε < δ} =, lm P ( {τ ε M} {τ ε ( δ, + δ} =. ε Proof. Gven M > 2, we se E ε M = {τ ε M}. We know from Lemma 8, ha here exss a gh famly of random varables (ζ ε M <ε 1 such ha X ε (X ( + εx (1,ε ε 2 ζ ε M, τ ε M. (28 Therefore, on E ε M, we can choose = τ ε above. We deduce ha ( 1 τ ε εx (1,ε τ ε ε 2 ζ ε M. (29 Up o a modfcaon of ζm ε, we deduce (whch s ( τ ε 2 2 εζm. ε (3 We now prove (. From (29, we deduce ha X (1,ε τ ε εζε M on Eε M. Snce X (1,ε s Lpschz connuous on he nerval [, M], we deduce from (3 ha here exss a gh famly of random varables (CM ε <ε 1 such ha X (1,ε = X (1,ε τ ε + X(1,ε X (1,ε τ ε εζε M + CM ε τ ε εζ ε M + CMε ε 1/2 (ζm ε 1/2. Tha s, for every δ >, lm P ( EM ε {X (1,ε ε > δ} =. We fnally prove (. From (28, we know ha εx (1,ε X ε ε 2 ζ ε. Therefore, on (EM ε, εx (1,ε ε 2 ζ ε. Ths proves ha, for every δ >, lm P( (EM ε {X (1,ε ε < δ} =. 16

18 Lemma 11 I holds ( M > 2, lm ε P{τ ε > M} = 1/2, ( δ >, lm ε P{τ ε ( δ, + δ} = 1/2. In parcular, τ ε converges n law owards 1 2 δ δ +. Proof. From Lemmas 9 and 1, for any M > 2 and any δ >, lm sup ε P{τ ε > M} lm sup P{X (1,ε δ} = P{X (1,1 δ}. ε Leng δ end oward, we oban lm sup ε P{τ ε > M} 1/2. Smlarly, lm sup P{τ ε M} P{X (1,1 } = 1 ε 2. We deduce (. Then, ( follows from ( n Lemma 1. We fnally clam Lemma 12 Se σ ε = nf{ : V ε }. Then, for all M >, lm M, σ ε < τ ε } =, ε (31 lm 2 ε M} = 1. ε (32 Proof. By Lemma 11, we can assume M > 2. We hen sar wh he proof of (31. By Markov propery, we noce ha P{τ ε M, σ ε < τ ε } 1 { πx(z 2 z 1 >, π v(z 2 z 1 =}P{τ ε M (Z 1,ε, Z 2,ε = (z 1, z 2 }dη ε (z 1, z 2, Γ 2 where η ε s he condonal law of (Z 1,ε ρ, ε Z2,ε ρ gven ε ρε M, wh ρ ε = nf(σ ε, τ ε, under he nal condon ((1, 2, ( 1, 2. Usng ( n Lemma 11, s plan o see ha he dsrbuons (η ε <ε 1 are gh. By Lemma 7, hs shows (31. Smlarly, we have P{τ2 ε M, τ ε < σ ε } 1 { πx(z 2 z 1 =, π v(z 2 z 1 <}P{τ ε M (Z 1,ε, Z 2,ε = (z 1, z 2 }dη ε (z 1, z 2, Γ 2 whch ends o by he same argumen as above. Snce lm ε P{τ ε 2 M, σ ε < τ ε } lm ε P{τ ε M, σ ε < τ ε } =, we deduce (32. (Keep n mnd ha P{τ ε = σ ε } =. 17

