Quantitative entropy estimates for 2D stochastic vortex model on the whole space under moderate interactions

Quan titativ e propagation of c haos for 2D sto c hastic v ortex mo del on the whole space under mo derate in teractions Alexandre B. de Souza ∗ Abstract W e deriv e the sto c hastic 2D vortex model on the whole Euclidean space from sto c hastic particle systems driven b y individual and en- vironmen tal noises, obtaining path wise quan titative b ounds in the sense of relativ e entrop y . The main nov elty is the application of the Donsk er–V aradhan inequalit y in the con text of mo derately in teracting particles to handle the nonlinearit y , as well as the use of lo calization tec hniques com bined with the probabilistic data setting, to deriv e es- timates for the quadratic v ariation terms. Moreo ver, w e prov e the existence of a suitable solution to the aforemen tioned model. MSC2010 sub ject classification: 49N90, 60H30, 60K35 Con ten ts 1 In tro duction 2 1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Statemen t of the main results . . . . . . . . . . . . . . . . . . 7 2 Pro ofs of main results 8 2.1 Time ev olution of the relativ e en tropy . . . . . . . . . . . . . . 8 2.2 Estimates for I I t + I V t : quadratic v ariation terms . . . . . . . 9 2.3 Estimates for I t : the nonlinear terms . . . . . . . . . . . . . . 12 2.4 Application of Gron wall’s Lemma . . . . . . . . . . . . . . . . 16 2.5 End of the pro of: Remo ving the stopping time . . . . . . . . . 17 ∗ Departamen to de Matem´ atica, Univ ersidade Estadual de Campinas, Brazil. a265040@dac.unicamp.br . 1 3 App endix 20 3.1 Pro of of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.1 Solving (1) for σ = 0 . . . . . . . . . . . . . . . . . . . 21 3.1.2 Solving (1) for σ  = 0 . . . . . . . . . . . . . . . . . . . 22 3.2 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1 In tro duction In this pap er, w e establish a quan titativ e deriv ation of the stochastic 2D v ortex mo del in the whole Euclidean space, whic h is a particular case of the follo wing sto c hastic F okker–Planc k equation d ρ t = ∆ ρ t dt + 1 2 D 2 ρ t ( σ σ ⊤ ) t dt − ∇ · ( ρ t ( K ∗ ρ t )) dt − ∇ ρ t · σ t dB t , (1) from a sto chastic mo derately interacting particle system giv en by dX i,N t = 1 N N X k =1  K ∗ V N   X i,N t − X k,N t  dt + √ 2 dW i,N t + σ t dB t , (2) where W i,N t and B t are indep endent standard R d -v alued Bro wnian motions, defined on a filtered probability space (Ω , F , ( F t ) t ≥ 0 , P ), K is a singular kernel and V N a suitable scalling. T o the b est of the author’s knowledge , this is the first time that (1) is deriv ed for 2D sto chastic v ortex mo del, by mo derately in teracting particles, in the whole space - extending what w as known b efore for bounded k ernels in [32], on R d , and in [39], for sub-Coulombian kernels, under p erio dic boundary conditions - with pathwise quantitativ e b ounds, in the sense of the relativ e en tropy functional. In the particle system (2), the sto c hastic term σ t dB t accoun ts for a source of randomness that affects all particles, mo deling a shared en vironmental fluctuation. Mo dels of this nature app ear in sev eral contexts, such as prob- lems in statistical mechanics, p opulation dynamics and biology , as well as in the analysis of n umerical Mon te–Carlo sc hemes and related applications; see, e.g., [4], [5], [24], [19], and [29]. The main goal of this pap er is to in vestigate the asymptotics in N of the empiric al me asur e defined b y S N t . = 1 N N X i =1 δ X i,N t , t ⩾ 0 , (3) 2 where δ a is the delta Dirac measure concen trated at a ∈ R d . F or that purp ose, considering V N . = N β V ( N β d · ), w e introduce the mol lifie d empiric al me asur e ρ N t . = V N ∗ S N t = Z R d V N ( · − y ) S N t ( dy ) , in order to apply the relativ e entrop y functional directly to the densities ρ N and ρ , since b y the Csiszar-Kullbac k-Pinsker inequalit y , we get ∥ S N t − ρ t ∥ 2 0 ≲ N − β d + ∥ ρ N t − ρ t ∥ 2 1 CKP ≲ N − β d + H  ρ N t | ρ t  , (4) where the metric in the left hand side of (4), is the Kantoro vic h-Rubinstein metric b et ween tw o probatilit y measures. It follows that, if there exists a constan t C = C ω > 