Francois Bolley

Arnaud Guillin

Florent Malrieu

François Bolley

Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation

Introduction and main results

We are interested in the long time behaviour and in a particle approximation of a distribution f t (x, v) in the space of positions x ∈ R d and velocities v ∈ R d (with d 1) evolving according to the Vlasov-Fokker-Planck equation

∂f t ∂t + v • ∇ x f t -C * x ρ[f t ](x) • ∇ v f t = ∆ v f t + ∇ v • ((A(v) + B(x))f t ), t > 0, x, v ∈ R d (1)
where

ρ[f t ](x) = R d f t (x, v) dv
is the macroscopic density in the space of positions x ∈ R d (or the space marginal of f t ). Here a • b denotes the scalar product of two vectors a and b in R d and * x stands for the convolution with respect to x ∈ R d :

C * x ρ[f t ](x) = R d C(x -y) ρ[f t ](y) dy = R 2d
C(xy) f t (y, v) dy dv.

Moreover ∇ x stands for the gradient with respect to the position variable x ∈ R d whereas ∇ v , ∇ v • and ∆ v respectively stand for the gradient, divergence and Laplace operators with respect to the velocity variable v ∈ R d .

The A(v) term models the friction, the B(x) term models an exterior confinement and the C(x-y) term in the convolution models the interaction between positions x and y in the underlying physical system. For that reason we assume that C is an odd map on R d . This equation is used in the modelling of the distribution f t (x, v) of diffusive, confined and interacting stellar or charged matter when C respectively derives from the Newton and Coulomb potential (see [START_REF] Bouchut | On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials[END_REF] for instance). It has the following natural probabilistic interpretation: if f 0 is a density function, the solution f t of (1) is the density of the law at time t of the R 2d -valued process (X t , V t ) t 0 evolving according to the mean field stochastic differential equation (diffusive Newton's equations)

dX t = V t dt dV t = -A(V t ) dt -B(X t ) dt -C * x ρ[f t ](X t ) dt + √ 2 dW t . (2)
Here (W t ) t 0 is a Brownian motion in the velocity space R d and f t is the law of (X t , V t ) in R 2d , so that ρ[f t ] is the law of X t in R d .

Space homogeneous models of diffusive and interacting granular media (see [START_REF] Benedetto | A non Maxwellian steady distribution for one-dimensional granular media[END_REF]) have been studied by P. Cattiaux and the last two authors in particular [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 PDE's[END_REF][START_REF]Convergence to equilibrium for granular media equations and their Euler schemes[END_REF], by means of a stochastic interpretation analogous to (2) and a particle approximation analogous to (4) below. They were interpreted as gradient flows in the space of probability measures by J. A. Carrillo, R. J. McCann and C. Villani [START_REF] Carrillo | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF][START_REF]Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF] (see also [START_REF]Optimal transport, old and new[END_REF]), both approaches leading to explicit exponential (or algebraic for non uniformly convex potentials) rates of convergence to equilibrium. Also possibly time-uniform propagation of chaos was proven for the associated particle system.

Obtaining rates of convergence to equilibrium for (1) is much more complex, as the equation simultaneously presents hamiltonian and gradient flows aspects. Much attention has recently been called to the linear noninteracting case of (1), when C = 0, also known as the kinetic Fokker-Planck equation. First of all a probabilistic approach based on Lyapunov functionals, and thus easy to check conditions, lead D. Talay [START_REF] Talay | Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme[END_REF], L. Wu [START_REF] Wu | Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems[END_REF] or D. Bakry, P. Cattiaux and the second author [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré[END_REF] to exponential or subexponential convergence to equilibrium in total variation distance. The case when A(v) = v and B(x) = ∇Ψ(x), and when the equilibrium solution is explicitely given by f ∞ (x, v) = e -Ψ(x)-|v| 2 /2 is studied in [START_REF] Hérau | Short and long time behavior of the Fokker-Planck equation in a confining potential and applications[END_REF], [START_REF] Hérau | Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential[END_REF] and [START_REF] Villani | Hypocoercivity[END_REF]Chapter 7]: hypocoercivity analytic techniques are developed which, applied to this situation, give sufficient conditions, in terms of Poincaré or logarithmic Sobolev inequalities for the measure e -Ψ , to L 2 or entropic convergence with an explicit exponential rate. We also refer to [START_REF] Esposito | Stability of the front under a vlasov-fokker-planck dynamics[END_REF] for the evolution of two species, modelled by two coupled Vlasov-Fokker-Planck equations.