19 3 Srucure of he nose In hs secon, we nvesgae he meanng of Assumpon (A.4. Frs, we ranslae no an ellpcy propery of he nose. Second, we dscuss some general examples nspred from urbulence heory. Fnally, we prove ha ellpcy of he nose lfs up o hypoellpcy of any mollfed verson of Eq. (13. In he whole secon, we shall make use of he noaons n ( Ellpcy of he nose When F = and x Γ x,n, Span {A k (x} k N\{} R Nd so ha he velocy componen n (13 moves along a resrced number of drecons only. On he conrary, when x Γ x,n, he nose generaed a x s non-degenerae because of he src posvy of Q ( x, ỹ on Γ x,n n (A.4: Lemma 13 Q sasfes (A.4 f and only f Span {A k (x} k N\{} = R Nd, x = ( x 1,..., x N Γ x,n. (33 Proof. For any v = ( v 1,..., v N R Nd \ {}, we have k=1 A k (x, v 2 R = Nd k=1 ( N =1 σ k(x, v R d 2 = N,j=1 Q(xj, x v, v j R d >. Below, we exhb neresng examples of srcly posve covarance funcons Q( x, ỹ ha are space-homogeneous. Precsely, we shall assume ha here s a symmerc d d marx-valued funcon Q( x, such ha Q( x, ỹ = Q( x ỹ = Q(ỹ x, wh he followng specral represenaon: Q ( x = e k x Q (k dk, x R d, (34 R d where he specral densy Q akes values n he space of non-negave real symmerc d d marces, wh coordnaes n L 1 (R d, and sasfes Q ( k = Q (k, k R d. (Above, k x s a shoren noaon for k, x R d. In hs framework, we have he general creron: Lemma 14 Assume ha Q has he followng propery: for any R d -valued rgonomerc polynomal v(k of he form v(k = N j=1 vj e k xj, for some (x 1,..., x N, (v 1,..., v N R Nd and for 2 = 1, he a.e. equaly Q (k v (k, v (k C d = for a.e. k R d 18

20 mples v (k = for any k R d, where, C d denoes he Herman produc n C d. Then Q ( x, ỹ s srcly posve on Γ x,n. (Keep n mnd ha, for any u, u C d, u, u C d = d j=1 ūj (u j, ū denong he conjugae of u. We wll also wre u, u C d = ū, u R d wh an abuse of noaon. Proof. The proof follows from he deny: N ( Q x j, x l v l, v j = Q (k v (k, v (k R d C d dk, (35 R d j,l=1 where v (k = N j=1 vj e k xj. Indeed, v (k = for any k R d mples (v 1,..., v N = snce v (k s a (vecor valued rgonomerc polynomal drven by parwse dfferen vecors x 1,..., x N. (See Remark 15 below. Remark 15 Le f : R d C be of he form f (k = N ( a j + k, v j C e x j k, k R d, d j=1 where a j C, v j C d and (x 1,..., x N Γ x,n. If here s a Borel se A R d of posve Lebesgue measure such ha f = on A, hen a j = and v j = for any j = 1,..., N. Indeed, by a sandard exenson of he prncple of analyc connuaon, f(k = for any k R d. Gven a smooh funcon ϕ : R d C wh compac suppor, we denoe by ˆϕ s Fourer ransform. We have f(k ˆϕ(kdk = and hus N R d j=1 [aj ϕ(x j v j, ϕ (x j R d] =. Snce he pons x are all dfferen, we may consruc a funcon ϕ such ha ϕ(x j = ā j and ϕ(x j = v j. By Lemma 14 and Remark 15, we ge as a frs example: Proposon 16 If Q (k s srcly posve defne on a Borel subse of R d of posve Lebesgue measure, hen Q ( x, ỹ s srcly posve on Γ x,n. 3.2 Isoropc random felds Proposon 16 does no cover mporan examples, as he followng one, whch appears n he leraure abou dffuson of passve scalars: 19

21 Example 17 We say ha Q s soropc f Q(U x = UQ( xu, for any x R d and U O(R d, where O(R d s he se of orhogonal marces of dmenson d (U denoes he ranspose of U. Ths s he case when Q(k n (34 has he form Q(k = π k f ( k, ha s Q (x = e k x π k f ( k dk, R d where π k = 1, f d = 1, and π k = (1 p I d + k 2 (pd 1 k k for some p [, 1], f d 2, and f : [, + R s n L 1 ([, + and sasfes f (r for a.e. r >. (36 The marx Q (k s symmerc, sasfes Q ( k = Q (k, and s almoseverywhere non-negave because (we resrc he proof o d 2 k 2 π k w, w C d = (1 p k 2 w 2 + (pd 1 k, w C d 2 (1 p k, w C d 2 + (pd 1 k, w C d 2 = p (d 1 k, w C d 2, (37 (Inequaly s here gven n C d bu only he R d par s useful o prove nonnegavy of Q(k. The full nequaly n C d wll be used nex. We refer o [22], [29] and [33] for references where hs form (for parcular choces of f s used or nvesgaed. Ths class of covarances s relaed o he Bachelor regme of he Krachnan model, where f(r = (r 2 + r 2 (d+ϖ/2 wh ϖ = 2, see [11]. In he lm r, he covarance of he ncremens of he nose s scale nvaran wh scalng exponen equal o 2. The urbulen regme of he Krachnan model ( ϖ < 2 s n conras no ncluded n our man fnal resul because of he regulary properes we requre on Q. Proposon 18 If here exss a Borel se A [, such ha Leb 1 (A > and f (r > for r A, hen Q( x, ỹ s srcly posve on Γ x,n. Proof. Sep 1. From Lemma 14 s suffcen o prove ha he condon f ( k π k v (k, v (k C d = for a.e. k R d mples v (k = for any k R d, when v (k has he form v (k = N j=1 vj e k xj for some (v 1,..., v N R Nd and (x 1,..., x N Γ x,n. Snce f on A, 2

