Propagation of chaos for a subcritical Keller–Segel model

www.imstat.org/aihp 2015, Vol. 51, No. 3, 965–992

DOI:10.1214/14-AIHP606

© Association des Publications de l’Institut Henri Poincaré, 2015

Propagation of chaos for a subcritical Keller–Segel model

David Godinho

and Cristobal Quiñinao

aLaboratoire d’Analyse et de Mathématiques Appliquées, CNRS UMR 8050, Université Paris-Est, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France. E-mail:david.godinho-pereira@u-pec.fr

bThe Mathematical Neuroscience Laboratory, CIRB and INRIA Bang Laboratory, Collège de France, 11, place Marcelin Berthelot, 75005 Paris, France / UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France.

E-mail:cristobal.quininao@college-de-france.fr

Received 20 June 2013; revised 13 December 2013; accepted 23 January 2014

Abstract. This paper deals with a subcritical Keller–Segel equation. Starting from the stochastic particle system associated with it, we show well-posedness results and the propagation of chaos property. More precisely, we show that the empirical measure of the system tends towards the unique solution of the limit equation as the number of particles goes to infinity.

Résumé. Cet article traite de l’équation de Keller–Segel dans un cadre sous-critique. À l’aide du système de particules en lien avec cette équation, nous montrons des résultats d’existence et d’unicité, puis la propagation du chaos pour ce dernier. Plus précisément, nous montrons que la mesure empirique du système tend vers l’unique solution de l’équation limite lorsque le nombre de particules tend vers l’infini.

MSC:65C35

Keywords:Keller–Segel; Propagation of chaos; Well-posedness

1. Introduction and main results

The subject of this paper is the convergence of a stochastic particle system to a nonlinear and nonlocal equation which can be seen as a subcritical version of the classical Keller–Segel equation.

1.1. The subcritical Keller–Segel equation Consider the equation:

∂f_t(x)

∂t =χ∇x·

(K∗f_t)(x)f_t(x)

+_xf_t(x), (1.1)

wheref:R+×R²→Randχ >0. The force field kernelK:R²→R²comes from an attractive potential:R²→R and is defined by

K(x):= x

|x|^α+1 = −∇

1

α−1|x|^1−α

(x)

, α∈(0,1). (1.2)

The standard Keller–Segel equation correspond to the critical caseK(x)=x/|x|²(i.e., more singular at x=0) and it describes a model of chemotaxis, i.e., the movement of cells (usually bacteria or amoebae) which are attracted

by some chemical substance that they produce. This equation has been first introduced by Keller and Segel in [15,16].

Blanchet, Dolbeault and Perthame showed in [4] some nice results on existence of global weak solutions if the non- negative parameterχ(which is the sensitivity of the bacteria to the chemo-attractant) is smaller than 8π/MwhereM is the initial mass (hereM will always be 1 since we will deal with probability measures). For more details on the subject, see [12,13].

1.2. The particle system

We consider the following system of particles

∀i=1, . . . , N, X_t^i,N=X^i,N₀ −χ N

N j=1,j=i

_t

0

K

X_s^i,N−X_s^j,N ds+√

2B_tⁱ, (1.3)

where(Bⁱ)_i=1,...,N is an independent family of 2D standard Brownian motions andK is defined in (1.2). We will show in the sequel that there is propagation of chaos to the solution of the following nonlinear S.D.E. linked with (1.1) (see the next paragraph)

Xt=X0−χ _t

0

R²K(Xs−x)fs(dx)ds+√

2Bt, (1.4)

wheref_t=L(X_t)(L(X_t)denotes the law ofX_t).

1.3. Weak solution for the P.D.E.

For any Polish spaceE, we denote byP(E)the set of all probability measures onEwhich we endow with the topology of weak convergence defined by duality against functions ofC_b(E). We give the notion of weak solution that we use in this paper.