C. Villani's approach extends to the selfconsistent situation when C derives from a nonzero potential U (see [START_REF] Villani | Hypocoercivity[END_REF]Chapter 17]): replacing the confinement force B(x) by a periodic boundary condition, and for small and smooth potential U , he obtains an explicit exponential rate of convergence of all solutions toward the unique normalized equilibrium solution e -|v| 2 /2 .

In this work we consider the case when the equation is set on the whole R d , with quadratic-like friction A(v) and confinement B(x) forces, and small Lipschitz interaction C(x): in the whole paper we make the following Assumption. We say that Assumption (A) is fullfilled if there exist nonnegative constants α, α ′ , β, γ and δ such that

|A(v) -A(w)| α|v -w|, (v -w) • (A(v) -A(w)) α ′ |v -w| 2 , B(x) = β x + D(x) where |D(x) -D(y)| δ|x -y| and |C(x) -C(y)| γ|x -y| for all x, y, v, w in R d .
Convergence of solutions will be measured in terms of Wasserstein distances: let P 2 be the space of Borel probability measures µ on R 2d with finite second moment, that is, such that the integral R 2d (|x| 2 +|v| 2 ) dµ(x, v) be finite. The space P 2 is equipped with the (Monge-Kantorovich) Wasserstein distance d of order 2 defined by

d(µ, ν) 2 = inf (X,V ), (Y,W ) E |X -Y | 2 + |V -W | 2
where the infimum runs over all the couples (X, V ) and (Y, W ) of random variables on R 2d with respective laws µ and ν. Convergence in this metric is equivalent to narrow convergence plus convergence of the second moment (see [START_REF]Optimal transport, old and new[END_REF]Chapter 6] for instance).

The coefficients A, B and C being Lipschitz, existence and uniqueness for Equation [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré[END_REF] with squareintegrable initial data are ensured by [START_REF] Meléard | Asymptotic behaviour of some interacting particle systems: Mc Kean-Vlasov and Boltzmannn models[END_REF]. It follows that, for all initial data f 0 in P 2 , Equation (1) admits a unique measure solution in P 2 , that is, continuous on [0, +∞[ with values in P 2 .

Assumption (A) is made in the whole paper. Under an additional assumption on the smallness of γ and δ, we shall prove a quantitative exponential convergence of all solutions to a unique equilibrium: Theorem 1. Under Assumption (A), for all positive α, α ′ and β there exists a positive constant c such that, if 0 γ, δ < c, then there exist positive constants C and C ′ such that

d(f t , ft ) C ′ e -Ct d(f 0 , f0 ), t 0 (3)
for all solutions (f t ) t 0 and ( ft ) t 0 to (1) with respective initial data f 0 and f0 in P 2 .

Moreover (1) admits a unique stationnary solution µ ∞ and all solutions (f t ) t 0 converge towards it, with

d(f t , µ ∞ ) C ′ e -Ct d(f 0 , µ ∞ ), t 0.
For instance, for α = α ′ = β = 1, the general proof below shows that the nonnegative γ and δ with γ + δ < 0, 26 are admissible. In the linear free case when γ = δ = 0, the convergence rate is given by C = 1/3, and for instance for γ and δ with γ + δ = 0, 1 we obtain C ∼ 0, 27.

Compared to Villani's results, convergence is here proven in the (weak) Wasserstein distance, not in L 1 norm, or relative entropy as in the noninteracting case -the latter being a stronger convergence since, in this specific situation, the equilibrium measure e -Ψ(x)-|v| 2 /2 satisfies a logarithmic Sobolev inequality, hence a transportation inequality. We refer to [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF], [START_REF] Ledoux | The concentration of measure phenomenon[END_REF] or [START_REF]Optimal transport, old and new[END_REF]Chapter 22] for this and forthcoming notions.