22 holds π k v (k, v (k C d = for k n a Borel subse A R d of posve measure. We now prove ha hs mples v (k. We focus on he condon π k w, w C d = for some w C d. When d = 1, mples w =. For p (, 1], d > 1, nequaly (37 mples p (d 1 k, w 2 C =, and hus k, w 2 d C =. Fnally, n he case p =, d d > 1, for all w C d we have k 2 π k w, w C d = ( k w k, w C d 2 and hus π k w, w C d = mples ha w = λk, for some λ R, f k. Gong back o he man lne of he proof, we have π k v (k, v (k C d = for all k A. Dependng on he values of p and d, hs mples a leas one of he hree followng condons: v(k = for all k A, or k, v (k C d = for all k A, or v (k k for all k A (excep maybe a k =. By Remark 15, vj =, for j = 1,..., N, n he wo frs cases. In he hrd case, we noce ha v(k k may be wren as N j=1 vj l e l vj l e l, k C de xj k =, for 1 l, l d, where (e l 1 l d s he canoncal bass of C d. By Remark 15 agan, hs also mples v j =, for j = 1,..., N. (See also [17, Theorem 4.7] and [1]. We are now able o gve examples for whch Theorem 2 apples: Proposon 19 In he case when d = 1, consder Q as n Example 17, wh f L 1 (R +, R + sasfyng he assumpon of Proposon 18 ogeher wh + k 4 f(kdk < +, hen Condons 3 and 4 n Secon 2 are sasfed wh σ(x = sgn(x (Q( Q(x/2, for x R. Proof. Go back o he framework of Secon 2 and recall (16 and (17. Exsence and unqueness n Condon 3 follow from Theorem 22 below. In order o prove (18, we consder an arbrary random nal condon (Z 1,ε, Z 2,ε Γ 2, ndependen of he nose ((W k k 1. For any, dx ε = sgn(x ε d + ε k 1 = sgn(x ε ( d + ε ρ ε db ε, wh ρ ε = Q( Q(X ε and db ε = ( 1 {X ε } 1 {X ε <} ( σ k (X 2,ε σ k (X 1,ε 1 {ρ ε >} 2ρ ε k 1 21 σ k (X 2,ε σ k (X 1,ε 2 dw k dw k + 1 {ρ ε =}dw 1.

23 I s well-checked ha d B ε = d. By Lévy s Theorem, (B ε s a Brownan moon w.r.. he augmened flraon generaed by he nal condon (Z 1,ε, Z 2,ε and by he nose ((W k k 1. We now nvesgae he properes of σ. Clearly, σ( =. Below, we prove ha σ s C 2 wh bounded dervaves and ha σ(1 >. We have Q( Q(x = x 2 1 cos(kx f( k dk = x 2 k 2 ϕ(kxf( k dk, x 2 R wh ϕ(u = u 2 (1 cos(u. Clearly, ϕ s nfnely dfferenable wh bounded dervaves. Therefore, he funcon Φ : R x R k2 ϕ(kxf( k dk s wce connuously dfferenable wh bounded dervaves. A x =, Φ( >, so ha Φ s wce connuously dfferenable n he neghborhood of. Then, he funcon σ, whch reads σ(x = x Φ(x, for x R, s wce connuously dfferenable n he neghborhood of. Away from, he funcon R x Q( Q(x has posve values so ha s square roo s also wce connuously dfferenable and σ s wce connuously dfferenable as well. The dervaves of order one and wo of σ are bounded snce he dervaves of order one and wo of Q are bounded and Q( Q(x Q( > as x +. Moreover, σ(1 s clearly posve. 3.3 Hypoellpcy of he N-pon moon Ellpcy of he nose urns no hypoellpcy of he sysem, n he followng sense (he proof s sandard and s hus lef o he reader: Proposon 2 Assume ha, F and σ k, for any k N \ {}, are of class C 1 on R d and ha, for every x = (x 1,..., x N Γ x,n, Span {A k (x} k N\{} = R Nd. Then, for every z R 2Nd of he form z = (x, v wh x = (x 1,..., x N Γ x,n, v R Nd, we have Span{A k (z, [A k, F](z} k N\{} = R 2Nd. (Here, [, ] sands for he Le bracke of vecor felds. Here s he precse formulaon of hypoellpcy n our framework: Proposon 21 In addon o (A.1 4, assume ha F s Lpschz connuous on he whole R d. Then, for every nal condon Z = z R 2Nd, Eq. (13 adms a unque srong soluon. Moreover, he mappngs ϕ : R 2Nd z Z subjec o Z = z,, form a sochasc flow of homeomorphsms on R 2Nd. Fnally, for any >, he margnal law of he 2Nd-dmensonal vecor Z s absoluely connuous wh respec o he Lebesgue measure when z = (x, v sasfes x Γ x,n. 22 R