Definition 1.1. We say thatf =(f_t)_t≥0∈C([0,∞),P(R²))is a weak solution to(1.1)if

∀T >0, _T

0

R²

R²

K(x−y)f_s(dx)fs(dy)ds <∞, (1.5)

and if for allϕ∈C_b²(R²),allt≥0,

R²ϕ(x)f_t(dx)=

R²ϕ(x)f₀(dx)+ _t

0

R²xϕ(x)f_s(dx)ds

−χ _t

0

R²

R²K(x−y)· ∇xϕ(x)fs(dy)fs(dx)ds. (1.6) Remark 1.2. We can see easily that if(X_t)_t≥0is a solution to(1.4),then settingf_t=L(X_t)for anyt≥0,(f_t)_t≥0is a weak solution of(1.1)in the sense of Definition1.1provided it satisfies(1.5).Indeed,by Itô’s formula,we find that forϕ∈C_b²(R²),

ϕ(X_t)=ϕ(X₀)−χ t

0

∇xϕ(X_s)·

R²K(X_s−y)f_s(dy)ds +

_t

0

√2∇xϕ(X_s)· dBs+ _t

0

xϕ(X_s)ds.

Taking expectations,we get(1.6).

1.4. Notation and propagation of chaos

ForN ≥2, we denote by Psym(E^N)the set of symmetric probability measures onE^N, i.e., the set of probability measures which are laws of exchangeableE^N-valued random variables.

We consider for anyF ∈Psym((R²)^N)with a density (a finite moment of positive order is also required in order to define the entropy) the Boltzmann entropy and the Fisher information which are defined by

H (F ):= 1 N

(R²)^N

F (x)logF (x)dx and I (F ):= 1 N

(R²)^N

|∇F (x)|² F (x) dx.

We also define (xi∈R²stands for theith coordinate ofx∈(R²)^N), fork≥0, M_k(F ):= 1

N

(R²)^N

N i=1

|x_i|^kF (dx).

Observe that we proceed to the normalization by 1/N in order to have, for anyf∈P(R²), H

f^⊗N

=H (f ), I f^⊗N

=I (f ) and Mk

f^⊗N

=Mk(f ).

We introduce the spaceP1(R²):= {f ∈P(R²),M₁(f ) <∞} and we recall the definition of the Wasserstein distance:

iff, g∈P1(R²), W1(f, g)=inf

R²×R²|x−y|R(dx,dy)

,

where the infimum is taken over all probability measuresR onR²×R²withf for first marginal andgfor second marginal. It is known that the infimum is reached. See, e.g., Villani [21] for many details on the subject.

We now define the notion of propagation of chaos.

Definition 1.3. Let X be someE-valued random variable. A sequence(X^N₁, . . . , X^N_N)of exchangeable E-valued random variables is said to beX-chaotic if one of the three following equivalent conditions is satisfied:

(i) (X₁^N, X^N₂)goes in law to2independent copies ofXasN→ +∞;

(ii) for allj≥1,(X^N₁, . . . , X^N_j )goes in law toj independent copies ofXasN→ +∞;

(iii) the empirical measureμ^N

X^N :=_N¹ _N

i=1δ_XN

i ∈P(E)goes in law to the constantL(X)asN→ +∞.

We refer to [19] for the equivalence of the three conditions or [11], Theorem 1.2, where the equivalence is estab- lished in a quantitative way.

Propagation of chaos in the sense of Sznitman holds for a system ofN exchangeable particles evolving in time if when the initial conditions (X₀^1,N, . . . , X^N,N₀ ) are X₀-chaotic, the trajectories ((X_t^1,N)_t≥0, . . . , (X^N,N_t )_t≥0)are (X_t)_t≥0-chaotic, where(X_t)_t≥0is the (unique) solution of the expected (one-particle) limit model.

We finally recall a stronger (see [11]) sense of chaos introduced by Kac in [14] and formalized recently in [6]: the entropic chaos.

Definition 1.4. Letf be some probability measure onE.A sequence(F^N)of symmetric probability measures onE^N is said to be entropicallyf-chaotic if

F₁^N→f weakly inP(E) and H F^N

→H (f ) asN→ ∞, whereF₁^Nstands for the first marginal ofF^N.