However our result holds in the noncompact case with small Lipschitz interaction, and can be seen as a first attempt to deal with more general case. Moreover it shows existence and uniqueness of the equilibrium measure, and in particular does not use its explicit expression (which is unknown in our broader situation). It is also not only a result on the convergence to equilibrium, but also a stability result of all solutions. Let us finally note that it is based on the natural stochastic interpretation (2) and a simple coupling argument, and does not need any hypoelliptic regularity property of the solutions.

The particle approximation of solutions to (1) consists in the introduction of a large number N of R 2d -valued processes (X i,N t , V i,N t ) t 0 with 1 i N , no more evolving according to the force field C * x ρ[f t ] generated by the distribution f t as in [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré[END_REF], but by the empirical measure μN

t = 1 N N i=1 δ (X i,N t ,V i,N t
) of the system: if (W i • ) i 1 with i 1 are independent standard Brownian motions on R d and (X i 0 , V i 0 ) with i 1 are independent random vectors on R 2d with law f 0 in P 2 and indepedent of (

W i • ) i 1 , we let (X (N ) t , V (N ) t ) t 0 = (X 1,N t , . . . , X N,N t , V 1,N t , . . . , V N,N t ) t 0 be the solution of the following stochastic differential equation in (R 2d ) N :            dX i,N t = V i,N t dt dV i,N t = -A(V i,N t ) dt -B(X i,N t ) dt - 1 N N j=1 C(X i,N t -X j,N t ) dt + √ 2 dW i t , 1 i N (X i,N 0 , V i,N 0 ) = (X i 0 , V i 0 ). (4)
The mean field force

C * x ρ[f t ] in (2) is replaced by the pairwise actions 1 N C(X i,N t -X j,N t
) of particle j on particle i. Since this interaction is of order 1/N , it may be reasonable that two of these interacting particles (or a fixed number k of them) become less and less correlated as N gets large.

In order to state this propagation of chaos property we let, for each i

1, ( Xi t , V i t ) t 0 be the solution of the kinetic McKean-Vlasov type equation on R 2d      d Xi t = V i t dt d V i t = -A( V i t ) dt -B( Xi t ) dt -C * x ρ[ν t ]( Xi t ) dt + √ 2 dW i t , ( Xi 0 , V i 0 ) = (X i 0 , V i 0 ). ( 5
)
where ν t is the distribution of ( Xi t , V i t ). The processes ( Xi t , V i t ) t 0 with i 1 are independent since the initial conditions and driving Brownian motions are independent. Moreover they are identically distributed and their common law at time t evolves according to (1), so is the solution f t of (1) with initial datum f 0 . In this notation, and as N gets large, the N processes (X i,N t , V i,N t ) t 0 look more and more like the N independent processes ( Xi t , V i t ) t 0 :

Theorem 2 (Time-uniform propagation of chaos). Let (X i 0 , V i 0 ) for 1 i N be N independent R 2d - valued random variables with law f 0 in P 2 (R 2d ). Let also (X i,N t , V i,N t ) t 0,1 i N be the solution to (2) and ( Xi t , V i t ) t 0 the solution to (5) with initial datum (X i 0 , V i 0 ) for 1 i N.
Under Assumption (A), for all positive α, α ′ and β there exists a positive constant c such that, if 0 γ, δ < c, then there exists a positive constant C, independent of N , such that for i = 1, . . . , N

sup t 0 E X i,N t -Xi t 2 + V i,N t -V i t 2 C N •
Here the constant C depends only on the coefficients of the equation and the second moment of f 0 .