0 suc h that for any N ∈ N , sup t ∈ [0 ,T ] H ( ρ N t | ρ t ) ≤ C ( H ( ρ N 0 | ρ 0 ) + N − θ ) , (5) where θ is an explicit p ositive parameter, by (4) and (5), we hav e, in partic- ular, propagation of chaos for the marginals of the empirical measure of the particle system, see [1, Section 8.3] and [45]. In order to derive (5), w e apply for the first time in the context of mo d- erately interacting particles, the Donsker–V aradhan inequalit y together with the dissipation pro vided b y the Fisher information, to handle the nonlinear- it y . In addition, w e employ localization techniques com bined with the prob- abilistic data setting, to derive estimates for the quadratic v ariation terms in the whole space, as the estimate established in [39], under p erio dic b oundary conditions, do es not apply in the present framew ork. Moreov er, we pro v e the existence of a suitable solution to the equation (1). Related w orks Quan titative estimates for singular interactions ha v e b een substantially de- v elop ed in recen t years. The relative en tropy metho d for establishing quan- titativ e propagation of chaos in McKean–Vlasov systems with singular inter- action kernels was first introduced in [23]. It was developed in the con text of general first-order systems with interaction kernels in W − 1 , ∞ , a framew ork that includes the p oin t v ortex mo del appro ximating the tw o-dimensional Na vier–Stokes equation. W e also refer to [42], where the mean-field limit and quan titative propagation of c haos for McKean–Vlasov equations with singu- lar interaction k ernels are established via the mo dulated energy metho d. In con trast to the relativ e en tropy approac h, which operates at the lev el of the 3 join t la w of the particle system, the mo dulated energy method is formulated directly in terms of the empirical measure asso ciated with the particles. Since then, the aforemen tioned w orks ha ve pa v ed the w ay for establishing quan titative propagation of c haos results. In [2] and [3], the strategies giv en in [23] and [42], w ere adapted to get quan titative deriv ation for the Pat- lak–Keller–Segel mo del, in optimal sub critical regimes. Additionally , [20], b y means of a logarithmic Sob olev inequalit y , deriv ed uniform in time prop- agation of chaos for the 2D vortex mo del, adapting the strategy in [23]. Alternativ ely , with sharp estimates, global in time deriv ation of that mo del, based on relaxation estimates for the limiting equation and inequalities of Riesz mo dulated energy , was derived by [12], see also [41],[11] and [31], for the mo dulated energy metho d. Regarding sub-Coulombian kernels, [6] ob- tained the deriv ation of the mean-field appro ximation for Landau-lik e equa- tions, while [40] obtained global-in-time mean-field con vergence for gradien t (conserv ative) diffusive flo ws. By extending the BBGKY hierarc hy giv en in [30], for singular kernels, [47] obtained the optimal conv ergence rate for the relative entrop y b etw een the k-particle marginals and the correspond- ing tensorized law for weakly interacting diffusions with b ounded interaction k ernels. With this in view we observ e that, the relativ e entrop y metho d has pro v en effectiv e for deriving quantitativ e estimates, for singular kernels, under p eri- o dic boundary conditions, while the modulated energy method, which can be applied in the whole space, has primarily been dev elop ed for deterministic in- teraction. How ev er, more recently , quan titative propagation of c haos for the 2D v ortex model and the 2D Log Gas on the whole space w as obtained in [48] and [49], [7], resp ectiv ely . In these works, the authors exploit Li–Y au-type estimates and Hamilton-type heat kernel estimates to derive suitable b ounds for the limiting solution and its deriv atives up to second order, whic h allo ws them to extend the strategy given in [23], to R d . Quan titative estimates in the whole space were derived, in the context of mo derately in teracting particles 1 with singular kernels - including the Biot- Sa v art, the Keller-Segel and the Dirac delta measure kernels - b y [36] and [37], based on a semigroup approach in tro duced in [17], with further adv ancemen ts in [8], [21], [38], and [27]. It is imp ortant to note that these w orks do not rely on the relative entrop y method, but instead establish quantitativ e estimates in strong functional top ologies. The connection b etw een the time evolution of the relativ e en tropy , of the join t distribution, and mo derate in teracting regime, on the whole space was carried out by [10], based on a combination b et w een the relativ e en tropy and the regularised L 2 -estimate in [35], deriving 1 The moderate in teract mo del w as introduced b y Oelsc hlager in [34],[33] and [35]. 