24 Proof. Unque srong solvably and homeomorphsm propery of he flow may be found n [3] and [2, Chaper 4, Secon 5]. When he coeffcens (σ k k N\{} are smooh, wh dervaves of any order n l 2 (N \ {}, and F s smooh as well, absolue connuy hen follows from Proposon 2 and a suable verson of Hörmander s Theorem for sysems drven by an nfne-dmensonal nose. See for example [3, Theorem 4.3]. Here he coeffcens are no smooh. Anyhow, absolue connuy follows from he Bouleau and Hrsch creron drecly. By Proposon 2.2 n [3], (Z s dfferenable n he sense of Mallavn wh + k=1 E Dk s Z 2 ds < +, for any. We also know ha D k r Z = Y (Y r 1 A k (Z r, r, (38 he equaly holdng rue n R 2Nd, where (Y s an R 2Nd 2Nd -valued process, soluon a lnear SDE of he form Y = I 2Nd + + k=1 α k (sy s dw k s + α (sy s ds,, (39 he processes (α k (s s, k N, beng bounded and progressvely-measurable and he nfne-dmensonal process (( α k (s s k N\{} beng bounded n l 2 (N \ {}. When he coeffcens F and A k, k N \ {}, n he compac formulaon (14 are smooh, holds α (s = F(Z s and α k (s = A k (Z s, k N \ {}. We hen use he followng noaon: gven a square marx M of sze 2Nd 2Nd, we denoe by [M] x,x, [M] x,v, [M] v,x and [M] v,v he blocks of sze Nd Nd correspondng o he decomposon of a vecor z R 2Nd no coordnaes x = Π x (z and v = Π v (z n R Nd. Wh hs noaon, [α k (s] x,v and [α k (s] v,v are zero snce A k s ndependen of v. Smlarly, [α k (s] x,x s zero snce Π x (A k and [α (s] x,x = [α (s] v,v = snce Π x (F v and Π v (F s ndependen of x. Moreover, [α (s] x,v = I Nd. These relaonshps keep rue n he Lpschz seng by a mollfcaon argumen. Fnally, as n he fne-dmensonal framework, we can check ha, a.s., for any >, Y s nverble, he nverse beng of fne polynomal momens of any order. For r small, Y r = I 2Nd +o r (1, o r (1 sandng for he Landau noaon and almos-surely convergng o wh r. Therefore, by he equales [α (s] x,x = [α k (s] x,x = [α k (s] x,v = [α (s] v,v = [α k (s] v,v = and [α (s] x,v = I Nd, we deduce ha [Y r ] x,v = I Nd + ro r (1 and [Y r ] v,v = I Nd + o r (1 and ha [Y r ] x,v may expanded as [Y r ] x,v = ri Nd + ro r (1. 23