We can observe that since the entropy is lower semi continuous (so thatH (f )≤lim infNH (F^N)) and is convex, the entropic chaos (which requires limNH (F^N)=H (f )) is a stronger notion of convergence which implies that for allj≥1, the density of the law of(X₁^N, . . . , X_j^N)goes tof^⊗jstrongly inL¹asN→ ∞(see [3]).

1.5. Main results

We first give a result of existence and uniqueness for (1.1).

Theorem 1.5. Letα∈(0,1).Assume thatf₀∈P1(R²)is such thatH (f₀) <∞.

(i) There exists a unique weak solutionf to(1.1)such that f∈L^∞_loc

[0,∞),P1

R²

∩L¹_loc

[0,∞);L^p R²

for somep > 2

1−α. (1.7)

(ii) This solution furthermore satisfies that for allT >0, _T

0

I (f_s)ds <∞, (1.8)

for anyq∈ [1,2)and for allT >0,

∇xf ∈L^2q/(3q−2)

0, T;L^q R²

, (1.9)

for anyp≥1, f∈C

[0,∞);L¹ R²

∩C

(0,∞);L^p R²

, (1.10)

and that for anyβ∈C¹(R)∩W_loc^2,∞(R)such thatβis piecewise continuous and vanishes outside a compact set,

∂tβ(f )=χ (K∗f )· ∇x

β(f )

+ xβ(f )

−β(f )|∇xf|²+χβ(f_s)f_s(∇x·K∗f_s), (1.11) on[0,∞)×R²in the distributional sense.

We denote byF₀^Nthe law of(X^i,N₀ )_i=1,...,N. We assume that for somef₀∈P(R²), F₀^N∈Psym((R²)^N) isf₀-chaotic;

sup_N≥2M₁(F₀^N) <∞, sup_N≥2H (F₀^N) <∞. (1.12) Observe that this condition is satisfied if the random variables(X^i,N₀ )_i=1,...,N are i.i.d. with lawf₀∈P1(R²)such that H (f₀) <∞. The next result states the well-posedness for the particle system (1.3).

Theorem 1.6. Letα∈(0,1).

(i) LetN ≥2 be fixed and assume that M₁(F₀^N) <∞ and H (F₀^N) <∞.There exists a unique strong solution (X_t^i,N)_{t≥0,i=1,...,N} to(1.3).Furthermore,the particles a.s.never collapse,i.e.,it holds that a.s.,for anyt≥0and i=j,X_t^i,N=X^j,N_t .

(ii) Assume(1.12).If for allt≥0, we denote byF_t^N∈Psym((R²)^N)the law of(X^i,N_t )_i=1,...,N,then there exists a constant C depending onχ, sup_N≥2H (F₀^N)andsup_N≥2M1(F₀^N)such that for allt≥0andN≥2

H F_t^N

≤C(1+t ), M1

F_t^N

≤C(1+t ), t

0

I F_s^N

ds≤C(1+t ).

Furthermore for anyT >0, E

sup

t∈[0,T]

X_t^1,N≤C(1+T ). (1.13)

We also have

H F_t^N

+ t

0

I F_s^N

ds≤H F₀^N

+ χ N²

i=j

t 0

E divK

X_s^i,N−X_s^j,N

ds. (1.14)

We next state a well-posedness result for the nonlinear S.D.E. (1.4).

Theorem 1.7. Letα∈(0,1)andf₀∈P1(R²)such thatH (f₀) <∞.There exists a unique strong solution(X_t)_t≥0 to(1.4)such that for somep >2/(1−α),

(f_t)_t≥0∈L^∞_loc

[0,∞),P1

R²

∩L¹_loc

[0,∞);L^p R²

, (1.15)

wheref_t is the law ofX_t.Furthermore,(f_t)_t≥0is the unique solution to(1.1)given in Theorem1.5.

We finally give the result about propagation of chaos.