4 a propagation of c haos result for the viscous p orous medium equation. But w as in [39], on the torus, that the equation (1), w as deriv ed, for singular kernels and common noise, in the context of mo derate in teraction regime, by applying the relativ e en tropy directly to the mollified empirical measure. Earlier, also the same domain, the results established in [36] were extended to the setting with en vironmental noise in [26]. It is w orth high- ligh ting, that quan titative en trop y estimates for systems with individual and common noise, as sho wn in [43] for incompressible the Navier-Stok es equa- tions, [9] for the Hegselmann-Krause mo del, and [32] for mean-field systems with b ounded k ernels. F or more results, see [18], [15], [16], and [28]. In con trast to the aforementioned works, our application of the relativ e en tropy metho d pro vides path wise estimates, deriving b ounds directly at the tra jectory level of the particle system. In those w orks, the estimates are obtained at the lev el of the join t la w. In order to present our results, this work is organized as follows. In the next t wo subsections, 1.1 and 1.2, resp ectiv ely , we introduce the notations and assumptions used throughout the text, setting the stage for the subse- quen t discussions. In the subsection 1.3, we present our main results. The section 2 is dedicated to the pro of of our main results. T o ease the presenta- tion, the Appendix, provides the corresp onding suitable notion of a solution for the limiting equation, as w ell as some inequalities used throughout the pro ofs. 1.1 Notations F or d ≥ 1, the space of probability densit y functions on R d is denoted by P ( R d ). F or a normed v ector space U , w e denote its norm by ∥ · ∥ U , except for L p spaces, where we write ∥ · ∥ p , for p ≥ 1. The Kan torovic h-Rubinstein metric reads, for an y t wo probabilit y measures µ and ν on R d , ∥ µ − ν ∥ 0 . = sup  Z R d ϕ d ( µ − ν ) ; ∥ ϕ ∥ ∞ , ∥ ϕ ∥ Lip ≤ 1  . (6) Let f and g b e p ositive probabilit y density functions on R d . The relativ e en tropy (or Kullback–Leibler divergence) of f with resp ect to g is defined as H ( f | g ) . = Z R d f ( x ) ln  f ( x ) g ( x )  dx. Also the Fisher information of f with respect to g is given b y I ( f | g ) . = Z R d f ( x )     ∇ ln f ( x ) g ( x )     2 dx = Z R d ( g ( x )) 2 f ( x )     ∇  f ( x ) g ( x )      2 dx. 5 1.2 Assumptions ( A V ) Let V N : R d → R b e the scaling giv en b y V N ( y ) = N β V ( N β d y ), β ∈ (0 , 1), with V : R d → R satisfying |∇ V | ≤ C d V , (7) and for α > d/ 2 V ( y ) ≤ C d (1 + | y | 2 ) − α . (8) ( A ∇· K ) It holds ∇ · K = 0 and there exists K 0 ∈ L ∞ , with ∥ K 0 ∥ ∞ ≤ 1 / 4, such that K = ∇ · K 0 . (9) ( A σ ) The co efficient σ : [0 , T ] → R d × d is measurable and b ounded. ( A ρ 0 ) The initial condition ρ 0 is tak en such that ρ 0 ∈ W 2 , ∞  R d  and |∇ ln ρ 0 ( x ) | 2 ≤ e C 1 (1 + | x | 2 ) , (10)   ∇ 2 ln ρ 0 ( x )   ≤ e C 2 (1 + | x | 2 ) (11) and ρ 0 ( x ) ≤ e C 3 exp  − e C − 1 3 | x | 2  . (12) ( A X 0 ) The initial particle system ( X i,N 0 ) N i =1 , N ∈ N is a sequence of i.i.d with la w ρ 0 . Remark 1. F or an example of mol lifier V : R d → R satisfying  A V  , take V ( y ) . = C d exp  − p 1 + | y | 2  with C d . =  R R d exp  − p 1 + | y | 2  dy  − 1 . Remark 2. The Biot-Savart kernel in dimension two is given by K ( x ) = 1 2 π x ⊥ | x | 2 (13) and then, it is diver genc e in the sense of distributions, of a L ∞ matrix K 0 , namely K = ∇ · K 0 , with K 0 ( x 1 , x 2 ) =  1 2 π arctan  x 1 x 2  I . (14) It fol lows that, this singular kernel satisfies the Assumption  A ∇· K  . 