Raphael Maillet

A NOTE ON THE LONG TIME BEHAVIOUR OF STOCHASTIC MCKEAN-VLASOV EQUATIONS WITH COMMON NOISE

Keywords: Mathematics Subject Classification (2010): MSC Non-linear McKean-Vlasov, Stochastic Partial Differential Equations, Interacting particle system, Common noise, Conditional propagation of chaos, Asymptotic stability. 0

This paper presents an investigation into the long-term behaviour of solutions to a nonlinear stochastic McKean-Vlasov equation with common noise. The equation arises naturally in the mean-field limit of systems composed of interacting particles subject to both idiosyncratic and common noise. Initially, we demonstrate that the addition of common diffusion in each particle's dynamics does not disrupt the established stability results observed in the absence of common noise. However, our main objective is to understand how the presence of common noise can restore the uniqueness of equilibria. Specifically, in a non-convex landscape, we establish uniqueness and convergence towards equilibria under two specific conditions: (1) when the dimension of the ambient space equals 1, and (2) in the absence of idiosyncratic noise in the system.

INTRODUCTION

We consider the following non-linear Stochastic Partial Differential Equation on r0, `8q ˆRd , [START_REF] Bashiri | On the long-time behaviour of McKean-Vlasov paths[END_REF] d t m t " ∇ ¨ˆσ 2 `σ2 0 2 ∇m t `mt bpt, ¨, m t q ˙dt ´σ0 Dm t ¨dB 0 t .

This SPDE is posed on a filtered probability space `Ω0 , F 0 , F 0 , P 0 ˘, B 0 is a d-dimensional F 0 -Brownian motion, the drift b : r0, `8q ˆRd ˆP `Rd ˘Ñ R d depends in time, space and measure, σ and σ 0 are two non-negative constants. This paper aims to study the long-term behavior of solutions of Equation ( 1), and more precisely, to understand the effect of the common noise on the asymptotic stability. We assume that the drift term b has a specific linear structure with respect to the measure variable, with two continuously differentiable functions V and W such that bpt, x, µq " ´∇V pxq ´∇W ˚µpxq, where ˚stands for the convolution operator. This assumption is typical for studying long-term behavior in McKean-Vlasov type equations, see e.g [START_REF] Bashiri | On the long-time behaviour of McKean-Vlasov paths[END_REF], [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF], [START_REF] Del Moral | Uniform propagation of chaos and creation of chaos for a class of nonlinear diffusions[END_REF], [START_REF] Herrmann | Non-uniqueness of stationary measures for self-stabilizing processes[END_REF] or [START_REF] Tugaut | Convergence to the equilibria for self-stabilizing processes in double-well landscape[END_REF]. In the following, we are interested in getting existence and uniqueness results for the invariant measure of the probability measure valued process pm t q. As the problem under consideration falls within the realm of McKean-Vlasov type, one may expect existence of an invariant measure and uniqueness at least in some specific cases. In this paper, we verify whether, in the case where W is convex and V is uniformly convex, the introduction of common noise does not compromise the classical uniqueness results of [START_REF] Benachour | Nonlinear self-stabilizing processes-II: Convergence to invariant probability[END_REF]. When V is not convex, the matter becomes considerably more intricate, studied so far without the presence of common noise, only partial results are known. There exist cases in which the uniqueness of the invariant measure is not satisfied. Unlike linear elliptic equations, the presence of nonlinearity leads to the existence of multiple invariant measures. Specifically, it has been proven in [START_REF] Herrmann | Non-uniqueness of stationary measures for self-stabilizing processes[END_REF] that when the confinement potential uniformly convex outside of a ball centred in the origine, admits a double-well and the diffusion coefficient σ is sufficiently small, there exist exactly three invariant solutions of the following equation:

B t m t " σ 2 2 ∆m t `∇ ¨pm t p∇V `∇W ˚mt qq.

Since 2019, several papers [START_REF] Delarue | Restoring uniqueness to mean-field games by randomizing the equilibria[END_REF], [START_REF] Delarue | Selection of equilibria in a linear quadratic mean-field game[END_REF] or [START_REF] Delarue | Exploration noise for learning linear-quadratic mean field games[END_REF] investigate the restoration of uniqueness in mean-field games derived from deterministic differential games with a large number of players by introducing an external noise. In a similar manner, this paper explores the restoration of uniqueness of the invariant measure by introducing common noise to the system. More precisely, we prove existence and uniqueness of the invariant measure for process pm t q in the following cases: (1) when the confinement potential V is uniformly convex and the interaction potential is convex; (2) when the confinement potential V is not convex and the dimension of the ambiant space d " 1;

(3) when the potential V is not convex and there is no idiosyncratic noise in the system, i.e σ " 0.

Without uniform convexity assumptions and in higher dimension d ě 2, we can get uniqueness of the invariant measure for solutions of

d t m t " ∇ ¨ˆσ 2
for all initial condition P 0 , we get the existence of a constant C ą 0, such that for each time t ą 0, d PpR d q 1 `Pt , s P ˘ď Ce ´ηt , where P t " Lpm t q, and for any p ě 1, the Wasserstein-p distance on PpPpR d qq is defined by

d PpR d q p pP, Qq " inf πPΠpP,Qq ż PpR d q d R d p pµ, νqπpdµ, dνq, (3)
for any P, Q P PpPpR d qq, and where d p stands for the classical Wasserstein-p distance on R d . Obtaining uniqueness of the invariant measure in the general case of Equation ( 1) without strong convexity assumptions on the confinement potential appears to be a challenging problem which is not solved in this paper. In fact, the long-term behaviour of solutions to SPDEs with nonlinear drifts is still not fully understood and therefore, the uniqueness results obtained in this paper represent a step towards understanding the long-term behaviour of solutions to Equation [START_REF] Bashiri | On the long-time behaviour of McKean-Vlasov paths[END_REF], but are far from giving a complete understanding of the asymptotic stability for that kind of Stochastic McKean-Vlasov Equations with common noise.

Probabilistic setting & Motivation.

Let us consider a filtered probability space pΩ 1 , F 1 , F 1 , P 1 q. Then, we define the following product structure

Ω " Ω 0 ˆΩ1 , F , F, P,
where pF , Pq is the completion of the set pF 0 b F 1 , P 0 b P 1 q and F is the right continuous augmentation of pF 0 t b F 1 t q tě0 . We also consider a d-dimensional Brownian motion B 0 supported by pΩ 0 , F 0 , P 0 q, adapted to F 0 and another Brownian motion B supported by pΩ 1 , F 1 , P 1 q, adapted to F 1 and independent of F 0 . Let us now consider a probability measure on the space of probability measures P 0 P PpPpR d qq, we are able to define m 0 , a F 0 0 -measurable random variable with value in the space of probability measure PpR d q and such that Lpm 0 q " P 0 , in the sense that for any bounded measurable function F : PpR d q Ñ R, E P 0 rF pm 0 qs " xP 0 ; F y. We can now define on the whole probability space pΩ, F , F, Pq a random variable X 0 such that LpX 0 |F 0 0 q " m 0 almost surely. Let us define the stochastic process X evolving in R d , supported by pΩ, F , F, Pq, which dynamic is given by ( 4)

"
dX t " ´∇V pX t qdt ´∇W ˚mt pX t qdt `σdB t `σ0 dB 0 t X |t"0 " X 0 , where m t stands for the conditional law of the random variable X t , with respect to the σ-algebra F 0 t . Precisely, m t " L `Xt |F 0 t ˘almost surely, and B is a d-dimensional F 1 -Brownian motion independent of F 0 . The dynamic of the process pm t q tě0 is well known and given by the following Lemma (see for example [START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications I-II[END_REF] among other references).

Lemma 1. The measure valued process pm t q tě0 is solution in the weak sense of the following Stochastic Partial Differential Equation:

d t m t " ∇ ¨ˆσ 2 `σ2 0 2 ∇m t `mt bpt, ¨, m t q ˙dt ´σ0 Dm t ¨dB 0 t ,
with initial condition L pm 0 q " P 0 .

Equation (1) connects closely to the McKean-Vlasov Equation with common noise, a common model in stochastic dynamics. It shows how large systems of interacting particles evolve. In mathematical finance, this model is particularly useful for situations like inter-bank borrowing and lending systems (see [START_REF] Carmona | Mean field games and systemic risk[END_REF] or [START_REF] Giesecke | Large portfolio asymptotics for loss from default[END_REF]). Studying the long-term behavior of solutions to Equation ( 1) is important because it gives us insight of the behavior of the solution to Equation (4).

Literature. Stochastic Partial Differential Equations in the more general form,

d t m t " " ÿ i,j D 2 ij pa ij pt, ¨, m t qm t q `div pm t bpt, ¨, m t qq ı dt ´σ0 pt, ¨, m t qDm t ¨dB 0 t , (5)
have been extensively studied in recent decades as it naturally arises in several applications. Equation ( 5) is linked to the stochastic scalar conservations law of the form d t m t `∇ ¨pσ 0 p¨, u t qu t q ˝dW t " 0, [START_REF] Briand | Particles systems for mean reflected BSDEs[END_REF] where ˝stands for the Stratonovich stochastic integral. In the case where σ 0 px, µq " σ 0 pµpxqq, meaning that the diffusion coefficient depends in the measure in a local way, this class of equations has been introduced in [START_REF] Lions | Scalar conservation laws with rough (stochastic) fluxes[END_REF] paving the way to several papers dealing with well-posedness of solutions of (6) in various frameworks [START_REF] Lions | Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case[END_REF][START_REF] Gess | Scalar conservation laws with multiple rough fluxes[END_REF][START_REF] Friz | Stochastic scalar conservation laws driven by rough paths[END_REF][START_REF] Gess | Stochastic non-isotropic degenerate parabolic-hyperbolic equations[END_REF][START_REF] Fehrman | Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise[END_REF]. Uniqueness of the solutions to (1) is a well known result in the class of solutions admitting a square integrale density with respect to the Lesbegue measure, see [START_REF] Kurtz | Particle representations for a class of nonlinear SPDEs[END_REF], and has been shown recently without any further moments assumptions in [START_REF] Coghi | Stochastic nonlinear Fokker-Planck equations[END_REF].

In a slightly different context, a series of papers demonstrated the well-posedness for a large class of Stochastic Differential Equations similar to (1), called Mean Reflected Stochastic Differential Equations, see [START_REF] Briand | BSDEs with mean reflection[END_REF], [START_REF] Briand | Particles systems for mean reflected BSDEs[END_REF] and [START_REF] Briand | Forward and backward stochastic differential equations with normal constraints in law[END_REF]. More precisely, [START_REF] Briand | Particles systems for mean reflected BSDEs[END_REF] and [START_REF] Briand | Forward and backward stochastic differential equations with normal constraints in law[END_REF] state conditional propagation of chaos under regularity conditions on the drift and diffusion terms. This equations naturally appears when considering interacting particle systems with constraint on the empirical measure of the systems and then the study of such equations is particularly important for example for applications to Mean Field Games.

In this paper, we focus on the following specific Stochastic McKean-Vlasov Equation with common noise:

d t m t " ∇ ¨ˆσ 2 0 2 ∇m t `mt bpt, ¨, m t q ˙dt ´σ0 Dm t ¨dB 0 t . (7)
As mentioned before, extensive research has been conducted on the equation in question, and recent studies have made notable contributions to understanding its properties. For example [START_REF] Hammersley | Weak existence and uniqueness for McKean-Vlasov SDEs with common noise[END_REF] explores the existence and uniqueness of solutions for McKean-Vlasov Stochastic Differential Equations (SDEs) with common noise. Similarly, [START_REF] Marx | Infinite-dimensional regularization of McKean-Vlasov equation with a Wasserstein diffusion[END_REF] proposes a regularization approach in an infinite-dimensional setting for the McKean-Vlasov equation with Wasserstein diffusion, enhancing the understanding of solutions' regularity properties. Additionally, [START_REF] Kumar | Well-posedness and tamed schemes for McKean-Vlasov equations with common noise[END_REF] investigate the wellposedness and numerical methods for McKean-Vlasov equations with common noise, providing valuable insights on the stability and convergence of computational approaches for solving these equations.