25 Seng Z r = (Y r 1 A k (Z r R 2Nd and wrng Z r under he form Z r = ((X r, V r 1 N, we have Y r Z r = A k (Z r, so ha [Y r ] x,x X r + [Y r ] x,v V r =, [Y r ] v,x X r + [Y r ] v,v V r = A k (X r, ha s X r + rv r = ro r (1, and o r (1X r + V r = A k (X r + o r (1. We deduce V r = A k (X r + o r (1 and X r = ra k (X r + ro r (1, so ha, by (38, (Y 1 D k r Z = (ra k (x + ro r (1, A k (x + o r (1. (4 (The above equaly holds almos-surely, o r (1 beng random self. For a gven ω Ω for whch (4 holds rue, consder ζ = ((χ, ν 1 N R 2Nd such ha Dr k Z, ζ R d = for any r and k N \ {}. Changng ζ no ((Y 1 ζ, we deduce from (4 ha r N ( σ k x, χ R d + =1 N ( σ k x, ν R d = ro r (1 χ + o r (1 ν, =1 Leng r, we ge ν A k (x for any k N \ {}. By (A.4, ν =. Dvdng he above equaly by r and leng r, χ =. We complee he proof by Bouleau and Hrsch creron, see [31, Theorem 2.1.2]. 4 No coalescence of he sochasc dynamcs We now prove he man resul of he paper: Theorem 22 Under (A.1 4, for any z Γ N, here exss a unque soluon (Z (z o (13 wh z as nal condon. I sasfes P{, Z (z Γ N } = 1 and P{Leb 1 { : Π x (Z (z Γ x,n } = } = 1. The proof s spl no hree pars: we frs esablsh a pror esmaes for a regularzed verson of (13; usng a compacness argumen, we deduce ha srong unque solvably holds for Lebesgue almos-every sarng pon; akng advanage of he absolue connuy of he margnal laws of he regularzed sysem, we esablsh srong unque solvably for any z Γ N. 24

26 4.1 Smoohed sysem of equaons For every ε >, le F ε : R d R d be equal o F ousde B d (, ε, bu be smooh nsde B d (, ε, wh sup x R d F ε (x sup x R d F (x +1. Gven such an F ε, we consder Eq. (13, bu wh F ε nsead of F heren (or, equvalenly, he compac wrng (14 when drven by F ε, wh an approprae defnon of F ε n (15. By Proposon 21, he smoohed sysem s unquely solvable for every nal condon n R 2Nd, he soluon beng genercally denoed by (Z ε = (X ε, V ε, wh X ε = (X,ε 1 N and V ε = (V,ε 1 N, and he assocaed flow by ϕ ε : R 2Nd z Z ε wh Z ε = z,. By he a.e. equaly dv (x,v F ε = and dv (x,v A k = for all k N (he dvergence beng here compued n he phase space, we ge drecly: Lemma 23 For any, ϕ ε ( preserves he Lebesgue measure, ha s, for all measurable and non negave g, E R 2Nd g (ϕ ε (z dz = R 2Nd g (z dz. Proposon 24 Le log + : (, + r log + (r be he funcon equal o for r 1 and o log r for r (, 1. For every R 1, se h R (z = 1 { z R} z z j, z R 2Nd. (,j N log + Then, for any R, R, T > here exss a consan C such ha, for any ε >, [ ] E sup h R (ϕ ε (z dz C. (41 [,T ] B 2Nd (,R Proof. Sep 1. For a smooh funcon φ : R [, 1], wh suppor ncluded n (, 1 and wh 1 φ(rdr 2, le log+ φ : R + R + be he smooh funcon: log + φ (r = 1 r φ (s ds for r, so ha s d dr log+ φ (r 1 r for r >. As φ ncreases owards he ndcaor funcon of he nerval (, 1, log + φ (r ncreases owards log + (r. Gven he funcon R 2d z log + φ ( z, we have [ log + φ ( z ] 1 { z 1} C, z (42 [ ( ] 2 log + φ z C ( 1 + φ z 2 ( z 1 { z 1} (43 25

27 for a consan C ha s ndependen of he deals of φ. Gven R >, le θ R : R 2Nd [, 1] be a smooh funcon equal o 1 on B 2Nd (, R, equal o ousde B 2Nd (, R + 2, wh values n [, 1] and wh sup z R 2Nd θ R (z 1 and sup z R 2Nd 2 θ R (z 1. Defne h θ R φ (z = θ R (z (,j N log + φ ( z z j, z R 2Nd. Below, we prove ha, gven R, R >, here exss a consan C, ndependen of ε and he deals of φ and θ R n B 2Nd (, R + 2 \B 2Nd (, R, such ha ( E sup h θ R φ ϕ ε (z dz C. (44 d ( θ R (Z ε g ( Z,ε B(,R [,T ] Leng φ ncrease owards 1 (,1, (41 follows by monoonous convergence. Sep 2. We now prove (44. In he whole argumen, we use he compac formulaon (14. Wh he noaon g ( z = log + φ ( z and for a generc soluon (Z ε o he smoohed sysem, we have: { di 1 = di 11 + di 12 where di 11 = θ R (Z ε g ( Z,ε di 12 = θ R (Z ε 2 di 21 di 22 di 3 = = g ( Z,ε = g ( Z,ε k=1 Z j,ε = di 1 +di 2 +di 3, 2d α,β=1 Z j,ε Z j,ε 2 g ( Z,ε ( Z j,ε, d Z,ε 2 g ( Z,ε z α z β θr (Z ε, dz ε R 2Nd, N 2d,j =1 α,β=1 Z j,ε, R 2d [ (Z Z j,ε d,ε di 2 = di 21 + di 22 Z j,ε α, ( Z,ε (45 ] Z j,ε, β 2 θ ] R (Z (z α (z j ε d [(Z,ε α, (Z j,ε β, β [ ] Z j,ε, A k A j k (Z ε θ R 2d R (Z ε, A k (Z ε R 2Nd d. We frs ackle he muual varaons. By he deny [A k Aj k ](Zε = σ k (X,ε σ k (X j,ε and by (42, (12 and (A.3, di 3 C θ R (Z ε d. (46 26