6 1.3 Statemen t of the main results The first result, whose pro of is p ostp oned to the App endix, establishes that the limiting equation is w ell p osed in the appropriate functional framew ork. Indeed, it pro vides the regularit y prop erties required in the proof of the main results. Theorem 1. Assume ( A ∇· K ) , ( A σ ) , and ( A ρ 0 ) . Then, ther e exist T 1 > 0 , C 3 > 0 , such that if e C 3 < C 3 , • ther e exists a p athwise solution of (1) with initial c ondition ρ 0 , such that ρ ∈ L ∞  [0 , T 1 ]; W 2 , 1 ∩ W 2 , ∞ ( R d )  , P -a.s.; • the solution ρ verifies, P -a.s., that |∇ ln ρ ( t, x ) | 2 ≤ C 1 (1 + | x − X t | 2 ) , (15)   ∇ 2 ln ρ ( t, x )   ≤ C 2 (1 + | x − X t | 2 ) (16) and ρ ( t, x ) ≤ exp ( − 2 π | x − X t | 2 ) , (17) wher e dX t = σ t dB t , (18) for al l t ∈ [0 , T 1 ] and c onstants C 1 , C 2 dep endent on e C 1 , e C 2 and e C 3 . W e are no w able to extend what was kno wn b efore for b ounded k ernels in the whole space as in [32] and in [39], for sub-Coulombian k ernels, under p erio dic b oundary conditions. Our main result can b e stated as follows. Theorem 2. Assume ( A V ) , ( A ∇· K ) , ( A σ ) , ( A ρ 0 ) , ( A X 0 ) and let ρ b e a solution of (1), given by The or em 1. In addition, let the dynamics of the p article system b e given by (2) and lim N →∞ N θ H ( ρ N 0 | ρ 0 ) = 0 , P − a.s. wher e θ = min  1 − β (2 + 2 d + 2 α ); 1 2 − β  1 + 1 d   − δ with δ > 0 , such that θ > 0 , d ≥ 1 and β ∈  0 , 1 2  1+ 1 d   ∩  0 , 1  2+ 2 d +2 α   . Then, ther e exists a time T ∈ (0 , T 1 ) , such that lim N →∞ N θ sup t ∈ [0 ,T ] H ( ρ N t | ρ t ) = 0 , P − a.s. 7 As a corollary of the previous theorem, we establish propagation of chaos for the marginals of the empirical measure of the particle system. In the sense of [1, Section 8.3], it corresp onds to conv ergence of the empirical measure with resp ect to the Kantoro vic h–Rubinstein metric, given in (6). Corollary 1. Under assumptions of The or em 2, we have lim N →∞ N θ − δ sup t ∈ [0 ,T ]   S N t − ρ t   2 0 = 0 , P − a.s. Remark 3. We observe that, c omp ar e d with pr evious works on mo der ate p article systems without c ommon noise, we lose the or der of c onver genc e. This is b e c ause the estimation of the martingale term in our work is of or der 1 2 − β  1 + 1 d  , while in [36] is the or der 1 2 − β 2 , se e The or em 1.3 in [36]. When envir onmental noise is taken into ac c ount, the over al l fr amework r emains c omp ar able to that of [26]. F or the same r e ason, the r ange of the p ar ameter β is smal ler in our work. However, in the pr esent work we obtain an impr ove d c onver genc e r ate c omp ar e d to that establishe d in [39]. 2 Pro ofs of main results In this section w e presen t the pro ofs of our main results. In Subsection 2.1, b y applying the Itˆ o’s form ula, suc h as in [39], we deriv e an evolution equation for the relative entrop y functional of the regularized empirical measure with resp ect to the solution of (1). In Subsection 2.2, w e deal with the quadratic v ariation terms arising from the application of Itˆ o’s form ula. In Subsection 2.3, w e address the nonlinear contributions app earing in the ev olution equation for the relative entrop y functional obtained in the first step. Finally , in Subsections 2.4 and 2.5, com bining the previous estimates with Gron wall’s lemma and suitable lo calization tec hniques, w e close the argument and complete the pro of. 2.1 Time evolution of the relative entrop y By applying Itˆ o’s formula, following the same arguments dev elop ed in Section 2 of [39], w e derive the corresp onding evolution equation for the en tropy functional b etw een the regularized empirical measure ρ N and the solution ρ of (1): H ( ρ N t | ρ t ) − H ( ρ N 0 | ρ 0 ) = I t + I I t + I I I t + M N t + I V t , (19) 8 where I t = Z t 0 Z R d ρ N s ∇ ln  ρ s ρ N s  [ K ∗ ρ s − K ∗ ρ N s ] dx ds − Z t 0 Z R d ∇ ln  ρ s ρ N s   S N s , V N ( x − · )[ K ∗ ρ N s ( · ) − K ∗ ρ N s ( x )]  dx ds, (20) I I t + I V t = 1 2 N 2 N X i =1 Z R d Z t 0 1 ρ N s |∇ V N ( x − X i,N s ) | 2 ds dx, (21) I I I t = − 1 2 Z t 0 I  ρ N s | ρ s  ds, (22) and M N t = Z R d Z t 0  ln( ρ s ) − ln( ρ N s )  σ ⊤ s ∇ ρ N s dB s dx + Z R d Z t 0 σ ⊤ s ∇ ρ N s dB s dx + 1 N N X i =1 Z