Theorem 1.8. Letα∈(0,1).Assume(1.12).For eachN≥2,consider the unique solution(X^i,N_t )_i=1,...,N,t≥0to(1.3).

Let(Xt)t≥0be the unique solution to(1.4).

(i) The sequence(X_t^i,N)_{i=1,...,N,t≥0}is(X_t)_t≥0-chaotic.In particular,the empirical measureQ^N:=_N¹_N

i=1δ_(Xi,N t )_t≥0

goes in law toL((Xt)t≥0)inP(C((0,∞),R²)).

(ii) Assume furthermore that limNH (F₀^N)=H (f0). For all t ≥0, the sequence (X^i,N_t )i=1,...,N is then Xt- entropically chaotic. In particular, for any j ≥1 and any t ≥0, denoting by F_tj^N the density of the law of (X^1,N_t , . . . , X_t^j,N),it holds that

Nlim→∞F_tj^N−f_t^⊗j

L¹((R²)^j)=0.

We can observe that the condition limNH (F₀^N)=H (f0)is satisfied if the random variables (X^i,N₀ )i=1,...,N are i.i.d. with lawf₀such thatH (f₀) <∞.

1.6. Comments

This paper is some kind of adaptation of the work of Fournier, Hauray and Mischler in [8] where they show the propagation of chaos of some particle system for the 2D viscous vortex model. We use the same methods for a subcritical Keller–Segel equation. The proofs are thus sometimes very similar to those in [8] but there are some differences due to the facts that (i) there are no circulation parameter (M^N_i in [8]): this simplify the situation since we thus deal with solutions which are probabilities, (ii)α=1 so when we use Hardy–Littlewood–Sobolev’s inequality an extra change of variables for the time variable is needed (see Step 1 in the proof of Theorem1.5in Section6) and (iii) the kernel is not the same: it is not divergence-free and we thus have to deal with some additional terms in our computations (see the comments before Proposition3.1and in the proof of Theorem1.5). We can also notice that due to this fact, we have no already known result for the existence and uniqueness of the particle system that we consider.

The methods used to prove uniqueness for the Keller–Segel equation (1.1) and its associated S.D.E. (1.4), and to prove the entropic chaos are also different.

The proof of Theorem 1.5follows the ideas of renormalisation solutions to a P.D.E. introduced by DiPerna and Lions in [7] and developed since then. The key point is to be able to find gooda prioriestimates which allow us to approximate the weak solutions by regular functions, i.e., to use C^k functions instead ofL¹. Then, using these estimates, one can pass to the limit and go back to the initial problem. One can further see that the uniqueness result is proven based on coupling methods and the Wasserstein distance. This will allow us to use more general initial conditions than we could use in a strictly deterministic framework.

The proof of existence and uniqueness for the particle system (1.3) (Theorem1.6) use some nice arguments. Like for S.D.E.s with locally Lipschitz coefficients, we show existence and uniqueness up to an explosion time and the interesting part of the proof is to show that this explosion time is infinite a.s.

To our knowledge, there is no other work that give a convergence result of some particle system for a chemo- taxis model with a singular kernelK and without cutoff parameter. In [17] and [18], Stevens studies a particle sys- tem with two kinds of particles corresponding to bacteria and chemical substance. She shows convergence of the system for smooth initial data (lying inC_b³(R^d)) and for regular kernels (continuously differentiable and bounded together with their derivatives). In [9] and [10], Haskovec and Schmeiser consider a kernel with a cutoff parameter K_ε(x)=_|x|(|x^x_|+ε). They get some well-posedness result for the particle system and they show the weak convergence of subsequences due to a tightness result (observe that here we have propagation of chaos and also entropic chaos). In a recent work [5], Calvez and Corrias work on some one-dimensional Keller–Segel model. They study a dynamical particle system for which they give a global existence result under some assumptions on the initial distribution of the particles that prevents collisions. They also give two blow-up criteria for the particle system they do not state a convergence result for this system.

Finally, it is important to notice that the present method can not be directly adapted for the standard caseα=1 because in this last situation the entropy and the Fisher information are not controlled.