28 In order o deal wh he erm I 12 we need o analyze he muual varaon [(Z,ε Z j,ε α, (Z,ε Z j,ε β ]. Obvously, holds [(X,ε X j,ε p, (Z,ε Z j,ε β ] = for all p = 1,..., d and β = 1,..., 2d, snce X,ε X j,ε s of bounded varaon. Moreover, by (12, we have, for p, q = 1,..., d, [( d V,ε V j,ε p, ( ] V,ε V j,ε C X,ε X j,ε 2 d. By nequaly (43, we ge (renamng he consan C di 12 Cθ R (Z ε ( 1 + φ ( Z,ε Z j,ε Z,ε Z j,ε 2 X,ε X j,ε 2 d Cθ R (Z ε ( 1 + φ ( Z,ε Z j,ε d. q (47 Fnally, le us deal wh I 22. As above, he only erms o be non-zero n he varaon d[(z,ε α, (Z j,ε β ] are he erms d[(v,ε p, (V j,ε q ]. By boundedness of Q( x, x, we clam d[(v,ε p, (V j,ε q ] Cd and hus where di 22 ( Cg Z,ε Z j,ε 2 θ R (Z ε d. (48 Sep 3. Spl now di 11 and di 21 no di di 112 and di di 212, di 111 = θ R (Z ε g ( Z,ε di 112 = θ R (Z ε di 211 di 212 = g ( Z,ε = g ( Z,ε k=1 Z j,ε Z j,ε g ( Z,ε Z j,ε, F ε (Z ε F j ε (Z ε d, R 2d Z j,ε θr (Z ε, F ε (Z ε R 2Nd d, k=1, A k (Z ε A j k (Zε R 2d dw k θ R (Z ε, A k (Z ε R 2Nd dw k. By (42 and by boundedness of F ε on B 2Nd (, R, we have C θ R (Z ε Z,ε Z j,ε d, di 111 di211 Cg ( Z,ε, Z j,ε θr (Z ε d. (49 Sep 4. We now deal wh he marngale erms I 112 and I 212. By (42, (12, by boundedness of Q( x, x and Doob s nequaly, [ ] [ T ] E sup I T CE θr 2 (Zs ε ds, [,T ] [ ] [ T E I T CE g ( ] 2 Zs,ε Zs j,ε θr (Zs ε 2 ds. sup [,T ] 27

29 From he above bounds, ogeher wh (45, (46, (47, (48 and (49, and makng use of he followng esmaes max ( θ R (z, R θ (z, 2 R θ (z 1 { z R+2}, z R 2Nd, max ( g ( z, g 2 ( z C z, z R2d. we deduce (wh Z ε = z [ ( ( E sup θr (Z ε log + Z,ε φ Z j,ε ] ( θ R (z log + z φ z j [,T ] ( ( T 1 + C 1 + E 1 { Z ε s R+2} Z,ε s Zs j,ε + φ ( Z,ε s Zs j,ε ds. Sep 5. We now negrae on a ball B 2Nd (, R of R 2Nd wh respec o he nal condons. Applyng Lemma 23, we ge [ ( ( E sup θr (ϕ ε (z log + ϕ,ε φ (z ϕ j,ε (z ] dz B 2Nd (,R [,T ] ( θ R (z log + z φ z j dz B 2Nd (,R [ T ( + C R 2Nd 1 + E z z j + φ ( z z j ] dzds. { z R+2} By a sphercal change of varable, he negral of φ ( z z j by C 1 φ (r dr, whch s less han 2C. Ths complees he proof. Lemma 25 Gven R, T >, defne m as he normalzed produc measure V 1 2Nd (R Leb 2Nd P on O = B 2Nd (, R Ω. Then, { lm sup m nf nf ϕ,ε (z, ω ϕ j,ε (z, ω } ɛ =. ɛ [,T ] (,j N ε> (For a measurable funcon φ : O R w.r.. he produc σ-feld on O and a Borel se A R, m{φ(z, ω A} sands for m{(z, ω O : φ(z, ω A}. Proof. By boundedness of F ε and Q( x, x and by Markov nequaly, s well-seen ha, for any R >, here exss a consan C, only dependng on 28