However, to the best of our knowledge, little is known about the asymptotic behavior of the solutions of equations of the form [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF]. In the case without common noise, σ 0 " 0, where m is a deterministic flow of measures, past research has focused on various aspects of the solutions of (4), including existence, uniqueness ( [START_REF] Mckean | A class of Markov processes associated with nonlinear parabolic equations[END_REF], [START_REF] Funaki | A certain class of diffusion processes associated with nonlinear parabolic equations[END_REF], [START_REF] Graham | Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models[END_REF]), and stability. Over the past two decades, significant advancements have been made in understanding the convergence to equilibrium for solutions of the deterministic McKean-Vlasov equation. For example, see [START_REF] Carrillo | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF] or [START_REF] Carrillo | Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF] for proofs of an exponential convergence rate to equilibrium under strict convexity conditions on the potentials V and W . The case without strict convexity assumptions is more intricate. Nevertheless, through a thorough examination of the dissipation of the Wasserstein distance, [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF] showed an exponential convergence to equilibrium in a weakly-convex case. Recently, involving a coupling method issued from [START_REF] Lindvall | Coupling of multidimensional diffusions by reflection[END_REF], it has been shown using nice concentration properties from [START_REF] Eberle | Reflection couplings and contraction rates for diffusions[END_REF] that the convergence to equilibrium holds with an exponential speed in the case of a confinement potential that is only convex far from the origin, as seen in [START_REF] Durmus | An elementary approach to uniform in time propagation of chaos[END_REF]. The latter shows uniform in time propagation of chaos property, as introduced in [START_REF] Kac | Foundations of kinetic theory[END_REF] and [START_REF] Sznitman | Topics in propagation of chaos[END_REF], allowing one to conclude the uniqueness of the invariant measure and provide a rate of convergence to equilibrium.

Organisation of the paper. This paper has three main parts and primarily uses a probabilistic approach. In Section 2, we study the existence of an invariant measure for the process pm t q tě0 , controlled by Equation ( 1), and provide conditions on potentials V and W for this existence. We also give moment estimate for the invariant measures. Section 3 examines the uniqueness of invariant measures under a uniformly convex confinement potential V , adapting known results without common noise. We also discuss the Ornstein-Uhlenbeck process with common noise, where the invariant measure can be explicitly described. In both Sections 2 and 3, we show uniform-in-time propagation of chaos and convergence to equilibrium. Section 4 explores the same topic for non-convex potential V when d " 1 or σ " 0. Technical proofs are provided in the Appendix.

Definition and notation.

Throughout the paper, for a Polish space E we write PpEq for the space of Borel probability measures on E equipped with the topology of weak convergence and the corresponding Borel σ-algebra. In this paper, we consider a stochastic process pm t q tě0 with value in the space of probability measures P `Rd ˘. We denote by P t the law L pm t q of m t , for t ě 0, which is a probability measure on the space of probability measure. Then pm t q tě0 is a continuous P `Rd ˘-valued process, and P " pP t q tě0 belongs to C `r0, `8r; P `P `Rd ˘˘˘, the space of continuous functions from r0, `8r to P `P `Rd ˘˘. Moreover, for P, Q P P `P `Rd ˘˘, we denote by Π pP, Qq the set of transport plans between P and Q, and for any distance d : PpR d q ˆPpR d q Ñ r0; `8q, we define:

(8) d PpR d q d pP, Qq " inf ΛPΠpP,Qq ż PpR d q
dpµ, νqΛ pdµ, dνq .

In this paper, we mainly use a probability-based approach, often switching between the measurevalued stochastic process m and its probabilistic counterpart X, which solves Equation (4). At this point, it is worth noting that we are concerned with a stochastic process pm t q tě0 that takes values in the space of probability measures PpR d q. This means that at each time t ą 0, we are dealing with measures on the space of probability, rather than on the underlying space R d . To study this process, we use a probability-based approach and introduce several definitions that are specific to this setting. For instance, we define the notion of an invariant measure in PpPpR d qq, which is a probability measure that remains invariant under the evolution of the stochastic process. We also define the notion of a probability measure in L p pPpR d qq, which is a measure that satisfies a certain integrability condition. These definitions are essential for our analysis and detailed below.

Definition 1.

Let us consider a random variable X defined on the filtered probability space pΩ, F , F, Pq, and P P PpPpR d qq. According to Lemma 2.4 in [START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications I-II[END_REF], for P 0 ´a.e. ω 0 P Ω 0 , Xpω 0 , ¨q is a random variable on pΩ 1 , F 1 , P 1 q. By defining L 1 pXq : Ω 0 Q ω 0 Þ Ñ LpXpω 0 , ¨qq, we get a random variable from pΩ 0 , F 0 , P 0 q into PpR d q, providing a conditional law of X given F 0 . Finally, we say that LpXq " P whenever L 1 pXq is distributed with respect to P .

Definition 2 (Invariant measure). ' We say that s P P PpPpR d qq, is an invariant measure for the process pm t q tě0 , if whenever m 0 is distributed according to s P , then at each time, the law of m t is independent of t. More precisely, we say that s P is an invariant measure if and only if Lpm 0 q " s P ñ Lpm t q " s P , @t ą 0.

' We say that a stochastic process X on the probability space `Ω, F , F " pF t q tě0 , P ˘admits an invariant measure in P `P `Rd ˘˘if and only if, the measure valued stochastic process m t "

L `Xt |F 0 t ˘admits an invariant measure.

Definition 3. We say that a probability measure P P L p pPpR d qq for p ě 1, whenever ż

PpR d q ż R d
|x| p m pdxq P pdmq ă `8.

Definition 4. For any increasing, continuous and concave function such that f p0q " 0, such that px, yq Þ Ñ f p|x ´y|q defines a distande on R d , we define the following distance on PpR d q

d PpRq f pP, Qq " inf ΓPΠpP,Qq ż P2pRq d R f pµ, νqΓpdµ, dνq, with d R f pµ, νq " inf πPΠpµ,νq ş R f p|x ´y|q πpdx, dyq. 1.1.
The process m as a mean field limit. Thanks to the definition of the previous subsection, we are now ready to give an interpretation of the process pm t q in terms of mean field limit for interacting particle system. Let us consider P 0 P L 2 pPpR d qq, let also N ě 1 be an integer, and pX 1,N 0 , . . . , X N,N 0 q, N random variables which are conditionally independent and identically distributed with respect to F 0 0 , such that LpX i,N 0 q " P 0 for all i P t1, . . . , N u. We now define the following interacting particle system # dX i,N t " ´∇V pX i,N t q ´N ´1 ř N j"1 ∇W pX i,N t ´Xj,N t qdt `σdB i t `σ0 dB 0 t , X i,N |t"0 " X i,N 0 , @i P t1, . . . , N u ,

where the B i are independent d-dimensional F 1 -Brownian motion which are independent of F 0 . Then, we consider the mean-field limit system p s X 1 , . . . , s X N q driven by " d s

X i t " ´∇V p s X i t q ´∇W ˚mt `s X i t ˘dt `σdB i t `σ0 dB 0 t , s X i |t"0 " s X i 0 , @i P t1, . . . , N u ,
where `s X i 0 ˘i are conditionally iid random variables with respect to F 0 0 , such that L `s X i 0 ˘" P 0 , for all i P t1, . . . , N u. Our framework is exactly the same as the classical one for mean field games system with common noise. However here, the law of the initial conditions is random. Then, conditioning with respect to the σ-algebra F 0 0 , we get back to a more classical framework where the initial condition is a deterministic measure. More precisely, as stated in [START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications I-II[END_REF], under sufficient regularity conditions on the transport part b mainly Lipschitz continuity with respect to the space and measure variables, we get that for any fixed t ě 0, lim

N Ñ`8 Er| s X i t ´Xi,N t | 2 s `Erd R d 2 pm N t , m t qs " 0,
where d R d 2 p¨, ¨q stands for the classical Wasserstein distance on R d , and

m N t :" N ´1 ř N i"1 δ X i,N t
is the empirical measure of the interacting particle system.

EXISTENCE OF AN INVARIANT MEASURE FOR THE STOCHASTIC FLOW OF MEASURES.

Let us consider a stochastic process pm t q tě0 with value in the space of probability measures P `Rd ˘, and with dynamic given by [START_REF] Bashiri | On the long-time behaviour of McKean-Vlasov paths[END_REF]. Then, m is a weak solution of

d t m t " ∇ ¨´σ 2 `σ2 0 2
∇m t `mt p∇V `∇W ˚mt q ¯dt ´σ0 Dm t ¨dB 0 t . More precisely, for all t ě 0 and ϕ P C 8 c pR d q, dxm t , ϕy " xm t , L mt ϕy dt `σ0 xm t , p∇ϕq J y dB 0 t , where for any probability measure m P PpR d q, the operator L m acts on a smooth function ϕ of compact support by

L m ϕ " ´p∇V `∇W ˚mq ¨∇ϕ `σ2 0 `σ2 2 ∆ϕ,
where ∇, ∆ respectively stands for the gradient and laplacian operator, while ¨denotes the usual inner product in R d , and for any ϕ P C 8 c pR d q and any probability measure m,

xm; ϕy " ż R d ϕ dm.
In this section, we aim at giving conditions on the potentials V and W to ensure existence of an invariant measure for the process pm t q. Let us now consider the following assumptions:

Assumption (A1).

(1) There exists a continuous function κ : r0, `8q Ñ R, such that lim inf rÑ`8 κprq ą 0, and p∇V pxq ´∇V pyqq ¨px ´yq ě κp|x ´y|q|x ´y| 2 .

(2) ∇V is L V -Lipschitz continuous.

Assumption (A2).

(1) W is symmetric, i.e., W pxq " W p´xq for all

x P R d . (2) ∇W is L W -Lipschitz continuous.
In the following, we work under the set of assumptions (A1)&(A2). Under this set of assumptions, both equations ( 4) and ( 1) are well-posed, as shown in [START_REF] Dawson | Stochastic McKean-Vlasov equations[END_REF]. The assumption regarding confinement potential V primarily ensures convexity at infinity, which helps keep the process within a compact set with a high probability. We can moreover note that this implies the existence of m V ą 0 and M V ě 0, such that p∇V pxq ´∇V pyqq ¨px ´yq ě m V |x ´y| 2 ´MV .

In this section, we start with a result regarding uniform-in-time control of the process pX t q tě0 with initial condition in L 2 pPpR d qq dynamic given by (4). To prove the existence of an invariant measure for pm t q tě0 solution of (1), we will use the concept of intrinsic derivative for a functional defined on a space of measure, as seen in [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF].

Definition 5. Let us define C 2

b pPpR d qq as the collection of continuous, bounded functions F : PpR d q Ñ R with the following properties:

' There exists a unique continuous and bounded function B m F :

PpR d q ˆRd Ñ R such that lim hÑ0 F pm `hpm 1 ´mqq ´F pmq h " ż R d B m F pm, vqpm 1 ´mqpdvq, for all m, m 1 P PpR d q and ż R d B m F pm, vqmp dvq " 0, m P PpR d q; ' The mapping x Þ Ñ B m F pm, xq is continuously differentiable with uniformly bounded gradient D m F pm, xq in pm, xq; ' For any fixed x P R d , every component of the R d -valued function m Þ Ñ D m F pm, xq satis-
fies the same conditions as the first two bullet points, resulting in a continuous and bounded

D 2 m F pm, x, yq P R dˆd ; ' For m P PpR d q, we use D x D m F pm, vq to denote the Jacobian of the function x Þ Ñ D m F pm, xq,
which is assumed to be continuous and bounded in pm, xq.

Proposition 1. Considering a measure valued process pm t q, which dynamic is given by (1), and defining P t " L pm t q, we have that for any bounded and twice differentiable function F P C 2 b pPpR d q, Rq:

xP t ´P0 , F y " ż t 0 xP s , MF y ds, @t ą 0, (9)
where we define, for m P PpR d q,

MF pmq :" ż R d " D m F pm, xq ¨bpx, mq `σ2 `σ2 0 2 ∇ ¨pD m F pm, xqq ı mpdxq `σ2 0 2 ż R 2d Tr " D 2 mm F pm, x, yq ‰ mpdxqmpdyq,
and where for all measurable and bounded function Φ : PpR d q Ñ R, and all P P PpPpR d qq, xP ; Φy "

ż PpR d q
ΦpmqP pdmq.