R d Z t 0  ln( ρ s ) − ln( ρ N s )  ∇ V N ( x − X i s ) dW i s dx − 1 N N X i =1 Z R d Z t 0 ∇ V N ( x − X i s ) dW i s dx + Z R d Z t 0 ρ N s ρ s σ ⊤ s ∇ ρ s dB s dx. (23) 2.2 Estimates for I I t + I V t : quadratic v ariation terms W e no w derive estimates for the quadratic v ariation terms app earing in (19). The main nov elt y consists in introducing a suitable stopping time in order to handle the integral ov er the whole space arising in this term. It is imp or- tan t to emphasize that the argument used in [39], which relies on p erio dic b oundary conditions, is no longer av ailable in the present setting. First, taking in to accoun t Assumption ( A V ) in (21), w e ha ve I I t + I V t ≤ N 2 β /d 2 N 2 N X i =1 Z R d Z t 0 1 ρ N s | V N ( x − X i,N s ) | 2 ds dx. (24) 9 W e introduce the following stopping time: τ N = inf n t ≥ 0; ∃ i ∈ { 1 , ..., N } , | X i,N t | ≥ N β o ∧ T . (25) By deca y of ρ 0 in (12), along with Borel-Cantelli Lemma, almost surely , there exists N 0 = N 0 ( ω ) such that τ N ( ω ) > 0, for all N ≥ N 0 . Indeed, setting Λ N . = n ∃ i ∈ { 1 , ..., N } , | X i,N 0 | ≥ N β o = N [ i =1 n | X i,N 0 | ≥ N β o (26) b y Assumption ( A X 0 ), w e deriv e P  Λ N  (26) ≤ N X i =1 P  | X i,N 0 | ≥ N β  = N X i =1 Z | x |≥ N β ρ 0 ( x ) dx ≤ C d N X i =1 ( N β ) d − 2 e − e C − 1 3 ( N β ) 2 ≤ C d N ( N β ) d − 2 e − e C − 1 3 ( N β ) 2 (27) where in the second inequalit y w e are using (12) to get Z | x |≥ N β ρ 0 ( x ) dx ≤ e C 3 Z | x |≥ N β e − e C − 1 3 | x | 2 dx = C d Z ∞ r = N β e − e C − 1 3 r 2 r d − 1 dr ≤ C d ( N β ) d − 2 e − e C − 1 3 ( N β ) 2 . (28) It follo ws by Borel-Can telli Lemma, P ∞ \ M =1 ∞ [ N = M Λ N ! = 0 . (29) With this in view for all ω ∈ Λ 0 , with Λ 0 . = ∞ [ M =1 ∞ \ N = M (Λ N ) c (30) there exists N 0 = N 0 ( ω ), suc h that τ N ( ω ) > 0, for all N ≥ N 0 , since t 7→ X i,N t ( ω ) is contin uous, almost surely . 10 Let s ∈ (0 , τ N ). W e observe that, I N . = Z R d | V N ( x − X i,N s ) | 2 ρ N s ( x ) dx = Z | x |≤ 2 | X i,N s | | V N ( x − X i,N s ) | 2 ρ N s ( x ) dx + Z | x | > 2 | X i,N s | | V N ( x − X i,N s ) | 2 ρ N s ( x ) dx. (31) No w, if | x | ≤ 2 | X i,N s | , then | x | ≤ 2 N β for s ∈ (0 , τ N ), and since V N ≤ N β , w e derive for α > d/ 2 V N ( x − X i,N s ) ≤ N β  1 + | x | 2  α − α ≤ C α N 2 αβ + β  1 + | x | 2  − α . (32) On the other hand, if | x | > 2 | X i,N s | we hav e | x − X i,N s | > | x | 2 and by decay of V , giv en in (8), we get V N ( x − X i,N s ) (8) ≤ C α N β  1 + | N β d ( x − X i,N s ) | 2  − α ≤ C α N β 1 + | N β d x | 2 4 ! − α ≤ C α N β  1 + | x | 2 4  − α . (33) Therefore b y (31), (32) and (33), w e find I N ≤ C α Z | x |≤ 2 | X i,N s | | V N ( x − X i,N s ) | ρ N s ( x )  N 2 αβ + β  1 + | x | 2  − α  dx + C α Z | x | > 2 | X i,N s | | V N ( x − X i,N s ) | ρ N s ( x ) N β  1 + | x | 2 4  − α ! dx ≤ C α N 2 αβ + β Z R d | V N ( x − X i,N s ) | ρ N s ( x )  1 + | x | 2  − α dx + C α N 2 αβ + β Z R d | V N ( x − X i,N s ) | ρ N s ( x )  1 + | x | 2 4  − α dx = C α N 2 αβ + β Z R d | V N ( x − X i,N s ) | ρ N s ( x ) ϕ ( x ) dx, (34) with ϕ ( x ) . =  1 + | x | 2  − α +  1 + | x | 2 4  − α ! (35) 11 Hence b y (34) in (24) w e obtain I I t ∧ τ N + I V t ∧ τ N = 1 2 N 2 Z t ∧ τ N 0 N X i =1 Z R d 1 ρ N s ( x ) |∇ V N ( x − X i,N s ) | 2 dx ds ≤ N β N 2 β d 2 N 2 Z t ∧ τ N 0 N X i =1 Z R d 1 ρ N s ( x ) | V N ( x − X i,N s ) | 2 ds dx ≤ C α N β (2+ 2 d +2 α ) N 2 Z t ∧ τ N 0 N X i =1 Z R d | V N ( x − X i,N s ) | ρ N s ( x ) ϕ ( x ) dx ds ≤ C α N β (2+ 2 d +2 α ) N Z t ∧ τ N 0 Z R d 1 N P N i =1 V N ( x − X i,N s ) ρ N s ( x ) ϕ ( x ) dx ds ≤ C α N β (2+ 2 d +2 α ) N Z t ∧ τ N 0 Z R d ϕ ( x ) dx ds α>d/ 2 ≤ C α,T N β (2+ 2 d +2 α ) N . (36) 2.3 Estimates for I t : the nonlinear terms W e no w address the nonlinear terms in (19). The main no v elty is to use Donsk er-V aradhan inequality (79), in the con text of mo derate in teractions, along with the dissipation that comes from of Fisher information and As- sumption ( A ∇· K ), to tac kle with the singularit y of k ernel K . First, w e s et I t = Z t 0 Z R d ρ N s ∇ ln  ρ s ρ N s  [ K ∗ ρ s − K ∗ ρ N s ] dx ds − Z t 0 Z R d ∇ ln  ρ s ρ N s   S N s , V N ( x − · )[ K ∗ ρ N s ( · ) − K ∗ ρ N s ( x )]  dx ds . = I 1 t − I 2 