30 R and T, such ha, for any R >, m { sup [,T ] ϕ ε (z, ω > R } C R /R. Moreover, by Proposon 24, { [ m 1 { ϕ ε (z,ω R} ϕ,ε (z, ω ϕ j,ε (z, ω ] } > K C K, sup [,T ] (,j N log + C possbly dependng upon R as well. The proof s easly compleed. Lemma 26 Gven R, T >, keep he same defnon for m as above. Then, { lm sup sup m (z, ω O : (ɛ,a (,+ ε> <δ <1 [ Leb 1 ( [, T ] : nf π x ϕ,ε (z, ω ϕ j,ε (z, ω ] } δ ɛ > Aδ =. (,j N A Proof. The proof follows from Lemma 25 and Proposon 27 below. Indeed, by boundedness of F ε and Q( x, x, he probably ha he v-coordnae of (ϕ ε (z T s 1/4-Hölder connuous wh A as Hölder consan converges owards 1 as A ends o +, unformly n ε > and n z B 2Nd (, R. Proposon 27 Gven A, R, T >, le (ζ = (χ, ν T be a connuous pah wh values n R 2Nd such ha ζ = z B 2Nd (, R, (ν s a 1/4- Hölder connuous R d -valued pah wh A as Hölder consan, for 1 N, and [dχ /d] = ν, for [, T ] and {1,..., N}. Then, here exss a consan C, dependng on d, A, N, R and T only, such ha, for any ɛ, δ (, 1, nf [,T ] nf (,j N ζ ζ j ɛ mples Leb 1 ( [, T ] : nf (,j N χ χ j δ ɛ 5 C Cδ. Proof. Assume ha here exs δ (, ɛ, [, T and (, j N such ha χ χ j δ. Snce nf [,T ] nf (,j N ζ ζ j ɛ, we deduce ν ν j ɛ 2 δ 2. By Hölder propery of (ν T, here exss a consan C, ndependen of ɛ, and δ, such ha ν ν j ɛ2 δ 2 C( 1/4, T. Therefore, here exss one coordnae l {1,..., d} such ha ( ν ν j l ɛ2 δ 2 C( 1/4, T. d 29

31 For C( 1/4 < ɛ 2 δ 2, he rgh-hand sde s always posve so ha (ν ν j l canno vansh. By connuy, s of consan sgn. Therefore, ( χ χ j χ χ j ɛ2 δ ( 2 C( 1/4 δ, d l for C( 1/4 ɛ 2 δ 2. For δ ɛ/2, we deduce χ χ j ɛ( /(4 d δ, for C( 1/4 ɛ/4. Fnally, for 8 dδ/ɛ ɛ 4 /(4C 4, χ χ j δ. Modfyng C f necessary, we deduce ha χ χ j δ, (5 for Cδ/ɛ ɛ 4 /C and δ ɛ/2. Assume now w.l.o.g. ha C 2 and choose δ of he form δ ɛ 5 /C 2 wh δ 1. Defne he se I x (δ, ɛ = { [, T ] : χ χ j δ ɛ 5 /C 2}. By (5, I x (δ, ɛ [ + δ ɛ 4 /C, + ɛ 4 /C] I x (δ, ɛ =. Therefore, Leb 1 (I x (δ, ɛ δ ɛ 4 /C T C/ɛ 4 δ (T No coalescence for a.e. nal confguraon As a consequence of he prevous esmaes, we prove (he resul below mgh be compared wh [8] and [1] abou a.e. solvably of ODEs, bu heren unqueness s nvesgaed hrough unqueness of a regular Lagrangan flow: Theorem 28 Under Assumpons (A.1 3, for Lebesgue almos every z, equaon (13 has one and only one global srong soluon. Proof. Sep 1. We here consder Ξ = C([, +, R 2Nd C([, +, R N\{} endowed wh he produc σ-feld X of he Borel σ-felds. For R > and ε > and wh he same noaons as n Lemma 25, we endow he par (Ξ, X wh he probably Q ε defned on he cylnders as Q ε (A A 1 A k C([, +, R = m { (ϕ ε (z A, ( } W 1,..., W k A1 A k, where A s a Borel subse of C([, +, R 2Nd and A 1,..., A N are Borel subses of C([, +, R. The σ-feld X concdes wh he Borel σ-feld generaed by he sandard produc merc on he produc space Ξ. In parcular, 3