Moreover, s P is an invariant measure if and only if @ s P , MF D " 0, @F P C 2 b pPpR d q, Rq. [START_REF] Carrillo | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF] The proof of Proposition 1 is based on Section 1.2 of [START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF] and then postponed to the Appendix A.1.

The existence result.

We now present a result about the existence of an invariant measure for the equation of interest, adapting classical results for McKean-Vlasov equation without common noise where the flow of probability measures, represented as pm t q tě0 is deterministic. We begin this section with the following Lemma, the proof of which is classical and postponed to Appendix A.2.

Lemma 2. Under Assumptions (A1) & (A2), let us consider

P 0 P L 2 pPpR d qq.
Then, denoting by pX t q t the associated stochastic process with initial condition L pX 0 q " P 0 in the sense of Definition 1, and dynamic given by (4), we have the following uniform in time moment control:

sup tą0 E " |X t | 2 ‰ ă `8.
Now, we are ready to state the following Proposition:

Proposition 2. Under Assumptions (A1) & (A2), the dynamical system given by (1) admits at least an invariant measure s P P L 2 pPpR d qq, i.e such that ż

PpR d q ż R d |x| 2 mpdxq s P pdmq ă `8.
Proof of Proposition 2. Let us fix P 0 P L 2 pPpR d qq, i.e such that: ż

PpR d q ż R d |x| 2 mpdxqP 0 pdmq ă `8.
Step 1. Let us now consider a random flow of probability measures pm t q tě0 with dynamic given by ( 1) and initial condition P 0 P L 2 pPpR d qq. At each time t ě 0, we denote by P t the law of this measure valued process. For T ą 0, we define the process pQ T q T ą0 with value in L 2 pPpR d qq by

Q T " T ´1 ż T 0 P t dt.
This defines a sequence of probability measures on PpR d q. Let us show that pQ T q T ą0 admits at least a convergent subsequence. Thanks to Prohorov theorem, we only need to show that this sequence is tight. For R ą 0, let us consider the set

K R " ! m P PpR d q, ż R d |x| 2 mpdxq ď R
) .

This set is compact for the topology of weak convergence in PpR d q. Now, for T ą 0,

Q T pK R q " T ´1 ż T 0 P t pK R qdt " T ´1 ż T 0 ż PpR d q 1 tmPKRu P t pdmqdt " T ´1 ż T 0 E " 1 tmtPKRu ‰ dt ě 1 ´c R , where c " sup tą0 E " |X t | 2 ‰ ă `8
, thanks to Lemma 2. Hence, for any ε ą 0, there exists R ε ą 0 such that Q T pK Rε q ą 1 ´ε, for all T ą 0. This gives tightness of the sequence and then existence of a converging subsequence that we keep denoting by pQ T q T ą0 in the following.

Step 2. Let us denote by Q the limit of this converging subsequence, and show that Q is an invariant measure. Let T ą 0, and F P C 2 b pPpR d q, Rq, thanks to Proposition 1, we get that

xQ T , MF y " ż PpR d q ´żR d rD m F pm, xq ¨bpx, mq `σ2 `σ2 0 2 ∇ ¨pD m F pm, xqqsmpdxq `σ2 0 2 ż R 2d TrrD 2 mm F pm, x, yqsmpdxqmpdyq ¯QT pdmq " T ´1 ż T 0 ż PpR d q ´żR d rD m F pm, xq ¨bpx, mq `σ2 `σ2 0 2 ∇ ¨pD m F pm, xqqsmpdxq `σ2 0 2 ż R 2d TrrD 2 mm F pm, x, yqsmpdxqmpdyq ¯Pt pdmq " T ´1xP T ´P0 , F y.
Hence, for all F P C 2 b pPpR d qq, xQ, MF y " 0. This ensures that Q is an invariant measure.

Step 3. Finally we move on to the moment estimate. We know that there exists a subsequence of Q T which converges weakly to s P . Moreover, ż

PpR d q ż R d |x| 2 mpdxqQ T pdmq " 1 T ż T 0 ż PpR d q ż R d |x| 2 mpdxqP t pdmqdt, and
sup T ą0 1 T ż T 0 ż PpR d q ż R d |x| 2 mpdxqP t pdmqdt ă `8, as sup tą0 E " |X t | 2 ‰ ă `8. Moreover, the function m P P 2 `Rd ˘Þ Ñ ş R d |x| 2
mpdxq is lower semi continuous, and then, ż

PpR d q ż R d |x| 2 mpdxq s P pdmq ď lim inf T Ñ`8 1 T ż T 0 ż PpR d q ż R d |x| 2 mpdxqP t pdmqdt ă `8.
From the previous result we get the following Corollary Corollary 1. For P P L 2 pPpR d qq, we have that d

PpR d q 2
pP, s P q ă `8, where d

PpR d q 2
p¨, ¨q is defined in 3.

THE CASE OF A UNIFORMLY CONVEX CONFINEMENT POTENTIAL

In this section, we show that the process pm t q tě0 , driven by equation ( 1), has a unique invariant measure under strong convexity assumptions on the confinement potential. Moreover, our method allows us to find exponential rates of convergence toward the invariant measure for a specific set of initial conditions. We again consider P 0 P L 2 pPpR d qq and a random variable X 0 such that L pX 0 q " P 0 , following Definition 1. Next, we study the stochastic process pX t q driven by equation ( 4) with the initial condition X 0 . In this part of the paper, we consider strict convexity assumptions on the confinement potential V . To be specific, we adopt the following assumptions throughout this section:

Assumption (A4).

' V is uniformly convex, more precisely, there exists β ą 0 such that:

∇ 2 V ě β Id.
' ∇V is Lipschitz continuous. ' W is even, convex, and ∇W is globally Lipschitz continuous, with Lipschitz constant L W .

Thanks to the previous section, under Assumption (A4), a process X driven by equation ( 4) has an invariant measure. Specifically, there exists s P P L 2 pPpR d qq such that the process pm t q tě0 , governed by the dynamic in Equation ( 1) and with an initial condition of Lpm 0 q " s P , is invariant. This section starts with a key result that states uniform propagation of chaos uniformly in time. This result will then help us establish the uniqueness of the invariant measure and to give a rate of convergence to equilibrium.

3.1. Uniform in time propagation of chaos. In the case without common noise, the uniqueness of the invariant measure for the process X has been already been established see e.g [START_REF] Cattiaux | Probabilistic approach for granular media equations in the non uniformly convex case[END_REF], [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDEs[END_REF], and [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF]. In this section, our aim is to adapt this result to the case with common noise and obtain the uniqueness of the invariant measure for the probability measure valued stochastic process pm t q, and exponentially fast convergence to the equilibria. Theorem 1. Let us consider P, Q P L 2 pPpR d qq, and two particle systems X " pX 1 , . . . , X N q and X N " pX 1,N , . . . , X N,N q with dynamics, [START_REF] Carrillo | Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF] dX i t " ´∇V pX i t qdt ´∇W ˚mt pX i t qdt `σdB i t `σ0 dB 0 t , @i P t1, . . . , N u , and

(12) dX i,N t " ´∇V pX i,N t qdt ´1 N N ÿ j"1 ∇W pX i,N t ´Xj,N t qdt `σdB i t `σ0 dB 0 t , @i P t1, . . . , N u .
where pX i 0 q iPt1,...,N u are independent and identically distributed such that Lp s X 1 0 q " P in the sense of Definition 1, and where the same holds for the second system with LpX 1,N 0 q " Q. Then, under assumptions (A4), there exists a constant C ą 0 depending only on the dimension d and the probability measures P and Q, such that

E " d R d 2 pm P,N Xt , m Q,N X N t q ı ď C ´e´βt `1 2β ? N ´1
¯,

where m P,N Xt " 1 N ř N i"1 δ X i t and m Q,N X N t " 1 N ř N i"1 δ X i,N t .
The proof of this result closely follows the approach presented in [START_REF] Benachour | Nonlinear self-stabilizing processes-II: Convergence to invariant probability[END_REF] and [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDEs[END_REF], which shows the propagation of chaos for particle systems in cases without common noise. However, in our situation, we need to be careful with the interaction term and its dependency on the common noise.

Proof. We only sketch the proof, as it follows closely the proof of [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDEs[END_REF], Theorem 3.3. Let i P t1, . . . , N u, using Itô formula, we get

d dt |X i,N t ´Xi t | 2 " ´2pX i,N t ´Xi t q ¨p∇V pX i,N t q ´∇V pX i t qq ´2 N N ÿ j"1 pX i,N t ´Xi t q ¨p∇W pX i,N t ´Xj,N t q ´∇W ˚mt pX i t qq.
In order to control the second term, we make the following decomposition:

´2pX i,N t ´Xi t q ´N ´1 N ÿ j"1 ∇W pX i,N t ´Xj,N t q ´∇W ˚mt pX i t q " ´2pX i,N t ´Xi t q ´N ´1 N ÿ j"1 ∇W pX i,N t ´Xj,N t q ´N ´1 N ÿ j"1 ∇W pX i t ´Xj t q 2pX i,N t ´Xi t q ´N ´1 N ÿ j"1 ∇W pX i t ´Xj t q ´∇W ˚mt pX i t q " Ξ i,N t
`Υi,N t . Summing the first term over i shows and using the convexity of the interaction potential, we get

1 N N ÿ i"1 Ξ i,N t " ´2N ´2 N ÿ i,j"1 pX i,N t ´Xj,N t ´Xi t `Xj t qp∇W pX i t ´Xj t q ´∇W pX i t ´Xj t qq ď 0.
Moreover, the second term can be decomposed into two terms

Υ i,N t " ´2pX i,N t ´Xi t q ¨´pN ´1q ´1 N ÿ j"1 ∇W pX i t ´Xj t q ´∇W ˚mt pX i t q 2pX i,N t ´Xi t q ¨´´1 N ´1 N ´1 ¯N ÿ j"1 ∇W pX i t ´Xj t q
For the first one

Er|∇W ˚mt pX i t q ´1 N ´1 N ÿ j"1 ∇W pX i t ´Xj t q| 2 s " E " E "ˇˇˇ∇ W ˚mt pX i t q ´1 N ´1 N ÿ j"1 ∇W pX i t ´Xj t q ˇˇ2 ˇˇX i t , F 0 t ıı " E " Var " 1 N ´1 N ÿ j"1 ∇W pX i t ´Xj t q ˇˇX i t , F 0 t ıı ď 1 N ´1 ErEr|∇W pX i t ´Xj t q| 2 |X i t , F 0 t ss, for some j ‰ i ď L 2 W N ´1 ErEr|X i t ´Xj t | 2 |X i t , F 0 t ss ď 2L 2 W N ´1 Er|X 1 t | 2 s,
where the first inequality comes from the fact that Er∇W pX i t ´Xj t q|X i t , F 0 t s " ∇W ˚mt pX i t q. Then, thanks to Lemma 2, there exists a constant C ą 0, such that

E " pX i,N t ´Xi t q ¨´pN ´1q ´1 N ÿ j"1 ∇W pX i t ´Xj t q ´∇W ˚mt pX i t q ¯ı ď Er|X i,N t ´Xi t | 2 s 1{2 E "ˇˇˇ∇ W ˚mt pX i t q ´1 N ´1 N ÿ j"1 ∇W pX i t ´Xj t q ˇˇ2 ı 1{2 ď C ? N ´1 Er|X i,N t ´Xi t | 2 s 1{2 . Now, E "ˇˇˇ1 N ´1 ¨1 N N ÿ j"1 ∇W pX i t ´Xj t q ˇˇ2 ı ď 2L 2 W pN ´1q 2 Er|X 1 t | 2 s.
Finally, using once again Lemma 2 we get the existence of a constant C, such that:

Er|Υ i,N t |s ď C ´1 N ´1 `1 ? N ´1 ¯Er|X i,N t ´Xi t | 2 s 1{2 .
Finally, we get that

1 N N ÿ i"1 d dt Er|X i t ´Xi,N t | 2 s ď ´2β N N ÿ i"1 Er|X i t ´Xi,N t | 2 s `C ? N ´1 ´N ´1 N ÿ i"1 Er|X i,N t ´Xi t | 2 s ¯1{2 , for some constant C ą 0. Let us now denote v N ptq " N ´1 ř N i"1 Er|X i t ´Xi,N t | 2 s, then we have v 1 N ptq ď ´2βv N ptq `C ? N ´1 v N ptq 1{2
. This gives, using Grönwall Lemma:

(13) v N ptq 1{2 ď e ´βt v N p0q `C 2β ? N ´1 . Moreover, v N p0q " N ´1 ř N i"1 Er|X i 0 ´Xi,N 0 | 2 s ď Er|X i 0 | 2 `|X i,N 0 | 2 s is bounded uniformly in N , and as Erd R d 2 pm N Xt , m N X N t qs ď v N ptq 1{2
, we conclude the proof of Theorem 1.