1.7. Plan of the paper

In the next section, we give some preliminary results. In Section3, we establish the well-posedness of the particle system (1.3). In Section4, we prove the tightness of the particle system and we show that any limit point belongs to the set of solutions to the nonlinear S.D.E. (1.4). In Section5, we show that the P.D.E. (1.1) and the nonlinear S.D.E.

(1.4) are well-posed and we show the propagation of chaos. Finally, in the last section, we improve the regularity of the solution, give some renormalization results for the solution to (1.1) and we conclude with the entropic chaos.

2. Preliminaries

In this section, we recall some lemmas stated in [8] and [11] and we state a result on the regularity of the kernelK defined in (1.2). The first result tells us that pairs of particles which law have finite Fisher information cannot be too close.

Lemma 2.1 ([8], Lemma 3.3). ConsiderF ∈P(R²×R²)with finite Fisher information and (X₁, X₂)a random variable with lawF.Then for anyγ∈(0,2)and anyβ > γ /2there existsC_{γ ,β} so that

E

|X1−X2|^−γ

=

R²×R²

F (x₁, x₂)

|x₁−x₂|^γ dx1dx2≤Cγ ,β

I (F )^β+1 .

In the next lemma, we see that the Fisher information of the marginals of someF ∈Psym((R²)^N)is smaller than the Fisher information ofF.

Lemma 2.2 ([11], Lemma 3.7). For anyF ∈Psym((R²)^N)and1≤l≤N,I (F_l)≤I (F ),whereF_l∈Psym((R²)^l) denotes the marginal probability ofF on thelth block of variables.

The following lemma allows us to control from below the entropy of someF ∈Pk((R²)^N)by its moment of order kfor anyk >0.

Lemma 2.3 ([8], Lemma 3.1). For any k, λ∈(0,∞),there is a constantCk,λ∈Rsuch that for anyN≥1, any F∈Pk((R²)^N),

H (F )≥ −Ck,λ−λMk(F ).

The next result tells us that a probability measure onR²with finite Fisher information belongs toL^pfor anyp≥1 and its derivatives, toL^qfor anyq∈ [1,2).

Lemma 2.4 ([8], Lemma 3.2). For anyf∈P(R²)with finite Fisher information,there holds

∀p∈ [1,∞), fL^p(R²)≤C_pI (f )^1−1/p,

∀q∈ [1,2), ∇xf_L^q_(R²₎≤C_qI (f )^3/2−1/q. We end this section with the following result onK.

Lemma 2.5. Letα∈(0,1).There exists a constantC_α such that for allx, y∈R² K(x)−K(y)≤Cα|x−y|

1

|x|^α+1+ 1

|y|^α+1

.

Proof. We have

K(x)−K(y)= x

1

|x|^α+1− 1

|y|^α+1

+ x−y

|y|^α+1

≤ |x||x−y|(α+1)max 1

|x|^α+2, 1

|y|^α+2

+|x−y|

|y|^α+1. By symmetry, we also have

K(x)−K(y)≤ |y||x−y|(α+1)max 1

|x|^α+2, 1

|y|^α+2

+|x−y|

|x|^α+1. So we deduce that

K(x)−K(y)≤ |x−y|

(α+1)min

|x|,|y| max

1

|x|^α+2, 1

|y|^α+2

+ 1

|x|^α+1+ 1

|y|^α+1

≤ |x−y|

(α+1) 1

min(|x|,|y|)^α+1+ 1

|x|^α+1+ 1

|y|^α+1

≤(α+2)|x−y| 1

|x|^α+1+ 1

|y|^α+1

,

which concludes the proof.

3. Well-posedness for the system of particles

Let’s now introduce another particle system with a regularized kernel. We set, forε∈(0,1), K_ε(x)= x

max(|x|, ε)^α+1, (3.1)

which obviously satisfies|K_ε(x)−K_ε(y)| ≤C_α,ε|x−y|and we consider the following system of S.D.E.s

∀i=1, . . . , N, X^i,N,ε_t =X₀^i,N− χ N

N j=1,j=i

_t

0

K_ε

X_s^i,N,ε−X_s^j,N,ε ds+√

2B_tⁱ, (3.2)

for which strong existence and uniqueness thus holds.