t , (37) where I 1 t = − Z t 0 Z R d  ρ N s ∇ 2 ρ s ρ s  [ K 0 ∗ ρ s − K 0 ∗ ρ N s ] dx ds − Z t 0 Z R d  ∇ ρ s ∇  ρ N s ρ s  [ K 0 ∗ ρ s − K 0 ∗ ρ N s ] dx ds . = − I 1 , 1 t − I 1 , 2 t . (38) 12 No w by ϵ -Y oung inequality and conv olution inequality , w e deduce I 1 , 2 t = Z t 0 Z R d " ∇ ρ s p ρ N s ρ s ! ρ s p ρ N s ! ∇  ρ N s ρ s  # [ K 0 ∗ ρ s − K 0 ∗ ρ N s ] dx ds ≤ ϵ Z t 0 Z R d ρ 2 s ρ N s     ∇  ρ N s ρ s      2 dx ds + C ϵ Z t 0 Z R d |∇ ρ s | 2 ρ 2 s ρ N s   K 0 ∗ ( ρ s − ρ N s )   2 dx ds ≤ ϵ Z t 0 Z R d ρ 2 s ρ N s     ∇  ρ N s ρ s      2 dx ds + C ϵ Z t 0 ∥ K 0 ∥∥ ρ s − ρ N s ∥ 2 1 Z R d |∇ ρ s | 2 ρ 2 s ρ N s dx ds. (39) W e observe that b y Donsker-V aradhan inequalit y (79), for η >> 1 Z R d ρ N s |∇ ρ s | 2 ρ 2 s dx (79) ≤ η H ( ρ N s | ρ s ) + η ln  Z R d ρ s exp  |∇ ln ρ s | 2 η  dx  (15) , (17) ≤ η H ( ρ N s | ρ s ) + η ln  Z R d exp ( − 2 π | x − X t | 2 ) exp  C (1 + | x − X t | 2 ) η  dx  ≤ η H ( ρ N s | ρ s ) + C η , whic h implies by Csisz´ ar-Kullbac k-Pinsker inequality (78) and (39) I 1 , 2 t ≤ ϵ Z t 0 I ( ρ N s | ρ s ) ds + C ϵ,η ,K 0 Z t 0 H ( ρ N s | ρ s ) ds. (40) Regarding term I 1 , 1 t in (38), b y con volution inequalit y , we find I 1 , 1 t = Z t 0 ∥ K 0 ∥ ∞ ∥ ρ N s − ρ s ∥ 1 Z R d     ∇ 2 ρ s ρ s     ρ N s dx ds. (41) In addition, if C 1 , 2 . = max ( C 1 , C 2 ) since     ∇ 2 ρ s ( x ) ρ s ( x )     ≤ |∇ 2 ln ( ρ s )( x ) | + |∇ ln ( ρ s ( x )) | 2 (15) , (16) ≤ C 1 , 2 (1 + | x − X t | 2 ) , (42) 13 b y Donsker-V aradhan inequality (79), we obtain Z R d ρ N s     ∇ 2 ρ s ρ s     dx (79) ≤ η H ( ρ N s | ρ s ) + η ln   Z R d ρ s exp      ∇ 2 ρ s ρ s    η   dx   (42) , (17) ≤ η H ( ρ N s | ρ s ) + η ln  Z R d exp ( − 2 π | x − X t | 2 ) exp  C 1 , 2 (1 + | x − X t | 2 ) η  dx  η >> 1 ≤ η H ( ρ N s | ρ s ) + η ln (1) ≤ η H ( ρ N s | ρ s ) , (43) for η >> 1. Therefore, b y (43) in (41), w e get I 1 , 1 t ≤ C η ,K 0 Z t 0 H ( ρ N s | ρ s ) ds. (44) Th us, from (40) and (44), w e hav e the following estimate for the term I 1 t in (37): I 1 t ≤ C ϵ,K 0 ,η Z t 0 H ( ρ N s | ρ s ) ds + ϵ Z t 0 I ( ρ N s | ρ s ) ds. (45) No w, w e will fo cus on the term I 2 t in (37). First, w e recall that b y Assumption  A ∇· K  , K = ∇ · K 0 with K 0 ∈ L ∞ . Hence, w e arriv e at   K ∗ ρ N s ( · ) − K ∗ ρ N s ( x )   =   K 0 ∗ ∇ ρ N s ( · ) − K 0 ∗ ∇ ρ N s ( x )   ≤ Z R d | K 0 ( · − y ) − K 0 ( x − y ) | |∇ ρ N s ( y ) | dy ≤ Z R d 2 ∥ K 0 ∥ ∞ |∇ ρ N s ( y ) | dy = 2 ∥ K 0 ∥ ∞ Z R d |∇ ρ N s ( y ) | dy . (46) Additionally , ρ N s ∈ P  R d  , P -a.s. and then b y Holder’s inequalit y we obtain Z R d |∇ ρ N s | dx = Z R d |∇ ρ N s | p ρ N s p ρ N s dx Holder ≤  Z R d |∇ ρ N s | 2 ρ N s dx  1 2  Z R d ρ N s dx  1 2 =  Z R d |∇ ρ N s | 2 ρ N s dx  1 2 . (47) 14 No w from Leibniz’s rule w e deduce, for η >> 1 Z R d |∇ ρ N s | 2 ρ N s dx = Z R d    ∇  ρ s ρ N s ρ s     2 ρ N s dx ≤ 2 Z R d    ∇ ρ s  ρ N s ρ s     2 ρ N s dx + 2 Z R d    ρ s ∇  ρ N s ρ s     2 ρ N s dx ≤ 2 Z R d ρ N s |∇ ρ s | 2 ρ 2 s dx + 2 Z R d ρ N s     ∇ ln  ρ N s ρ s      2 dx ≤ 2 η H ( ρ N s | ρ s ) + 2 Z R d ρ N s     ∇ ln  ρ N s ρ s      2 dx, (48) where in the last inequalit y , w e are using Donsk er-V aradhan inequality (79): Z R d ρ N s |∇ ρ s | 2 ρ 2 s dx (79) ≤ η H ( ρ N s | ρ s ) + η ln  Z R d ρ s exp  |∇ ln ρ s | 2 η  dx  (15) , (17) ≤ η H ( ρ N s | ρ s ) + η ln  Z R d exp ( − 2 π | x − X t | 2 ) exp  C 1 (1 + | x − X t | 2 ) η  dx  η >> 1 ≤ η H ( ρ N s | ρ s ) + η ln (1) ≤ η H ( ρ N s | ρ s ) . F rom (46), (47) and (48), recalling I 2 t in (37), since by Assumption  A ∇· K  ∥ K 0 ∥ ∞ ≤ 1 / 4, we ha v e    S N s , V N ( x − · )[ K ∗ ρ N s ( · ) − K ∗ ρ N s ( x )]    ≤ 2 ∥ K 0 ∥ ∞  S N s , V N ( x − · )  ×  2 η H ( ρ N s | ρ s ) + 2 I  ρ N s | ρ s  1 2 ≤ 1 / 2 ρ N s  2 η H ( ρ N s | ρ s ) + 2 I  ρ N s | ρ s  1 2 , 15 whic h implies, by ϵ -Y oung and Holder inequalities, I 2 t . = Z t 0 Z R d ∇ ln  ρ s ρ N s   S N s , V N ( x − · )[ K ∗ ρ N s ( · ) − K ∗ ρ N s ( x )]  dx ds ≤ Z t 0 Z R d     ∇ ln  ρ s ρ N s      (1 / 2) ρ N s  2 η H ( ρ N s | ρ s ) + 2 I  ρ N s | ρ s  1 2 dx ds ≤ Z t 0 Z R d     ∇ ln  ρ s ρ N s      (1 / 2)( ρ N s ) 1 / 2 ( ρ N s ) 1 / 2  2 η H ( ρ N s | ρ s )  1 2 dx ds + Z t 0 Z R d     ∇ ln  ρ s ρ N s      (1 / 2) ρ N s  2 I  ρ N s | ρ s  1 2 dx