3.2. Uniqueness of the invariant measure. The main consequence of the previous result is the uniqueness of the invariant measure for the process pm t q tě0 driven by [START_REF] Bashiri | On the long-time behaviour of McKean-Vlasov paths[END_REF]. As recalled at the beginning of the section, we have already shown that under Assumptions (A1) and (A2), there exists an invariant measure s P . From Theorem 1, we get the following Corollary Corollary 2. Under Assumptions (A4), the stochastic process pm t q admits a unique invariant measure s P P P 2 `P `Rd ˘˘. Moreover, for each P 0 P L 2 pPpR d qq, there is an exponential convergence to the invariant measure:

d PpR d q 2 pP t , s P q ď e ´βt d
PpR d q 2 pP 0 , s P q 2 , where we recall that the Wasserstein-2 distance on PpR d q is defined by in (3). This implies uniqueness of the invariant measure and the convergence to this equilibria for a large class of initial conditions P 0 . In order to prove the previous result, we begin with a technical Lemma: Lemma 3. Let m and ρ be two probability measures valued random variables which are F 0 0 -measurable. Then, there exists a random variable ξ defined on the space pΩ, F , Pq and with value in PpR d q, such that almost surely:

ξ P arg min πPΠpm,ρq ż R d |x ´y| 2 πpdx, dyq.
The proof of this Lemma is postponed to Appendix A.3 and relies on mesurability arguments for set valued functions issued from [START_REF] Stroock | Multidimensional diffusion processes[END_REF].

Proof of Corollary 2. The proof of this result relies on the result and the proof of Theorem 1, but the important difference is the choice of the initial conditions. More precisely, for P 0 P L 2 pPpR d qq, we pick Γ P Π `P0 , s P ˘which is not empty. Let us consider a couple of probability measure valued random variables pm 0 , s m 0 q, and such that L ppm 0 , s m 0 qq " Γ. It means that Lpm 0 q " P 0 and Lp m0 q " P0 . Thanks to Lemma 3, we know that there exists ξ random variable such that almost surely,

ξ P arg min πPΠpm0,Ď m0q ż R d |x ´y| 2 πpdx, dyq.
We consider once again the particle system:

dX i
t " ´∇pX i t qdt ´∇W ˚mt pX i t qdt `σdB i t `σ0 dB 0 t , @i P t1, . . . , N u, and

dX i,N t " ´∇pX i,N t qdt ´1 N N ÿ j"1 ∇W pX i,N t ´Xj,N t qdt `σdB i t `σ0 dB 0 t , @i P t1, . . . , N u,
where the pX i 0 , X i,N 0 q for i P t1, . . . , N u are independent and such that LppX i 0 , X i,N 0 q|F 0 0 q " ξ. From Theorem 1, and more precisely Equation ( 13), we know that there exists a constant C ą 0, such that:

(14) Erd R d 2 pm N Xt , m N X N t q 2 s ď e ´βt Er|X 1 0 ´X1,N 0 | 2 s `CN ´1{2 .
Moreover, with the particular choice we made for the initial conditions, we get that

Er|X 1 0 ´X1,N 0 | 2 s " ErEr|X 1 0 ´X1,N 0 | 2 |F 0 0 ss " E "ż R d |x ´y| 2 ξpdx, dyq  " ż PpR d q d R d 2 pµ, νqΓpdµ, dνq.
Taking the infimum over all the transport plan Γ between P 0 and s P in ( 14), we get that

(15) E " d R d 2 pm N Xt , m N X N t q 2 ı ď e ´βt d PpR d q 2 pP 0 , s P q `C ? N .
Moreover,

d PpR d q 2 pP t , s P q ď E " d R d 2 p s m t , m t q 2 ı , (16)
where p s m t q is the stochastic flow of measure driven by (1) with initial condition s P and pm t q has the same dynamic with initial condition P 0 . Now combining ( 15) and ( 16), taking the limit N to infinity, we get d PpR d q 2 pP t , s P q ď e ´βt d PpR d q 2 pP 0 , s P q.

Remarks 1. Some remarks are in order: (1) We recall that thanks to Proposition 1, the distance d

PpR d q 2 pP 0 , s P q is finite because P 0 P L 2 pPpR d qq in the sense of Definition 3; (2) Common noise doesn't ruin the usual convergence results. It makes sense because the common noise mainly acts like a drift term in Equation (1) and should not make the usual convergence results go haywire.

Example.

In this section, we consider an Ornstein-Uhlenbeck process with common noise, that is taking V : x Þ Ñ |x| 2 {2 and W " 0. Let X be a real valued stochastic process evolving in R d , driven by the following stochastic differential equation: [START_REF] Delarue | Restoring uniqueness to mean-field games by randomizing the equilibria[END_REF] dX t " ´Xt dt `σdW t `σ0 dB 0 t . Then, the process associated process pm t q tě0 is solution of (18)

d t m t " ∇ ¨" σ 2 `σ2 0 2 ∇m t `mt x  dt ´σ0 Dm t ¨dB 0 t ,
Moreover, we consider the following initial condition L pX 0 q " P 0 (in the sense of Definition 1), for some P 0 P L 2 pPpR d qq. In this particular setting, we are able to explicitly describe the invariant measure.

Proposition 3. The unique invariant measure s P of the process on the space PpR d q is the image, by the function x P R d Þ Ñ N d p´x, σ 2 Idq P PpR d q of the measure γ σ0 ; where γ σ0 pdxq " p2σ 2 0 q ´1{2 e ´|x| 2 {2σ 2 0 dx, where for all µ P R d , Σ P M d pRq, N d pµ, Σq denotes a gaussian distribution in dimension d centered in µ and with variance-covariance matrix Σ.

The fact that in this case we are able to exhibit the invariant measure comes from the linearity of the equation and the linearity of the confinement forces which derives from a quadratic potential, as shown in the proof. The convergence to the equilibria holds at an exponential thanks to Theorem 1.

Proof of Proposition 3. Let be the process X 0 defined dynamic given by: "

dX 0 t " ´X0 t dt ´σ0 dB 0 t L `X0 0 ˘" N `0, σ 2 0 ˘.
The initial condition is such that the process X 0 is stationary. We now define the process p s m t q t with s m 0 , such that Lp m0 q " P as initial condition and staying

(19) d t s m t " ∇ ¨" σ 2 `σ2 0 2 ∇ s m t `s m t x ı dt ´σ0 D s m t ¨dB 0 t .
Let us define r m t :" pId ´σ0 B 0 t q7 s m t , where 7 stands for the pushforward operator. Then, applying Itô-Wentzell formula, we get that r m t satisfies ( 20)

" B t r m t " ∇ ¨r σ 2 2 ∇ r m t `r m t px `B0 t qs, r m 0 " s m 0 . Moreover, we get that m t :" N p´pX 0 t `σ0 B 0 t q, σ 2 Idq satisfies, (21) d t m t pxq " ´σ´2 px `X0 t `σ0 B 0 t qm t pxq dpX 0 t `σ0 B 0 t q " σ ´2px `X0 t `σ0 B 0 t q ¨X0 t m t pxq dt.
Then, as it shows that pm t q t satisfies (20) with m 0 " s m 0 and then ppId `σ0 B 0 t q7m t q t is solution of ( 19) with the same initial condition. Finally, s m t pdxq " pId `σ0 B 0 t q7m t pdxq " c expt´|x `X0 t | 2 {2σ 2 udx. As pX 0 t q t is stationary in R d , s m t is also in PpR d q, which shows the first part of the Proposition. The uniqueness of this invariant measure is then given by Corollary 2.

THE ONE DIMENSIONAL CASE

To get uniqueness of the invariant measure when the confinement potential is not convex , we need to establish a more advanced coupling than the one used in previous section. To do this, we follow [START_REF] Durmus | An elementary approach to uniform in time propagation of chaos[END_REF], [START_REF] Eberle | Quantitative Harris-type theorems for diffusions and McKean-Vlasov processes[END_REF], [START_REF] Eberle | Reflection couplings and contraction rates for diffusions[END_REF], or [START_REF] Conforti | Coupling by reflection for controlled diffusion processes: turnpike property and large time behavior of Hamilton Jacobi Bellman equations[END_REF]. For the sake of clarity, we recall here the assumptions in dimension d " 1:

Assumption (A1).

(1) There exists a continuous function κ : r0, `8q Ñ R, such that lim inf rÑ`8 κprq ą 0, and pV 1 pxq ´V 1 pyqqpx ´yq ě κp|x ´y|q|x ´y| 2 .

(

) V 1 is L V -Lipschitz continuous. 2

Assumption (A2).

(1) W is symmetric, i.e., W pxq " W p´xq for all x P R d . (2) W 1 is L W -Lipschitz continuous.

Remark 1.

As the function κ exhibits positive values far from the origin, its continuity implies that it possesses a lower bound. Hence, we can ensure the boundedness of the function κ ´" maxp´f, 0q.

This framework allows multi-well interactions potentials for which uniqueness of the invariant measure does not hold without common noise, see [START_REF] Herrmann | Non-uniqueness of stationary measures for self-stabilizing processes[END_REF].

4.1. Uniform in time propagation of chaos in dimension 1. This section introduces a quantitative study of the uniform propagation of chaos. Because of the non-convexity of the confinement potential, we cannot use the classical propagation of chaos approach from Section 3. Instead, we will apply a more advanced approximation method. However, this method does not produce satisfactory results in dimensions d ě 2 with common noise. For this reason, in this section, we focus on the case where d " 1, and will explain why in more detail later.

The approximation method. Following the framework of [START_REF] Eberle | Reflection couplings and contraction rates for diffusions[END_REF], we now define the constants: R 0 " inf ts ě 0, κprq ě 0, @r ě su , R 1 " inf s ě R 0 , sps ´R0 qκprq ě 4σ 2 0 , @r ě s ( .

Moreover, we consider ϕ, Φ, g : r0, `8q Ñ r0, `8q defined by

ϕprq " exp ˆ´1 2σ 2 0 ż r 0 sκ ´psqds ˙, Φprq " ż r 0 ϕpsqds, gprq " 1 ´ℓ 2 ż r^R1 0 Φpsq{ϕpsqds,
where κ ´" maxp0, ´κq and ℓ " ´şR1 0 Φpsqϕpsq ´1ds ¯´1 . We now define an increasing function f : r0, `8q Ñ r0, `8q by:

f prq " ż r 0 ϕpsqgpsqds.
The function f that has been constructed is clearly positive, non-decreasing and concave. Moreover, it satisfies [START_REF] Eberle | Reflection couplings and contraction rates for diffusions[END_REF] ϕpR 0 qr{2 ď f prq ď r.

This ensures that px, yq Þ Ñ f p|x ´y|q defines a distance which is equivalent to the Euclidean one. Below, we will use contraction properties in Wasserstein-1 distance based on the underlying distance f p|x ´y|q. These contraction property is a consequence of Proposition 4.

Proposition 4.