The following result will be useful for the proof of Theorem1.6. Its proof is very similar to the proof of [8], Proposition 5.1. Nevertheless, due to the fact that the kernel is not divergence-free, there is an additional term in the dissipation of entropy’s formula (3.3) which will lead to additional computations to control it.

Proposition 3.1. Letα∈(0,1).

(i) Let N ≥2 be fixed. Assume that M₁(F₀^N) < ∞ and H (F₀^N) <∞. For all t ≥0, we denote by F_t^N,ε ∈ Psym((R²)^N)the law of(X^i,N,ε_t )_i=1,...,N.Then

H F_t^N,ε

=H F₀^N

+ χ N²

i=j

_t

0

(R²)^N

divKε(x_i−x_j)F_s^N,ε(x)dsdx

− _t

0

I F_s^N,ε

ds. (3.3)

(ii) There exists a constant C which depends onχ,H (F₀^N)andM₁(F₀^N)(but not onε)such that for allt≥0and N≥2,

H F_t^N,ε

≤C(1+t ), M₁ F_t^N,ε

≤C(1+t ), _t

0

I F_s^N,ε

ds≤C(1+t ). (3.4)

Furthermore,

E

[sup0,T]

X_t^1,N,ε≤C(1+T ). (3.5)

Proof. Letϕ∈C_b²((R²)^N), andt≥0 be fixed. Using Itô’s formula, we compute the expectation ofϕ(X_t^1,N,ε, . . . , X_t^N,N,ε)and get (recall thatx_i∈R²stands for theith coordinate ofx∈(R²)^N)

d dt

(R²)^N

ϕ(x)F_t^N,ε(dx)= −χ N

(R²)^N

i=j

K_ε(x_i−x_j)· ∇x_iϕ(x)F_t^N,ε(dx) +

(R²)^Nxϕ(x)F_t^N,ε(dx). (3.6)

We deduce thatF^N,εis a weak solution to

∂_tF_t^N,ε(x)= χ N

i=j

divxi

F_t^N,ε(x)K_ε(x_i−x_j)

+ xF_t^N,ε(x). (3.7)

We are now able to compute the evolution of the entropy.

d dtH

F_t^N,ε

= 1 N

(R²)^N

∂_tF_t^N,ε(x)

1+logF_t^N,ε(x) dx

= χ N²

i=j

(R²)^N

divx_i

F_t^N,ε(x)Kε(xi−xj)

1+logF_t^N,ε(x) dx + 1

N

(R²)^N

xF_t^N,ε(x)

1+logF_t^N,ε(x) dx.

Performing some integrations by parts, we get d

dtH F_t^N,ε

= − χ N²

i=j

(R²)^N

K_ε(x_i−x_j)· ∇x_iF_t^N,ε(x)dx−I F_t^N,ε

= χ N²

i=j

(R²)^N

divKε(xi−xj)F_t^N,ε(x)dx−I F_t^N,ε

,

and (3.3) follows. Using that divK_ε(x)=_|x¹_|⁻α+^α11_{|x|≥ε}+_εα²+11_{|x|<ε}≤ _|x|²α+1 and the exchangeability of the parti- cles, we get

d dtH

F_t^N,ε

≤ 2χ N²

i=j

(R²)^N

F_t^N,ε(x)

|x_i−x_j|^α+1dx−I F_t^N,ε

≤2χ

(R²)^N

F_t^N,ε(x)

|x1−x2|^α+1dx−I F_t^N,ε

.