32 he noon of ghness s relevan for probably measures on he par (Ξ, X : s well-checked ha he famly (Q ε ε> s gh. Denong by Q he lm of some convergen sequence (Q εn n N for a decreasng sequence of posve reals (ε n n N convergng owards, we nvesgae he properes of he canoncal process under Q, denoed by ξ = ( ξ k, k N (ξ beng R 2Nd -valued and he (ξ k, k 1, beng R-valued. Clearly, he famly ((ξ k k N\{} s a famly of ndependen Brownan moons under Q. Moreover, he margnal law of Ξ ξ ξ s he unform dsrbuon on he ball B 2Nd (, R. For any ɛ >, he se {ξ Ξ : nf [,T ] nf (,j N π (ξ π j (ξ < ɛ} s open n Ξ. Usng he pore-maneau Theorem o pass o he lm n Lemma 25 and leng ɛ end o, we deduce ha, for any T >, Q { ξ Ξ : nf (,j N nf [,T ] } ( ( π ξ πj ξ = =. (51 Smlarly, he se {ξ Ξ : Leb 1 ( [, T ] : nf (,j N π,x (ξ π j,x (ξ < δ ɛ/a > Aδ } s open n Ξ. Usng he pore-maneau Theorem o pass o he lm n Lemma 26 and leng δ end o frs and hen (ɛ, A o (, +, we oban ( } ( ( Q {ξ Ξ : Leb 1 [, T ] : nf π,x ξ πj,x ξ = > =. (,j N Se now ξ = (χ, ν, wh χ = Π x (ξ and ν = Π v (ξ,. Se also ν = ν (52 Π v ( F(χ s ds,. (53 We clam ha ( ν s a square-negrable connuous marngale under Q w.r.. he flraon ( G = σ(ξs k, s, k N wh he muual varaons [ ( ν, ( ν ] j = Q ( (χ s, (χ s j ds,, j {1,..., d}, (54 [ ( ν, ξ k] ( = σ k (χ s ds, {1,..., d}, k N \ {}. (55 The proof s que sandard and consss n passng o he lm n he marngale properes characerzng he dynamcs of (Z ε. The only dffculy 31

33 s o pass o he lm along he mollfed drfs. For T >, we hus prove ha ( ( F ε (ξsds Q ε F(ξ ε sds Q, (56 T T where he lef- and he rgh-hand sdes ndcae he dsrbuons of he ndcaed processes under he ndcaed measures on C([, T ], R 2Nd and sands for he convergence n dsrbuon. By boundedness of F ε, we emphasze ha here exss a consan C >, ndependen of ε such ha, for any a > and any ε > ε, ( Q ε sup F ε (ξsds F(ξsds > a T Q ε ( CLeb 1 ( [, T ] : nf (,j N π,x (ξ π j,x (ξ ε The even n he rgh-hand sde s closed n Ξ, so ha ( lm sup Q ε sup F ε (ξsds F(ξsds > a ε T ( Q (CLeb 1 [, T ] : nf π,x (ξ π j,x (ξ ε (,j N a. a. (57 Leng ε end o n (57, we deduce from (52 ha he lef-hand sde s. Therefore, o prove (56, s suffcen o prove ( ( F(ξ s ds Q ε F(ξ s ds Q. ε T T By domnaed convergence Theorem, he map C([, T ], R 2Nd (ξ T ( F(ξ ( sds T C([, T ], R 2Nd s connuous a any pah ξ for whch Leb 1 [, T ] : nf(,j N π,x (ξ π j,x (ξ = =. By (52 agan, hs s rue a.s. under Q: by connuous mappng Theorem, we complee he proof of (56. Thus, ( ν n (53 sasfes he announced marngale propery. Sep 2. Denoe by (G he rgh-connuous verson of (G augmened wh Q-null ses. Clearly, ( ν s a square-negrable connuous marngale under Q w.r.. (G and boh (54 and (55 reman rue. In parcular, we can compue [ ν ] ( A k χ s dξ k s =,, k N\{} 32