The following inequalities holds for all r ą 0:

f 2 prq ´1 2σ 2 0 rκprqf 1 prq ď ´ℓf prq{2.
The proof of this Proposition is found in Appendix A.4. However, it is essential to emphasize that the function f was constructed to achieve a contraction inequality, ensuring uniform propagation of chaos over time. These techniques were greatly inspired by [START_REF] Lindvall | Coupling of multidimensional diffusions by reflection[END_REF] and further developed in [START_REF] Eberle | Reflection couplings and contraction rates for diffusions[END_REF]. In that sense, let us consider P, Q P L 4 pPpR d qq. Let us also consider δ ą 0, and once again two particle systems X δ " pX 1,δ , . . . , X N,δ q and X N,δ " pX 1,N,δ , . . . , X N,N,δ q with dynamic given by: (23) dX i,δ t " ´V where:

' The B i are independent Wiener processes in R, which all are independent of F 0 ; ' B 0 is a Wiener process, adapted to the filtration F 0 and indepependent of F 1 ; ' m δ t stands for the conditional law of each process X i,δ with respect to the σ-field F 0 t , at each time t ą 0; ' E N,δ t stands for the vector of differences at time t, more precisely, for i P t1, . . . , N u,

E i,N,δ t " X i,δ t ´Xi,N,δ t .
' The pairs pX i,N,δ 0 , X i,δ 0 q are conditionally independent and identically distributed with respect to the σ-algebra F 0 0 . Furthermore, for every i P t1, . . . , N u, the law of X i 0 is P according to Definition 1, and the law of X i,N 0 is Q, in the same manner. ' We define a non-decreasing and continuous function π, such that for x P R N , πpxq "

# 1 if N ´1 ř N i"1 |x i | ě 1 0 if N ´1 ř N i"1 |x i | ď 1{2
, and, consider a non negative function λ such that πpxq 2 `λpxq 2 " 1, @x P R N . Moreover, we extend π on the whole space, with the constraint that this remains a non decreasing and Lipschitz continuous function. Finally, we define π δ : x Þ Ñ π px{δq.

The condition on π and λ implies in particular that ´şt 0 π δ pE N,δ s qdB 0 s `şt 0 λ δ pE N,δ s qd B0 s indeed defines a Brownian motion thanks to Levy's characterisation of Brownian motion. Then, ( 23) and ( 24) defines a coupling of ( 11) and [START_REF] Cattiaux | Probabilistic approach for granular media equations in the non uniformly convex case[END_REF]. Before giving the first main result of the section, let us introduce the quantity [START_REF] Fehrman | Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise[END_REF] cpV, W, σ 0 q " ℓσ 2 0 ´4L W ϕpR 0 q ´1. This quantity naturally appears in the result of uniform in time propagation of chaos. In order to ensure the uniformity in time, let us denote for all V and W , the space D V,W " t σ 0 ě 0, cpV, W, σ 0 q ą 0u. Proposition 5. There exists s L ą 0, s σ 0 ą 0 such that for all σ ě 0, for any W which is L-Lipschitz continuous with L ă s L, rσ 0 ; `8q Ă D V,W .

The proof of this Proposition is postponed to Appendix A.6. We are now ready to give the main result of this section Theorem 2. Under Assumptions (A1) and (A2) there exist two systems pX, X N q, couplings of (11) and (12), which are susbsenquential weak limit as δ Ñ 0 of pX δ , X N,δ q. Let σ ě 0, W such that L W ă s L, and σ 0 ě s σ 0 , where s L, s σ 0 are defined in Proposition 5. Then, there exists a constant C ą 0 which does not depend on t and N , such that:

E " d R d f pm N Xt , m N X N t q ı ď C ´e´cpV,W,σ0qt `1 ? N ¯, where d R d f pµ, νq " inf πPΠpµ,νq ş R d f p|x ´y|qπpdx, dyq, and m N Xt (resp. m N X N t
) stands for the empirical measures of all the particles of the system X (resp. X N ) at time t.

Remark 2. Some remarks are in order: (1) As in [START_REF] Durmus | An elementary approach to uniform in time propagation of chaos[END_REF], we need to consider a slight distortion of the Wasserstein-1 distance. This allows for easy handling of the residual terms, due to the non-convexity of the confinement potential, through a functional inequality.

(2) The Lipschitz continuity assumption on the confinment force ∇V does not help convergence toward the invariant measure and mainly comes from technical needs. It is not a sharp condition and similar outcomes might be reached by assuming for example that ∇V has at most a polynomial growth.

(3) The first drawback of this result is the impossibility to take advantage of the potential convexity structure of the interaction potential W , as strong interactions within the system should ideally help in converging to equilibrium. This limit is shown by the term ´4LϕpR 0 q ´1 in cpV, W, σ 0 q, which penalizes the convergence rate. (4) The main issue with this method is that it does not work in higher dimensions. Using the same coupling techniques in dimensions d ě 2 requires a reflection with the orthogonal matrix Id ´2e i,N t pe i,N t q t in order to ensure that both process are remaining close in large time, as explained in [START_REF] Lindvall | Coupling of multidimensional diffusions by reflection[END_REF] . However, this is not possible in our situation as we cannot use a reflection matrix dependent on a single particle for the common noise. In the following section, we will discuss a specific case where we can overcome this difficulty.

In order to prove the previous quantitative and uniform in time propagation of chaos result, we begin with the following Proposition. Proposition 6. Under Assumptions (A1) and (A2), we have the following decomposition for all i P t1, . . . , N u:

d|E i,N,δ t | " ´ei,N,δ t ´V 1 pX i,δ t q ´V 1 pX i,N,δ t q ¯dt `Ai,N,δ t dt `2σ 0 π δ pE N,δ t qpe i,N,δ t q T dB 0 t ,
where

e i,N,δ t " # E i,N,δ t {|E i,N,δ t | if E i,N,δ t ‰ 0 0 otherwise , and
A i,N,δ t ď ˇˇW 1 ˚mδ t pX i,δ t q ´1 N N ÿ j"1 W 1 pX i,N,δ t ´Xj,N,δ t q ˇˇ.
The result above adapts Lemma 7 from [START_REF] Durmus | An elementary approach to uniform in time propagation of chaos[END_REF] for the common noise setting, and thus, the proof is postponed to Appendix A.5. This Proposition is important as under Assumptions (A1) and (A2) the main term helps with uniform control over time, while the secondary term is handled as a perturbation. The two martingale terms are centered, and the other terms guarantee concentration, even when the confinement potential is not convex. Now, let us present the following Lemma.

Lemma 4. Under assumptions (A1) and (A2), for all T ą 0, there exists a positive constant C independent of δ, such that for all 0 ă s ă t ă T ,

E " 1 N N ÿ i"1 |X i,δ t ´Xi,δ s | 4 `1 N N ÿ i"1 |X i,N,δ t ´Xi,N,δ s | 4 ı ď C|t ´s| 2 .
Proof of Lemma 4.

Step 1. Let us begin this proof by considering the existence of a finite moment of order 4 for every particle in both systems at all times. Let us do it only for the mean-field limit system as the proof is exactly the same for the interacting particle system. Then, from Itô's formula, we get for each i P t1, . . . , N u

d dt Er|X i,δ t | 4 s " ´p´E rpX i,δ t q 3 V 1 pX i,δ t qs `ErpX i,δ t q 3 W 1 ˚mδ t pX i,δ t qs 6pσ
2 `σ2 0 qErpX i,δ t q 2 s ď ´4m V ErpX i,δ t q 4 s `4M V ErpX i,δ t q 3 s `4ErpX i,δ t q 3 sV 1 p0q ´4ErpX i,δ t q 3 W 1 ˚mδ pX i,δ t qs `6pσ 2 `σ2 0 qpX i,δ t q 4 `6pσ 2 `σ2 0 q. Furthermore, we have knowledge that W 1 is Lipschitz continuous, which implies that the convolution W 1 ˚mδ t also is. Then, we get

d dt Er|X i,δ t | 4 s ď c 1 Er|X i,δ t | 4 s `c2
, where c 1 and c 2 are two constants independent of δ and t. Using Grönwall Lemma, this gives:

Er|X i,δ t | 4 s ď e c2t Er|X i,δ 0 | 4 s `c2 c ´1 1 e c1t .
Then, as P 0 , Q 0 P L 4 pPpR d qq we get that for each T ą 0, sup tPr0,T s Er|X i,δ t | 4 s ă `8.

Step 2. Let us now consider T ą 0, and 0 ă s ă t ă T .

Er|X i,δ t ´Xi,δ s | 4 s " E "ˇˇˇż t s ´V 1 pX i,δ r q ´W 1 ˚mδ r pX i,δ r qdr `σpB i t ´Bi s q `σ0 ! ż t s π δ pE N r qd B0 r `ż t s λ δ pE N r qdB 0 r )ˇˇˇ4ı ď E "ˇˇˇż t s ´V 1 pX i,δ r q ´W 1 ˚mδ r pX i,δ r qdr ˇˇ4 ı `Cpσ, σ 0 q|t ´s| 2 ,
where we used the Burkholder-Davis-Gundy inequality to obtain the constant Cpσ, σ 0 q, which does not depend on δ. Controlling the residual term is a straightforward process that involves using the result of Step 1 with the Lipschitz continuity of V 1 and W 1 . Everything works exactly the same way for the interacting particle system.

This result mainly ensures the existence of the coupling pX, X N q of Theorem 2, which we are now ready to prove.

Proof of Theorem 2. From Proposition 3 and using Itô's formula applied to the previously defined function f , we get [START_REF] Friz | Stochastic scalar conservation laws driven by rough paths[END_REF] df

p|E i,N,δ t |q " ´f 1 p|E i,N,δ t |qe i,N,δ t pV 1 pX i,δ t q ´V 1 pX i,N,δ t qqdt `Ai,N,δ t f 1 p|E i,N,δ t |qdt `2σ 2 0 f 2 p|E i,N,δ t |qπ δ pE N,δ t q 2 dt `2σ 0 e i,N,δ t f 1 p|E i,N,δ t |qπ δ pE N,δ t qdB 0 t . Moreover, ´f 1 p|E i,N,δ t |qe i,N,δ t pV 1 pX i,δ t q ´V 1 pX i,N,δ t qq `2σ 2 0 f 2 p|E i,N,δ t |qπ δ pE N,,δ t q 2 " 2σ 2 0 ´f 2 p|E i,N,δ t |qπ δ pE N,δ t q 2 ´1 2σ 2 0 f 1 p|E i,N,δ t |qe i,N,δ t pV 1 pX i,δ t q ´V 1 pX i,N,δ t qq ď 2σ 2 0 ´f 2 p|E i,N,δ t |qπ δ pE N,δ t q 2 ´1 2σ 2 0 |E i,N,δ t |κp|E i,N,δ t |qf 1 p|E i,N,δ t |q ď 2σ 2 0 ´´ℓ 2 f p|E i,N,δ t |qπ δ pE N,δ t q 2 `1 2σ 2 0 |E i,N,δ t |κ ´p|E i,N,δ t |qλ δ pE N,δ t q 2 ¯.
thanks to Proposition 4 and the fact that π 2 pxq `λ2 pxq " 1, for all x P R N . Then, averaging over all the particles, and using the fact that κ ´is bounded as a consequence of Remark 1, we get:

1 N N ÿ i"1 ´f 1 p|E i,N,δ t |qe i,N,δ t pV 1 pX i,δ t q ´V 1 pX i,N,δ t qq `2σ 2 0 f 2 p|E i,N,δ t |qπ δ pE N,δ t q 2 ď ´ℓσ 2 0 N N ÿ i"1 f p|E i,N,δ t |qπ δ pE N,δ t q 2 `|κ ´|8 N N ÿ i"1 |E i,N,δ t |λ δ pE N,δ t q 2
Plugging the last inequality into (26), we have

1 N N ÿ i"1 d dt E " f p|E i,N,δ t |q ı ď ´ℓσ 2 0 N N ÿ i"1 E " f p|E i,N,δ t |q ı `1 N N ÿ i"1 ErA i,N,δ t s `ℓσ 2 0 δ `|κ ´|8 δ.
We now need to control one last term N ´1 ř N i"1 ErA i,N,δ t s. This part of the proof follows the same lines as the one of [START_REF] Durmus | An elementary approach to uniform in time propagation of chaos[END_REF] and we get

1 N N ÿ i"1 Er|A i,N,δ t |s ď 4L W ϕpR 0 q ´1 N N ÿ i"1 Erf p|E i,N,δ t |qs `M L W N ´1{2 ,
for some M ą 0. The last inequality is not unexpected, as it is anticipated that the interaction term in Equation 12will converge at a rate of N 1{2 towards the conditional expectation Er∇W pX i t ´X t q|F 0 t , X i t s for any independent replication X of X i . Our objective now is to consider the limit as δ tends to zero. Let us define s P N,δ T as the law of ptX i,δ , X i,N,δ u i"1,...,N q on Cpr0, T s, R 2N q. With Lemma 4 and Kolmogorov's continuity criterion, we establish the tightness of the sequence p s P N,δ T q δ , leading to the existence of a subsequence pδ n q n that tends to zero, such that s P N,δn T converges to s P N T , which is defined on Cpr0, T s, R 2N q. By a diagonalization argument and consistency of the family of probability tP N T , T ą 0u, we can extend this probability measure and define s P N on Cpr0, `8q, R 2N q. Now we can define two systems of particle X " pX 1 , . . . , X N q and X N " pX 1,N , . . . , X N,N q with law s P N on Cpr0, `8q, R 2N q. This argument that justifies the existence of the coupling is classical, see for example [START_REF] Durmus | Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement[END_REF] or [START_REF] Eberle | Sticky couplings of multidimensional diffusions with different drifts[END_REF]. Moreover, by uniqueness in law of the solutions, we get that pX, X N q is a coupling of ( 11) and [START_REF] Cattiaux | Probabilistic approach for granular media equations in the non uniformly convex case[END_REF]. Then, defining:

E i,N t " X i t ´Xi,N t , and
e i,N t " # E i,N t {|E i,N t | if E i,N t ‰ 0 0 otherwise , we get 1 N N ÿ i"1 d dt Erf p|E i,N t |qs ď ´´ℓσ 2 0 N ´4L W ϕpR 0 q ´1¯N ÿ i"1 Erf p|E i,N t |qs `M L W N ´1{2
Now, as σ 0 P D V,W , we can use the Grönwall lemma, and get that

1 N N ÿ i"1 Erf p|E i,N t |qs ď exp ´´pℓσ 2 0 ´4L W ϕpR 0 q ´1qt ¯1 N N ÿ i"1 Erf p|E i,N 0 |qs `´ℓσ 2 0 ´4L W ϕ pR 0 q ´1¯´1 M L W N ´1{2
Moreover, we know that

Erd R f pm N Xt , m N X N t qs ď 1 N N ÿ i"1
Erf p|E i,N t |qs.