Sinceα∈(0,1), we can use Lemma2.1withγ=α+1 andβsuch that^α+₂¹< β <1, which gives

(R²)^N

F_t^N,ε(x)dx

|x₁−x₂|^α+1≤C I

F_t2^N,εβ

+1 ,

whereF_t2^N,ε is the two-marginal ofF_t^N,ε. By Lemma2.2, we haveI (F_t2^N,ε)≤I (F_t^N,ε). Using thatCx^β ≤C+_6χ^x for a constantCsufficiently large, we thus get

d dtH

F_t^N,ε

≤C−2 3I

F_t^N,ε ,

and thus H

F_t^N,ε +2

3 t

0

I F_s^N,ε

ds≤H F₀^N

+Ct. (3.8)

We now computeM₁(F_t^N,ε). We first observe that

M₁ F_t^N,ε

= 1 N

(R²)^N

N i=1

|x_i|F_t^N,ε(dx)=EX^1,N,ε_t ,

since the particles are exchangeable. We will need to controlE[sup_[0,T]|X^1,N,ε_t |]in the sequel. We have E

[sup0,T]

X^1,N,ε_t ≤C

EX¹₀+E

[sup0,T]

B_t¹ +E

sup

t∈[0,T]

1 N

j=1

t 0

K_ε

X_s^1,N,ε−X_s^j,N,ε ds

≤C

EX¹₀+T + 1 N

j=1

_T

0 EK_ε

X^1,N,ε_s −X^j,N,εs ds

≤C

EX¹₀+T + _T

0

E

1

|X^1,N,εs −X^2,N,εs |^α

ds

. (3.9)

Using Lemma2.1withγ=αandβ such that^α₂< β <1 and recalling thatI (F_t^N,ε₂ )≤I (F_t^N,ε), we get M1

F_t^N,ε

≤C

M1

F₀^N +T +

_t

0

I F_t^N,εβ

ds

≤C M₁

F₀^N +T

+1 3

_t

0

I F_t^N,ε

ds, (3.10)

where we used thatCx^β≤C+^x₃ for a constantCsufficiently large. Summing (3.8) and (3.10), we thus find H

F_t^N,ε +M1

F_t^N,ε +1

3 _t

0

I F_s^N,ε

ds≤H F₀^N

+Ct+C 1+M1

F₀^N .

Since the quantitiesM₁andI are positive, we immediately getH (F_t^N,ε)≤C(1+t ). Using Lemma2.3, we have H (F_t^N,ε)≥ −C−M₁(F_t^N,ε)/2, so that

M1

F_t^N,ε +1

3 t

0

I F_s^N,ε

ds≤C(1+t )+M1

F_t^N,ε /2.

Using again the positivity ofM₁andI, we easily get (3.4). Coming back to (3.9), we finally observe that E

[sup0,T]

X_t^1,N,ε≤C

EX₀¹+T + _T

0

I F_s^N,ε

ds

≤C

1+EX¹₀+T ,

which gives (3.5) and concludes the proof.

We can now give the proof of existence and uniqueness for the particle system (1.3).

Proof of Theorem1.6. Like in [20], the key point of the proof is to show that particles of the system (1.3) a.s. never collide. We divide the proof in three steps. The first step consists in showing that a.s. there are no collisions between particles for the system (3.2). In the second step, we deduce that the particles of the system (1.3) also never collide, which ensures global existence and uniqueness for (1.3). In the last step, we establish the estimates about the entropy, Fisher information and the first moment. We fixN≥2 and for allε∈(0,1), we consider(X^i,N,ε_t )_{i=1,...,N,t≥0} the unique solution to (3.2).