ds Y oung ≤ ϵ Z t 0 Z R d ρ N s     ∇ ln  ρ s ρ N s      2 dx ds + C ϵ,η Z t 0 H ( ρ N s | ρ s ) ds + √ 2 / 2 Z t 0  I  ρ N s | ρ s  1 2 Z R d     ∇ ln  ρ s ρ N s      ( ρ N s ) 1 / 2 ( ρ N s ) 1 / 2 dx ds Holder ≤ ( ϵ + √ 2 / 2) Z t 0 I  ρ N s | ρ s  ds + C ϵ,η Z t 0 H ( ρ N s | ρ s ) ds. (49) Finally , b y (45) and (49) in (37), for all ϵ > 0 and η >> 1, w e conclude with the follo wing estimate for the nonlinear term: I t ≤ C ϵ,K 0 ,η Z t 0 H ( ρ N s | ρ s ) ds + (2 ϵ + √ 2 / 2) Z t 0 I ( ρ N s | ρ s ) ds. (50) 2.4 Application of Gronw all’s Lemma In this subsection, w e conclude our estimates, b y applying Gron wall’s in- equalit y . Indeed, w e put (22) , (36) , (50) , into (19), to get H ( ρ N t ∧ τ N | ρ t ∧ τ N ) − H ( ρ N 0 | ρ 0 ) ≤ C ϵ,K 0 ,η Z t ∧ τ N 0 H ( ρ N s | ρ s ) ds + (2 ϵ + √ 2 / 2) Z t ∧ τ N 0 I ( ρ N s | ρ s ) ds − Z t ∧ τ N 0 I ( ρ N s | ρ s ) ds + C α,T N β (2+ 2 d +2 α ) N + M N t . (51) 16 With the same computations as in [39], for all δ ∈ (0 , θ 1 ) there exists a random v ariable A 0 , with finite momen ts suc h that sup t ∈ [0 ,T ] | M N t | ≤ A 0 N − θ 1 + δ , whic h allows us to express (51), for ϵ << 1, as H ( ρ N t ∧ τ N | ρ t ∧ τ N ) − H ( ρ N 0 | ρ 0 ) ≲ Z t ∧ τ N 0 H ( ρ N s | ρ s ) ds + C α,T N − θ 2 + A 0 N − θ 1 + δ , (52) with θ 2 . = 1 − β (2 + 2 d + 2 α ). F rom Gron w all’s Lemma in (52), recalling (26), (30) and (29), for all ω ∈ Λ 0 , there exists N 0 ( ω ) such that for N ≥ N 0 ( ω ), τ N ( ω ) > 0 and sup t ∈ [0 ,T ] H ( ρ N t ∧ τ N | ρ t ∧ τ N ) ≲  H ( ρ N 0 | ρ 0 ) + N − θ + A 0 N − θ  (53) with θ . = min  1 − β (2 + 2 d + 2 α );  1 2 − β  1 + 1 d   − δ  . 2.5 End of the pro of: Remo ving the stopping time W e no w remov e the stopping time τ N in tro duced in (53) by exploiting the probabilistic framework pro vided by Assumptions ( A ρ 0 ) and  A X 0  . These assumptions ensure sufficien t control on the initial data and on the particle system, allowing us to pass from the lo calized estimates to global-in-time b ounds with high probability , and finally eliminate the stopping time. First, since K = ∇ · K 0 , K 0 ∈ L ∞ and V N = N β V ( N β /d · ), by (2), w e ha ve sup t ∈ [0 ,T ] | X i,N t | ≤ | X i,N 0 | + T ∥∇ V ∥ 1 ∥ K 0 ∥ ∞ N β + sup t ∈ [0 ,T ]     Z t 0 √ 2 dW i,N s     + sup t ∈ [0 ,T ]     Z t 0 σ s dB s     , (54) 17 whic h implies for e C K 0 ,V . = (1 − T ∥∇ V ∥ 1 ∥ K 0 ∥ ∞ ) { τ N < T } ⊂ N [ i =1 ( sup t ∈ [0 ,T ] | X i,N t | ≥ N β ) = ( ∃ i ; sup t ∈ [0 ,T ] | X i,N t | ≥ N β ) ⊂ ( ∃ i ; | X i,N 0 | + sup t ∈ [0 ,T ]     Z t 0 √ 2 dW i,N s     + sup t ∈ [0 ,T ]     Z t 0 σ s dB s     ≥ e C K 0 ,V N β ) ⊂ L N 1 ∪ L N 2 ∪ L N 3 , (55) with L N 1 . = n ∃ i ; | X i,N 0 | ≥ C K 0 ,V N β o , (56) L N 2 . = ( ∃ i ; sup t ∈ [0 ,T ]     Z t 0 dW i,N s     ≥ C K 0 ,V N β ) (57) and L N 3 . = ( ∃ i ; sup t ∈ [0 ,T ]     Z t 0 σ s dB s     ≥ C K 0 ,V N β ) (58) with C K 0 ,V . = (1 − T ∥∇ V ∥ 1 ∥ K 0 ∥ ∞ ) / 3 √ 2 > 0, for T < (1 / ∥∇ V ∥ 1 ∥ K 0 ∥ ∞ ). W e note that, with the same computations like those led (27), w e deduce P  L N 1  ≤ N ( C K 0 ,V N β ) d − 2 e − ( C K 0 ,V N β ) 2 (59) W e no w tak e care ab out the L N 2 . Indeed, w e note that W i,N t = R t 0 dW i,N s , and since | W i,N t | 2 = d X j =1 ( W i,N ,j t ) 2 , if | W i,N t | ≥ a , then there exists j = 1 , ..., d , such that | W i,N ,j t | ≥ a √ d . There- fore,  sup t ≤ T | W i,N t | ≥ a  ⊂ d [ j =1  sup t ≤ T | W i,N ,j t | ≥ a √ d  , (60) 18 w e obtain P  L N 2  (57) , (60) ≤ N X i =1 d X j =1 P  sup t ≤ T | W i,N ,j t | ≥ C K 0 ,V N β √ d  ≤ r T 2 π 4 dN C K 0 ,V N β √ d × exp −  C K 0 ,V N β √ d  2 / (2 T ) ! , (61) where in the second inequalit y , we are using (8.3)’ in Chapter 2 of [25]. No w we will derive an analogous estimate for L N 3 . First, we observe that, if σ = 0 w e are done, thus supp ose σ  = 0. Since σ is b ounded, the function defined by t 7→ R t 0 σ 2 s ds is contin uous and increasing. Hence, recalling (18), b y Theorem 4.6, in Chapter 3 of [25], for all j ∈ { 1 , ..., d } , we get sup t ≤ T | X j t | = sup t ≤ T | e B j ⟨ X ⟩ t | ≤ sup t ≤∥ σ ∥ 2 ∞ T | e B j t | , (62) where ( e B t ) t ≥ 0 is the time-change, whic h in particular, is a one dimensional standard Bro wnian motion. In addition, since | X t | 2 = d X j =1 | X j t | 2 for X j t = Z t 0 σ s dB j s , if | X t | ≥ a , then there exists j = 1 , ..., d , suc h that | X j t | ≥ a √ d . Therefore, for all a > 0, w e find  sup t ≤ T | X t | ≥ a  ⊂ d [ j =1  sup t ≤ T | X j t | ≥ a √ d  (62) ⊂ d [ j =1 ( sup t ≤∥ σ ∥ 2 ∞ T | e B j t | ≥ a √ d ) . (63) 19 It follo ws that, b y (62) and (63) P  L N 3  (58) , (63) ≤ N X i =1 d X j =1 P sup t ≤∥ σ ∥ 2 ∞ T | e B j t | ≥ C K 0 ,V N β √ d ! ≤ r ∥ σ ∥ 2 ∞ T 2 π 4 dN C K 0 ,V N β √ d × exp −  C K 0 ,V N β √ d  2 / (2 ∥ σ ∥ 2 ∞ T ) ! , (64) where in the second inequalit y , we are using (8.3)’ in Chapter 2 of [25]. Hence, b y (59), (61) and (64) in (58), we find X N ∈ N P  τ N < T  < ∞ (65) and b y Borel-Cantelli Lemma, w e arrive at P ∞ \ M =1 ∞ [ N = M { τ N < T } ! = 0 . (66) Finally , b y (30) and (66), for T < min ( T 1 ; (1 / ∥∇ V ∥ 1 ∥ K 0 ∥ ∞ )) for all ω ∈ Λ, with Λ . = ∞ [ M =1 ∞ \ N = M (Λ N ) c ! ∩ ∞ [ M =1 ∞ \ N = M { τ N = T } ! , (67) there exists N 0 ( ω ) such that for N ≥ N 0 ( ω ), τ N ( ω ) = T and (53) holds. It follo ws that, almost surely lim N →∞ N θ sup t ∈ [0 ,T ] H ( ρ N t | ρ t ) = 0 , whic h conclude the pro of of Theorem. 3 App endix This appendix contains the pro of of Theorem 1 together with sev eral auxiliary inequalities used in the previous sections. W e b egin b y establishing Theorem 1, and subsequently derive the additional estimates required throughout the pap er. 20 3.1 Pro of of Theorem 1 3.1.1 Solving (1) for σ = 0 When σ = 0, a solution for (1) in the sense of Theorem 1, with exception of (17), was constructed in Sections 3 and 4 of [48] and [49]. With this in view, w e only need to verify that, it is p ossible to tak e ˜ C 3 sufficien tly small, in order to get that Gaussian decay . Pr o of of (17). By Theorem 3 in [13] along with (12), we obtain there exists C > 0 such that for all ( t, x ) ∈ [0 , T 1 ] × R d ¯ ρ ( t, x ) ≤ C e C 3 Z R d 1 t d/ 2 exp − e C 3 | x − y | 2 + 8 t | y | 2 8 t e C 3 ! dy . (68) No w since | x − y | 2 = | x | 2 − 2 x · y + | y | 2 . w e find e C 3 | x − y | 2 + 8 t | y | 2 = e C 3 | x | 2 − 2 e C 3 x · y + e C 3 | y | 2 + 8 t | y | 2 = e C 3 | x | 2 − 2 e C 3 x · y + ( e C 3 + 8 t ) | y | 2 . (69) Additionally , we get ( e C 3 + 8 t ) | y | 2 − 2 e C 3 x · y . = ( e C 3 + 8 t ) | y | 2 − 2 e C 3 e C 3 + 8 t x · y ! . and since | y | 2 − 2 ax · y = | y − ax | 2 − a 2 | x | 2 , with a = e C 3 e C 3 + 8 t . w e arrive at  ( e C 3 + 8 t ) | y | 2 − 2 e C 3 x · y  = ( e C 3 + 8 t ) | y − ax | 2 − e C 2 3 e C 3 + 8 t | x | 2 . (70) Therefore b y (69) and (70) w e deriv e exp − e C 3 | x − y | 2 + 8 t | y | 2 8 t e C 3 ! = exp  − | x | 2 e C 3 + 8 t  exp − e C 3 + 8 t 8 t e C 3 | y − ax | 2 ! . (71) 21 and letting z = y − ax we ha ve ¯ ρ ( x, t ) ≤ C e C 3 1 t d/ 2 e − | x | 2 e C 3 +8 t Z R d exp − e C 3 + 8 t 8 t e C 3 | z | 2 ! dz ≤ C e C 3 1 t d/ 2 e − | x | 2 e C 3 +8 t π 8 t e C 3 e C 3 + 8 t ! d/ 2 ≤ C e C 3 ( π 8 e C 3 ) d/ 2 1 ( e C 3 + 8 t ) d/ 2 exp  − | x | 2 e C 3 + 8 t  (72) since Z R d e − a | z | 2 dz =  π a  d/ 2 . It follo ws that taking e C 3 < C 3 . = min  1 2 π ; 1 (8 π ) d/ 2 C  , (73) for t ∈ (0 , T ] such that t < T 1 . = min e C 3 8 ; 8(1 − 2 π e C 3 ) 2 π ! (74) w e end up with ¯ ρ ( x, t ) ≤ exp ( − 2 π | x | 2 ) . (75) 3.1.2 Solving (1) for σ  = 0 W e assume ˜ ρ solves the deterministic PDE ∂ t ˜ ρ − ∆ ˜ ρ + ∇ · ( ˜ ρ t K ∗ ˜ ρ t ) = 0 (76) No w we define ρ ( t, x ) . = ˜ ρ ( t, x − X t ), where dX t = σ t dB t , (77) Hence, b y applying the Itˆ o’s formula, since that σ do es not dep ends on spatial v ariable, w e find that ρ solves (1), in the sense of Theorem 1. 22 3.2 Inequalities In this section, w e present several inequalities used throughout the text. The first one is the classical Csisz´ ar-Kullbac k-Pinsk er inequality , which is a fundamen tal result in information theory . It quantifies ho w control of the relativ e entrop y implies control of the total v ariation distance. Lemma 3 (Csisz´ ar-Kullbac k-Pinsk er inequality) . It holds for f , g ∈ P  R d  , ∥ f − g ∥ 2 1 ≲ H ( f | g ) . (78) No w, we recall the Donsk er–V aradhan inequality (see [23]) whic h follows from the v ariational represen tation of the relativ e en tropy . 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