Then,

Erd R f pm N Xt , m N X N t qs ď e ´cpV,W,σ0qt 1 N N ÿ i"1 Erf p|E i,N 0 |qs `cpV, W, σ 0 q ´1M L W N ´1{2 .
and finally

(27) Erf p|E 1,N 0 |qs " ErErf p|X 1 0 ´X1,N 0 |q|F 0 0 ss ď CEr|X 1,N 0 | `|X 1
0 |s ă `8, thanks to Assumption Lemma 2, which concludes the proof. 4.2. Uniqueness of the invariant measure. As in the previous section with convex potential, we can deduce from the propagation of chaos result the uniqueness of the invariant measure for the process pm t q tě0 driven by [START_REF] Bashiri | On the long-time behaviour of McKean-Vlasov paths[END_REF], whenever the intensity of the common noise is large enough. Corollary 3. Whenever d " 1 and Assumptions (A1) and (A2) hold, assuming that W is such that L ă L, and σ 0 ě σ0 , where L and σ0 are defined in Proposition 5, the stochastic process pm t q admits a unique invariant measure s P P PpPpRqq. Moreover, for each P 0 P L 4 pPpRqq,we get

d PpRq f pP t , s P q ď e ´cpV,W,σ0qt d PpRq f pP 0 , s P q,
where d

PpRq f p¨, ¨q is defined in Definition 5.

It is interesting to highlight here that the presence of common noise allows to get uniqueness, in fact in the case without common noise, when considering non-convex confinement potential, it has been shown that uniqueness of the invariant measure (in PpR d q) does not hold. More precisely, considering the deterministic flow of measure pm t q t which is weak solution of

B t m t " σ 2 2 B 2 xx m t `Bx pm t pV 1 `W 1 ˚mt qq, (28)
it is known, see e.g [START_REF] Herrmann | Non-uniqueness of stationary measures for self-stabilizing processes[END_REF], that whenever σ is small enough, there is exactly three invariant measures. This is roughly due to the fact that whenever the intensity of the noise is not important enough, the associated process X, solution of dX t " ´V 1 pX t qdt ´W 1 ˚mt pX t qdt `σdB t stays stuck in the minimum of V due to the lack of convexity. Here the presence of a strong enough common noise allows to recover uniqueness but in the space PpPpR d qq.

MULTIDIMENSIONAL NON-CONVEX CASE WITH σ " 0

In this section, we return to the general dimension case where d ě 2. It turns out that in some specific situations, we can use the presence of interaction to adapt the coupling introduced in Section 4 and achieve an exponential rate of convergence to the unique invariant measure when there is no idiosyncratic noise (σ " 0). More precisely, let us consider m with dynamic given by d t m t " ∇ ¨´σ 2 0 2 ∇m t `mt p∇V `∇W ˚mt q ¯dt ´σ0 Dm t ¨dB 0 t .

In the following of the section, let us consider the following Assumption:

Assumption (A6). The potential V satisfies Assumption (A1) and there exists α ą 0, such that W pxq " α|x| 2 {2.

5.1. Existence of an invariant measure. We begin this section with a non quantitative proof of existence and uniqueness of the invariant measure.

Proposition 7. Under Assumption (A6), for α large enough, there exists a unique invariant measure s P in the sense of Definition 1, and this measure is supported by Dirac masses. More precisely,

s P pdmq " ż R d δ δa m 0 pdaq
where m 0 is the probability measure on R d solution of ´σ2 0 2 ∆m 0 ´∇ ¨p∇V m 0 q " 0. Proof of Proposition 7. If s P pdmq " ş R d δ δa m 0 pdaq, then for all twice differentiable function in the sense of Lions derivarives F P C 2 pPpR d q, Rq (29)

IpF q :" ż P2pR d q "ż R d ˆDm F pm, xq ¨p´∇V pxq ´∇W ˚mpxqq `σ2 0 2 div x D m F pm, xq ˙qmpdxq `σ2 0 2 ż R 2d Tr " D 2 mm F pm, x, yqq ‰ mpdxqmpdyq  s P pdmq " ż R d " D m F pδ a , aq ¨p´∇V paqq `σ2 0 2 div x D m F pδ a , aq `σ2 0 2 Tr " D 2 mm F pδ a , a, aq ‰  m 0 pdaq,
where, if we define ϕpaq " F pδ a q, then ∇ϕpaq " D m F pδ a , aq, ∆ϕpaq " Tr " D 2 mm F pδ a , a, aq `D2 xm F pδ a , aq ‰ .

Then,

(30) IpF q " ż R d " ´∇ϕpaq ¨∇V paq `σ2 0 2 ∆ϕpaq  m 0 pdaq " 0, as m 0 is solution of σ 2 0
2 ∆m 0 `∇ ¨p∇V m 0 q " 0. 5.2. Uniqueness and convergence to the equilibirum. When σ " 0, we can explicitly find the unique invariant measure. Moreover, we can use the strong interaction structure to achieve exponential rates of convergence to this invariant measure. To do this, we build a coupling similar to the one in the previous section, taking advantage of the fact that in this setting, each particle is close to the average of all particles in the system. Consequently, we get the following result Theorem 3. Whenever σ " 0 and under Assumptions (A4), for any initial conditions P 0 , Q 0 P L 4 pPpR d qq, there exists a constant C that does not depend on t such that

d PpR d q f pP t , Q t q ď C ´e´ℓσ 2 0 t `e´pα´2LV qt ¯,
where f and ℓ are defined in Section 4, and the distance d

PpR d q f
p¨, ¨q is defined in [START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications I-II[END_REF].

Remark 3. In the light of Remark 2, we can now observe that the rate of convergence is not penalised by the presence of a strong interaction. In fact, in this case we are able to fully take advantage of the quadratic structure of the interaction potential.

The key observation to prove the previous result is that when considering an interacting particle system with each particle's dynamic given by dynamic [START_REF] Eberle | Quantitative Harris-type theorems for diffusions and McKean-Vlasov processes[END_REF], the interaction acts like an attractor gathering all the particles together. The coupling of Section 4 fails due to the dependence of the reflection matrix on a single particle. To address this dependence, we replace it with a dependence on the empirical mean of all system particles. By showing that the resulting error becomes negligible as the number of particles and time increase, we show the expected result.

Proof of Theorem 3. Let us consider two measures P 0 , Q 0 on P `Rd ˘, which admit a moment of order four. In the light of the proof of Proposition 2, and Remark 2, let us introduce the two following particle systems for δ ą 0, X δ " pX 1,δ , . . . , X N,δ q and Y N,δ " pY 1,N,δ , . . . , Y N,N,δ q, with dynamic dX i,δ t " ´∇V pX where s

E N,δ t " N ´1 ř N i"1 X i,δ t ´N ´1 ř N i"1 Y i,N,δ t
, and

s e N,δ t " # s E N,δ t {| s E N,δ t | if | s E N,δ t | ‰ 0 0 otherwise .
Moreover, the functions π and λ are defined in Section 4. Finally we assume that the X i,δ 0 are conditionally iid with respect to the common noise with law P 0 . Analogously, the same is assumed for the Y i,N,δ with law Q 0 . Proposition 8. Under Assumption (A4), we get:

Er 1 N N ÿ i"1 |X i,δ t ´s X N,δ t | 2 s ď Er 1 N N ÿ i"1 |X i,δ 0 ´s X N,δ 0 | 2 se ´2pα´2LV qt , and
E " 1
N N ÿ i"1 |Y i,N,δ t ´s Y N,δ t | 2 ı ď E " 1 N N ÿ i"1 |Y i,N,δ 0 ´s Y N,δ 0 | 2 ı e ´2pα´2LV qt ,
for α ą 2L V and where s

X N,δ t " N ´1 ř N i"1 X i,δ t and s Y N,δ t " N ´1 ř N i"1 Y i,N,δ t
stand for the empirical means of each system at time t.

The proof is straightforward using the Lipschitz continuity of ∇V and the quadratic structure of the interacting potential.

Proof. Let us only perform this calculation for the mean field limit system of N particles X δ " pX 1,δ , . . . , X N,δ q. Let now t ą 0 and i P t1, . . . , N u,

d|X i,δ t ´X N,δ t | 2 " ´2pX i,δ t ´X N,δ t q ¨´∇V pX i,δ t q ´1 N N ÿ i"1 ∇V pX i,δ t q ¯dt ´2pX i,δ t ´X N,δ t q ¨´∇W ˚mδ t pX i,δ t q ´1 N N ÿ i"1 ∇W ˚mδ t pX i,δ t q ¯dt ď 2L V |X i,δ t ´X N,δ t | 2 dt `2 L V N N ÿ j"1 |X j,δ t ´X N,δ t | |X i,δ t ´X N,δ t |dt ´2α|X i,δ t ´X N,δ t | 2 dt.
Then, averaging over i and using Cauchy-Schwarz inequality, we get that

d dt 1 N N ÿ i"1 Er|X i,δ t ´X N,δ t | 2 s ď 2p2L V ´αq 1 N N ÿ i"1 Er|X i,δ t ´X N,δ t | 2 s.
Then, one can conclude using Grönwall Lemma.