Step1. Letτ_ε:=inf{t≥0,∃i=j,|X_t^i,N,ε−X^j,N,ε_t | ≤ε}. The aim of this step is to prove that limε→0P[τ_ε< T] = 0 for allT >0. We fixT >0 and introduce

S_t^ε:= 1 N²

i=j

logX_t^i,N,ε−X_t^j,N,ε. (3.11)

For anyA >1, we have P[τ_ε< T] ≤P

[inf0,T]S_t^ε_∧τ

ε≤S^ε_τ

ε

≤P

∃i,∃t∈ [0, T],X^i,N,ε_t > A +P

∀i,∀t∈ [0, T],X_t^i,N,ε≤A, inf

[0,T]S_t^ε_∧τ

ε≤S_τ^ε

ε

≤NE[sup_[0,T]|X^1,N,ε_t |]

A +P

[inf0,T]S_t^ε_∧τε≤logε

N² +log 2A

≤C(1+T )N

A +P

[inf0,T]S_t^ε_∧τ

ε≤logε

N² +log 2A

, (3.12)

where we used (3.5). We thus want to computeP[inf_[0,T]S_t^ε_∧τ

ε≤ −M]for all (large)M >0. Using Itô’s formula, that K_ε(x)=K(x)for any|x| ≥ε(see (3.1)) and that(log|x|)=0 on{x∈R²,|x|> ε}, we have

logX^i,N,ε_t∧τ

ε −X_t^j,N,ε_∧τ

ε =logX^i,N

0 −X₀^j,N+M_t^i,j,ε_∧τ

ε

− χ N

_t∧τε

0 k=i,j

K

X_s^i,N,ε−X_s^k,N,ε

−K

Xs^j,N,ε−X_s^k,N,ε

+2K

X^i,N,ε_s −X^j,N,ε_s

· X^i,N,ε_s −X_s^j,N,ε

|X^i,N,ε_s −X^j,N,ε_s |²ds

=:logX^i,N₀ −X^j,N₀ +M_t^i,j,ε_∧τε +R^i,j,ε_t∧τε, where M_t^i,j,ε is a martingale. Setting S₀ := _N¹2

i=jlog|X^i,N₀ −X₀^j,N|, M_t^ε := _N¹2

i=jM_t^i,j,ε_∧τ

ε and R_t^ε :=

1 N²

i=jR_t^i,j,ε_∧τε, we thus have S_t^ε_∧τ

ε=S0+M_t^ε+R^ε_t, so that

P

[0,Tinf]S_t^ε_∧τ

ε≤ −M

≤P(S₀≤ −M/3)+P

[0,Tinf]M_t^ε≤ −M/3

+P

[inf0,T]R^ε_t ≤ −M/3

. (3.13)

Using first Lemma 2.5 and that |K(x)| = |x|^−α, and then exchangeability, we clearly have for some constant C independent ofN andε,

E

[sup0,T]

R^ε_t≤ C χ N³

i=j

k=i,j

E

1

|X^i,N,ε_s −X^k,N,ε_s |^α+1

+E

1

|X^j,N,ε_s −X^k,N,ε_s |^α+1

+E

1

|X^i,N,ε_s −X^j,N,ε_s |^α+1

ds

≤Cχ _T

0

E

1

|X^1,N,εs −Xs^2,N,ε|^α+1

ds

≤Cχ _T

0

1+I F_s2^N,ε

ds

≤C(1+T ), (3.14)

where we used Lemma2.1, the fact thatI (F_t2^N,ε)≤I (F_t^N,ε)by Lemma2.2, and finally Proposition3.1. We thus get P

[0,Tinf]R_t^ε≤ −M/3 ≤P

sup

[0,T]

R^ε_t≥M/3

≤C(1+T )

M . (3.15)

We now want to computeP(inf_[0,T]M_t^ε≤ −M/3). Using that log|x| ≤ |x|, we have S_t^ε≤ 1

N²

i=j

X_t^i,N,ε+X^j,N,ε_t ≤ 2 N

i

X_t^i,N,ε.

Consequently, M_t^ε≤S_t^ε_∧τ

ε+ sup

s∈[0,T]

R^ε_s−S0

≤ 2 N

i

sup

s∈[0,T]

X^i,N,ε_s + sup

s∈[0,T]

R_s^ε−S₀=:K^ε−S₀=:Z^ε.

We have P

[0,Tinf]M_t^ε≤ −M/3 ≤P

Z^ε≥ M/3

+P

[0,Tinf]M_t^ε≤ −M/3, Z^ε<

M/3

. (3.16)