Considering the function f introduced in Section 4, we can write:

f p|X i,δ t ´Y i,N,δ t |q ď f p| s Y N,δ t ´Y i,N,δ t |q `f p|X i,δ t ´s X N,δ t |q `f p| s X N,δ t ´s Y N,δ t |q. (32)
By taking the expectation and averaging, we can control the first two terms using Proposition 8. We now need to tackle the third term. To deal with this problem, we use an approach similar to the one employed in the proof of Theorem 2. Taking advantage of the presence of π δ , we can again apply the stochastic dominated convergence theorem, as shown in the proof of Theorem 2. Consequently, we obtain

d| s X N,δ t ´s Y N,δ t | " ´s e N,δ t ¨´1 N N ÿ i"1
∇V pX i,δ t q ´∇V pY i,N,δ t q ¯dt (33)

´s e N,δ t ¨´1 N N ÿ i"1 ∇W ˚mt pX i,δ t q ¯dt `2σ 0 π δ p s E N,δ
t qps e N,δ t q t dB 0 t . Now, we can write

df p| s X N,δ t ´s Y N,δ t |q " ´f 1 p| s E N,δ t |qs e N,δ t ¨´1 N N ÿ i"1 ∇V pX i,δ t q ´∇V pY i,N,δ t q ¯dt ´f 1 p| s E N,δ t |qs e N,δ t ¨´1 N N ÿ i"1 ∇W ˚mδ t pX i,δ t q ¯dt `2σ 0 f 1 p| s E N,δ t |qπ δ p s E N,δ t qps e N,δ t q t dB 0 t `2σ 2 0 f 2 p| s E N,δ t |qπ δ p s E N,δ t q 2 dt.
To control the first term, we carry out the following decomposition

´f 1 p| s E N,δ t |q s E N,δ t ¨p 1 N N ÿ i"1 ∇V pX i,δ t q ´∇V pY i,N,δ t qq " ´f 1 p| s E N,δ t |qs e N,δ t ¨´1 N N ÿ i"1 ∇V pX i,δ t q ´∇V p s X N,δ t q f 1 p| s E N,δ t |qs e N,δ t ¨´1 N N ÿ i"1 ∇V p s Y N,δ t q ´∇V pY i,N,δ t q f 1 p| s E N,δ t |qs e N,δ t ¨p∇V p s X N,δ t q ´∇V p s Y N,δ t qq.
Using the fact that f 1 is globally bounded and Proposition 8, we get the existence of C 1 ą 0, independent of N, t and δ, such that

´E" f 1 p| s E N,δ t |qs e N,δ t ¨´1 N N ÿ i"1 ∇V pX i,δ t q ´∇V pY i,N,δ t q ¯ı ď C 1 e ´pα´2LV qt ´E" f 1 p| s E N,δ t |qs e N,δ t ¨p∇V p s X N,δ t q ´∇V p s Y N,δ t qq ı .
Finally, thanks to the law of large number, we get the existence of a constant C 2 such that

E "ˇˇˇ1 N N ÿ i"1 ∇W ˚mt pX i,δ t q ˇˇı ď C 2 N ´1{2 .
We ı .

Finally, using Proposition 4, we conclude that

d dt Erf p| s X N,δ t ´s Y N,δ t |qs ďC 1 e ´pα´2LV qt `C2 N ´1{2 ´ℓσ 2 0 f p| s E N,δ t
|q `ℓσ 2 0 δ `|κ ´|8 δ, Then, taking the limit δ Ñ 0 exactly as in the proof of Theorem 2, we get the existence of two systems X and X N which are couplings of ( 11) and [START_REF] Cattiaux | Probabilistic approach for granular media equations in the non uniformly convex case[END_REF], such that

d dt E " f `| s X N t ´s Y N t | ˘‰ ďC 1 e ´pα´2LV qt `C2 N ´1{2 ´ℓσ 2 0 f p| s E N,δ t |q.
Then, using Grönwall lemma, we get that for α large enough,

E " f `| s X N t ´s Y N t | ˘‰ ď C 3 ´e´pα´2LV qt `e´ℓσ 2 0 t `N ´1{2 ¯,
for some constant C 3 that does not depends on t and N . Plugging this into Equation [START_REF] Hammersley | Weak existence and uniqueness for McKean-Vlasov SDEs with common noise[END_REF] gives

E " 1 N N ÿ i"1 f p|X i t ´Y i,N t |q ı ď C ´e´pα´2LV qt `e´ℓσ 2 0 t `N ´1{2 ¯.
Finally, using the exact same methodology as in Corollary 2, we conclude to the expected result.

APPENDIX A.

A.1. Proof of Proposition 1. Equation ( 9) comes from Section 1.5 in [START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF] (Equation (1.15)). Hence, if pm t q is an invariant measure in the sense of Definition 2, then P straightforwardly is a solution of Equation [START_REF] Carrillo | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF]. Conversly, let us consider a probability measure P , solution of Equation [START_REF] Carrillo | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF]. More precisely, for all F P C 2 b pPpR d qq @ s P , MF D " 0.

Let us consider T ą 0, then ż

PpR d q }∇V `∇W ˚m} 2 L 2 pmq P pdmq ď C ´1 `żPpR d q ż R d |∇V pxq| 2 mpdxq P pdmq `żPpR d q ż R d |∇W ˚mpxq| 2 mpdxq P pdmq ď C ´1 `żPpR d q ż R d |x| 2 mpdxq P pdmq `żPpR d q |∇W ˚mp0q| 2 P pdmq ď C ´1 `żPpR d q ż R d |x| 2 mpdxq P pdmq ¯,
for some constant C that may change from line to line and depends only on the Lipschitz constant of ∇V and ∇W . Then using Proposition 2, we get that ż PpR d q }∇V `∇W ˚m} 2 L 2 pmq P pdmq ă `8.

Applying Theorem 1.5 in [START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF], we get the existence of a process pµ t q P Cpr0, T s, PpR d qq such that µ has dynamic given by (1) and for all t P r0, T s, Lpµ t q " P . Then by weak uniqueness of the solutions of (1), we get that P is an invariant measure for m in the sense of Definition 2.

A.2. Proof of Lemma 2. We consider the process pX t q, driven by the dynamic (4) and with initial condition X 0 such that LpX 0 q " P 0 , in the sense of Definition 1. Let us denote in the following m 2 ptq " E " |X t | 2 ‰ . Then expanding using Ito formula and taking the time derivative, gives: m 1 2 ptq " ´2E rX t ¨p∇V pX t q `∇W ˚mt pX t qqs `pσ 2 `σ2 0 qd " ´2E r X t ¨p∇V pX t q ´∇V p0qqs ´2∇V p0qE rX t s ´2E rX t ¨∇W ˚mt pX t qs `pσ 2 0 `σ2 qd.

Moreover, if r X t an independent copy of X t , then:

∇W ˚mt pX t q " ż R d
∇W pX t ´yqm t pdyq " Er∇W pX t ´r X t q|X t , F 0 0 s. Now, the fact that W is even gives 2ErX t ∇W ˚mt pX t qs " ErpX t ´r X t q ¨∇W pX t ´r X t qs. This decomposition is the key, the end of the proof is straightforward using Assumption made on the potential W (see Assumptions (A2)).

A.3. A measurable selection result. In this part, we will mainly prove Lemma 3:

Lemma 5. Let m and ρ be two probability measure valued random variables which are F 0 0 -measurable. Assuming that both random measures m and ρ admit a moment of order two almost surely, then there exists a random variable ξ, such that almost surely: Proof. The purpose of the proof is to show that there exists a measurable function φ : PpR d q PpR d q Ñ PpR d q, such that for all pm, ρq P PpR d q ˆPpR d q Ñ PpR d q, φpm, ρq P Π opt pm, ρq, the set of minimizers for the transport problem. First of all, it is straightforward that Π opt is never empty. Then, we need to show that the set valued function

Φ :

PpR d q ˆPpR d q Ñ 2 PpR d q pm, ρq Þ Ñ Πpm, ρq, is measurable, where 2 A stands for the set of subsets of A. Considering the graph Γ Φ , of Φ: Γ Φ " tppm, ρq, ξq , ξ P Φpm, ρqu , it is clear that it is a closed set, and then measurable. In particular, the multi-application Φ is measurable. Moreover, by Lemma 12.1.7 in [START_REF] Stroock | Multidimensional diffusion processes[END_REF], the application Ψ which associate to any compact set K P PpRq 2 the set ΨpKq " arg inf πPK ż R 2 f p|x ´y|qπpdx, dyq is measurable. Then the function

Λ :

PpR d q ˆPpR d q Ñ EnspPpR d qq pm, ρq Þ Ñ Π opt pm, ρq, is a compound of two mesurables functions. Finally by the measurable selection theorem, there exists a measurable function φ such that for all pm, ρq P PpR d q ˆPpR d q, φpm, ρq P Π opt pm, ρq.

A.4. Proof of Proposition 4. This proof is given in [START_REF] Eberle | Reflection couplings and contraction rates for diffusions[END_REF], we repeat it here for the reader convenience. We begin with the easy case, which is whenever r ă R 1 . In fact in this case we have: f 2 prq " ϕ 1 prqgprq `ϕprqg 1 prq " ´1 2pσ 2 `σ2 0 q rκ ´prqf prq ´ℓ 2 Φprq ď ´1 2pσ 2 `σ2 0 q rκ ´prqf prq ´ℓ 2 f prq.

This gives:

f 2 prq ´1 2 pσ 2 `σ2 0 q rκprqf 1 prq ď ´1 2pσ 2 `σ2 0 q rf prq pκ ´prq `κprqq ´ℓ 2 f prq ď ´ℓ 2 f prq, because κ ´prq ´κprq ě 0. The case where r ě R 1 is more intricate. We can easily see that f 2 prq " 0, and (34) ´1 2pσ 2 `σ2 0 q rκprqf 1 prq ď ´1 4pσ 2 `σ2 0 q rκprqϕpR 0 q ď ´rϕpR 0 q pR 1 pR 1 ´R0 qq ´1 .

We easily show that the function r Þ Ñ r{Φprq is non-decreasing on rR 1 , `8r. Then, it comes that rR ´1 1 ď ΦprqΦpR 1 q ´1, which gives:

´1 2pσ 2 `σ2 0 q rκprqf 1 prq ď ´ϕpR 0 qΦprq pΦpR 1 q pR 1 ´R0 qq ´1 .

Moreover,

Φprq " ż r 0 ϕpsqds " pr ´R0 qϕpR 0 q `ΦpR 0 q, and ż R1 R0 Φpsqϕpsq ´1 ds " ΦpR 0 qϕ ´1pR 0 qpR 1 ´R0 q `1 2 pR 1 ´R0 q 2

" pR 1 ´R0 qϕpR 0 q ´1 ˆΦpR 0 q `1 2 pR 1 ´R0 qϕpR 0 q ě pR 1 ´R0 qΦpR 1 qϕpR 0 q ´1{2. Finally, Let us now consider ψ a : x Þ Ñ px `aq 1{2 , using once again Itô formula, we get: dψ a p|E i,N,δ t | 2 q " ´2ψ 1 a p|E i,N,δ t | 2 qE i,N,δ t pV 1 pX i,δ t q ´V 1 pX i,N,δ t qqdt ´2ψ pV 1 pX i,δ t q ´V 1 pX i,N,δ t qe i,N t dt.

´ϕpR
Moreover, once again using dominated convergence theorem and the definition of π δ , we get:

lim aÑ0 ż T 0 π δ pE i,N,δ t q 2 ! 4σ 2 ψ 1 a p|E i,N,δ t | 2 q `8σ 2 |E i,N,δ t | 2 ψ 2 a p|E i,N,δ t | 2 q
) dt " 0, and the same holds for the term that is issued from the common noise. The observed outcomes may initially seem unexpected; however, the presence of π δ helps such approximations by inhibiting the differential process from staying close to the origin, which explains the absence of Local time at 0.

A.6. Proof of Proposition 5. We know that:

γ :" lim inf rÑ`8 κprq ą 0, then, there exists r 0 ą 0, such that κprq ě γ{2, @r ě r 0 . It is clear that R 0 ď r 0 , and then R 1 ď inf s ě r 0 , sps ´R0 qκprq ě 4 `σ2 `σ2 0 ˘( . Now, using the definition of r 0 , we get R 1 ď infts ě r 0 , sps ´R0 q ě 8 γ `σ2 `σ2 0 ˘u. This gives us that

R 1 ď R 0 `aR 2 0 `32pσ 2 `σ2 0 q{γ 2 _ r 0 .
Then, as Φprq ě rϕpR 0 q, @r ě 0 and ϕpsq ď 1, @s ě 0, we can write that ℓ " p

ż R1 0 Φpsqϕpsq ´1 dsq ´1 ě 2ϕpR 0 q R 2 1 . (35)
Moreover, for σ 0 large enough, one can claim that

R 2 1 ď 2R 2 0 `32pσ 2 `σ2 0 q{γ. (36)
Finally, combing [START_REF] Kumar | Well-posedness and tamed schemes for McKean-Vlasov equations with common noise[END_REF] and [START_REF] Kurtz | Particle representations for a class of nonlinear SPDEs[END_REF], we have the following bound which is valid for σ 0 large enough: cpV, W, σ 0 q ě γ `σ2 `σ2 0 2γR 0 `32pσ 2 `σ2 0 q ϕpR 0 q ´4L W ϕpR 0 q ´1.

The right term appears to be (for each fixed values of σ) a non decreasing function of σ 2 0 . Then, as the right term converges to γ{32ϕpR 0 q ´4L W ϕpR 0 q ´1, there exists a s σ 0 , such that for all σ 0 ě s σ 0 , cpV, W, σ 0 q ě γ{64ϕpR 0 q´2L W ϕpR 0 q ´1. This allows to conclude that for L W ă s L :" γϕpR 0 q 2 {128, and σ 0 ą s σ 0 , cpV, W, σ 0 q ą 0. Hence, σ 0 P D V,W .

ξ

  P arg min πPΠpm,ρq ż R d |x ´y| 2 πpdx, dyq.

  0 qΦprqpΦpR 1 qpR 1 ´R0 qq ´1 ď ´1 2Φprq We begin this proof with the Itô decomposition of |E i,N t | 2 :

						´ż R1	Φpsqϕpsq ´1ds	¯´1
						R0
				ď	´ℓ 2	f prq,
	d|E i,N t | 2 " ´2E i,N,δ t ´2E i,N,δ t	pV 1 pX i,δ t q ´V 1 pX i,N,δ t ´W 1 ˚mδ t pX i,δ t q ´N ´1 N qqdt ÿ j"1	W 1 pX i,N,δ t	´Xj,N,δ t	q ¯dt
	`4σ 0 π δ pE N,δ t	qE i,N,δ t	dB 0 t `4σ 2 0 π δ pE N,δ

because Φprq ě f prq for all r ě 0; A.5. Proof of Proposition 6. t q 2 dt.

  As |2rψ a pr 2 q| ď 1{2, we can use the dominated convergence theorem and get

				1 a p|E i,N,δ t	|qE i,N,δ t	´W 1 ˚mδ t pX i,δ t q ´N	´1 N ÿ j"1	W 1 pX i,N,δ t	´Xj,N,δ t	q ¯dt
			`4σ 0 E i,N,δ t `8σ 2 0 |E i,N,δ ψ 1 a p|E i,N,δ t t | 2 ψ 2 a p|E i,N,δ | 2 qπ δ pE N,δ t t | 2 qπ δ pE N,δ qdB 0 t `4σ 2 0 ψ 1 a p|E i,N,δ t t q 2 dt.	| 2 qdt
	lim aÑ0	ż T 0	2ψ 1 a p|E i,N,δ t	| 2 qpV 1 pX i,δ t q ´V 1 pX i,N,δ t	qqE i,N,δ t	dt "	ż T 0

Acknowledgments

Date: June 29, 2023.

Acknowledgments. The author would like to thank Pierre Cardaliaguet for suggesting the problem and fruitful discussions during the preparation of this work.


Yves Bernard

Jean Loup Christen

Camilo Hernandez

Adel Razek

J.-L Christen

From piezoelectric actuator to piezomotor

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Piezoelectricity is a well known phenomenon that allows a direct generation of motion when an electrical field is applied with a rather high power density. However, the resulting strain has an order of magnitude of some micrometers for centimeter-sized piezoelectric elements. This particularity creates the need for complementary mechanical systems that allow obtaining suitable strokes for general applications. Piezoactuators are devices that amplify the motion of active piezoelectric elements to the range of millimeters. They are the brick elements of piezomotors which are devices that transmit motion with potentially unlimited stroke. A review of the main types of commercially available piezoelectric motors is presented in [START_REF] Spanner | An Overview of Piezoelectric Motors[END_REF]. Several surveys over specific types of piezomotors are available in [START_REF] Uchino | Acoustic Transduction --Materials and devices[END_REF] [START_REF] Morita | Miniature piezoelectric motors[END_REF]. Piezoelectric motors show characteristics such as high actuation forces, low speeds, which often render the use of gear unnecessary and high efficiency at a small scale. These features make of them an interesting solution to many applications in which the available space is limited. Additionally, the presence of friction forces in the transmission of motion often ensures a blocked position at rest. As for the small displacement amplitudes of some kinds of motors, it may lead to a very accurate motion capability. Application fields include watches, camera zoom actuators, valves for precise injection systems, and micropositionning devices.

PIEZOELECTRIC ACTUATORS

Piezoelectricity can be described as a linear phenomenon linking mechanical properties such as stress (T) and strain (S) to electric properties such as electric field (E) and electrical displacement (D) in some particular classes of materials, the most widely used being lead-zirconate-titanate (PZT) ceramics. When first poled, PZT ceramics are subject to longitudinal, transverse and shear efforts depending on the direction of the applied electrical field. Considering strain and electrical displacement as independent variables, the equations of piezoelectricity read:

⎪ ⎩ ⎪ ⎨ ⎧ E ε + dT = D E d + T s = S T t E
where d is a 3*6-matrix with only 5 non-zero coefficient when computed in the frame where the third direction is that of poling. The coefficient d 33 then accounts for longitudinal displacement, d 31 =d 32 for transverse displacement and d 15 =d 24 for shear strain. It is to be noted that longitudinal and transverse displacements always occur simultaneously when the electrical field is applied along the poling direction, whereas shear strain appears alone when the field is applied orthogonally. The order of magnitude of these coefficients is 10 -10 m.V -1 .

The reverse piezoelectric effect is used in actuators that provide a limited displacement with potentially high forces. Piezoelectric ceramics can be used directly when they are sandwiched between two electrodes, as bulktype piezoactuators. In order to achieve displacements of some tens of nanometers, the voltage applied to the piezoelement must be high (hundreds to thousands of Volts). This voltage has an upper limit imposed by the disruptive field of the material and a lower negative limit at which the natural polarization of the ceramic is reversed. Due to these constraints, several techniques are often employed to amplify the movement and reduce the applied voltage. Most piezoelectric actuators can be characterized by their free stroke and blocking force at a given voltage. In this case, a linear behavior between the maximum values of these parameters is observed.

1) Multilayers

The multilayer actuators are composed of stacked layers of piezoelectric ceramic of some tens of micrometers sandwiched between very thin electrodes, mounted electrically in parallel and mechanically in series. Compared to the bulk-type actuator, this configuration allows an increment in the electric field applied to a single layer resulting in higher strains for lower applied voltages. Typical strokes of these actuators are in the range of the tens of micrometers. Blocking forces can reach the 10 kN range. Due to layers bonding characteristics, these devices are much weaker when they are subject to tensional forces than bulk piezoelectric ceramics. Therefore, they cannot be used at resonance. In addition, the capacitance of a multilayer actuator is rather high (up to tens of µF) leading to important currents in dynamic operation.

2) Benders

The benders use piezoelectric transverse (d 31 ) mode. The extension or contraction of the ceramic occurs along one of the surfaces of a beam, causing a bending motion. Benders can be manufactured in different ways: unimorphs (a single piezoelement glued to a passive beam), parallel bimorphs (two piezoelements poled in the same direction, glued on each side of a passive grounded electrode, and powered with opposite voltages) and serial bimorphs (two piezoelements poled in opposite directions, glued to each other). The latter bimorphs present a smaller stiffness due to the absence of a passive beam. Consequently, they can provide higher strokes. The piezoelements can also be made of multilayer piezoelectric ceramics in order to limit the input voltage. Depending on the stiffness and the length of the beam, the achieved free stroke of this kind of actuators can reach several millimeters, but with very low actuation forces (less than 2 N).

3) Flextensional

The flextensional actuators work with the piezoelectric multilayer stacks clamped in a passive elastic structure, whose deformation amplifies that of the ceramic in longitudinal mode. There are several designs of flextensional structures, most of which are elliptic or rhomboidal. The amplification factor in terms of displacement ranges between 2 and 6 according to their axis ratio. This allows higher strains in the direction of motion in detriment of mass and size in the orthogonal directions. Depending on the dimension of the flexible structure and the stack, the developed force can reach the kN range. 4) Langevin transducers The Langevin transducers [START_REF] Sashida | An introduction to Ultrasonic Motors Oxford Science Publication[END_REF] are made of one piece of massive piezoelectric ceramic sandwiched between two masses whose shape and acoustic impedance are chosen to amplify the motion at the resonance. Due to the piezoelectric ceramic's weak resistance to tensional forces a prestress is applied by bolting all three parts together. This kind of actuator is optimized for amplification purposes at a specific resonant frequency and can only be used in longitudinal (d 33 ) mode. As they work at resonance, the blocked force and free displacement cannot be defined for the Langevin transducers, which are instead characterized by their operating frequency (in the 10-100 kHz range) and output power, that can reach the 100 W range for use in acoustic devices.

All the actuators presented in this part show interesting characteristics in terms of actuation force, but suffer from very limited strokes when it comes to macroscopic applications. They are either used alone (e.g. in adaptive optics), but more often assembled in more complex structures intended to provide high displacements.

DIFFERENT STRATEGIES TO ACHIEVE A SLIDING MOTION

Converting microscopic displacements into macroscopic ones can be done by transmitting periodically the movement generated in a stator to a slider through friction forces. The different types of piezomotors can be classified according to the movement of the contact point between the slider and the stator, which can be hysteretic or reversible. The latter category can be subdivided in permanent or discrete motion. The "contact point" is defined here as one particle in the stator that has had contact with the slider and has transmitted its movement to it, at a given instant in the past. For non-hysteretic movements of the contact points, there must be a dissymmetry between backward and forward phases of the movement, in order not to cancel the transmitted motion.

Figure 1: Stick-slip motor ([6])

Motors with permanent contact points rely on the stick-slip effect, which is based on the difference between static (µ s ) and dynamic (µ d ) friction coefficients. The principle is shown on figure 1. The piezoelement is driven slowly during the forward phase and rapidly during the backward phase. The forward force applied to the slider is given by:

F F = µ s N
where N is the normal preload. The backward force is given by: F B = µ d N.

Due to the difference between µ d and µ s the movement can be achieved either in the backwards or in the forward direction.

Non-permanent-contact-point motors such as the impact-drive motor or the ultrasonic standing-wave motor use a geometric dissymmetry to achieve motion. The operating principle of this motor is shown on figure 2. On the forward phase, the slider is driven by the stator with an angle α that is inside the friction cone of the slider-stator interface, so the stator sticks to the slider while imparting motion. On the backward phase, the stator tip drives the slider with an angle of π 2 α , which is outside the friction cone, so that the stator slips and similarly transmits a lower force in the backward direction. The main inconvenient of that kind of motors is the fact that it can only be driven one way. Finally, motors with hysteretic contact-point trajectory use this phenomenon to avoid exerting a force during the backward phase. One example of such a structure is the Inchworm-type motors, which use two piezoelements to clamp alternatively the slider and one between them to extend and contract. This results in a rectangular cycle during which one step is done. The process is explained on figure 3. This motor works in a quasi-static regime, but many other motors use resonance to achieve an elliptic motion of the contact point, for example in the famous traveling-wave ultrasonic motor.

EXAMPLES OF REALISATIONS AT LGEP

The LGEP has designed two piezomotors for different applications.

1) Inchworm

This work concerns the study of tissue regeneration that takes place between the existing gap of a cut bone while it is progressively separated. The actuation principle used to move one part of the bone with respect to the other the displacement has to be discontinuous performing small steps at a time in order to maintain a daily fixed gap between the two bone parts. The forces required are very weak because they only stretch the tendons that are elongated in a short distance. An inchworm (figure 4) has been designed and tested using multilayer actuators [START_REF] Sunyoto | Design and realization of a linear piezoelectric actuator for orthopaedic applications[END_REF].

CONCLUSION

In this article, the evolution of the basic piezoelectric ceramic and its extraordinary ability to generate microscopic displacements into a more elaborated structure capable of producing large or even unlimited strokes is presented.

As first stage element in the evolution there is the piezoelectric actuator. Here, different geometries have been reviewed (multilayer, benders, flextensional and Langevin). According to their principle, they favour certain aspects of the piezoelectric actuation like the force or displacement. For example, when high forces are required, it is recommended to use piezostacks whereas for important strokes benders can be the solution.

The linking between piezoelectric actuators and piezoelectric motors which compose the last element of the evolutionary chain has been effectuated by studying the motion of the contact point between the motiongenerating element and the moved one. In this manner, it has been possible to classify piezomotors. The classification unveiled in this paper presents the advantage of embracing all sorts of piezoelectric motors. In addition, with the proposed categorization it is possible to establish easily differences and relationships between motors presenting very different operating principles.

Figure 2 :

 2 Figure 2: Impact-drive motor (US Patent 20050127789)

Figure 3 :

 3 Figure 3: The operating principle of an Inchworm t

Figure 4 :

 4 Figure 4: Designed structure and test (displacement versus time)

Figure 5 :

 5 Figure 5: Designed structure and modeling for the travelling-wave motor