Remark 3. In particular the law f

(1,N ) t at time t of any (X i,N t , V i,N t
) (by symmetry) converges to f t as N goes to infinity, according to

d(f (1,N ) t , f t ) 2 E X i,N t -Xi t 2 + V i,N t -V i t 2 C N •
Propagation of chaos at the level of the trajectories, and not only of the time-marginals, is estimated in [START_REF]Quantitative concentration inequalities on sample path space for mean field interaction[END_REF] and [START_REF] Meléard | Asymptotic behaviour of some interacting particle systems: Mc Kean-Vlasov and Boltzmannn models[END_REF] for a broad class of equations, but with non time-uniform constants.

We finally turn to the approximation of the equilibrium solution of the Vlasov-Fokker-Planck equation (as given by Theorem 1) by the particle system at a given time T .

Since all solutions (f t ) t 0 to (1) with initial data in P 2 converge to the equilibrium solution µ ∞ , we let (x 0 , v 0 ) in R 2d be given and we consider the Dirac mass δ (x 0 ,v 0 ) at (x 0 , v 0 ) as the initial datum f 0 . We shall give precise bounds on the approximation of µ ∞ by the empirical measure of the particles (X i,N t , V i,N t

) for 1 i N , all of them initially at (x 0 , v 0 ). In the space homogeneous case of the granular media equation, this was performed by the third author [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF][START_REF]Convergence to equilibrium for granular media equations and their Euler schemes[END_REF] by proving a logarithmic Sobolev inequality for the joint law f (N ) t of the N particles at time t. In turn this inequality was proved by a Bakry-Emery curvature criterion (see [START_REF] Bakry | Diffusions hypercontractives, Sém. de probabilités[END_REF]). The argument does not work here as the particle system has -∞ curvature, and we shall only prove a (Talagrand) T 2 transportation inequality for the joint law of the particles. Remark 4. At this stage we have to point out that, for instance when the force fields A, B and C are gradient of potentials, the invariant measure of the particle system, that is, the large time limit of the joint law of the N particles, is explicit and satisfies a logarithmic Sobolev inequality with carré du champ |∇ x f | 2 +|∇ v f | 2 ; however it does not satisfy a logarithmic Sobolev inequality with carré du champ |∇ v f | 2 (initiated by our dynamics), which would at once lead to exponential entropic convergence to equilibrium for the particle system.

Let us recall that a probability measure µ on R 2d is said to satisfies a T 2 transportation inequality if there exists a constant D such that

d(µ, ν) 2 ≤ D H(ν|µ)
for all probability measure ν; here

H(ν|µ) = log dν dµ dν
if ν ≪ µ and +∞ otherwise is the relative entropy of ν with respect to µ.

Theorem 5. Under the assumptions of Theorem 1, for all positive α, α ′ and β there exists a positive constant c such that if 0 γ, δ < c, then the joint law of the N particles (X

(i,N ) T , V (i,N ) T
) at given time T , all with deterministic starting points (x 0 , v 0 ) ∈ R 2d , satisfies a T 2 inequality with a constant D independent of the number N of particles, of time T and of the point (x 0 , v 0 ). It follows that there exists a constant D ′ such that

P 1 N N i=1 h(X i,N T , V i,N T ) - R 2d h dµ ∞ (h) ≥ r + D ′ 1 √ N + e -CT ≤ exp - N r 2 2D
for all N, T, r 0 and all 1-Lipschitz observables h on R 2d .

Here the constant C has been obtained in Theorem 1 and the constant D ′ depends only on the point (x 0 , v 0 ) and the coefficients of the equation. Remark 6. Such single observable deviation inequalities were obtained in [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF] for the space homogeneous granular media equation; they were upgraded in [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non compact spaces[END_REF] to the very level of the measures, and to the level of the density of the equilibrium solution. The authors believe that such estimates can also be obtained in the present case. 1 Long time behaviour for the Vlasov-Fokker-Planck equation This section is devoted to the proof of Theorem 1, which is based on the stochastic interpretation (2) of (1) and a coupling argument. It uses the idea, also present in [START_REF] Talay | Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme[END_REF] and [START_REF] Villani | Hypocoercivity[END_REF], of perturbing the Euclidean metric on R 2d in such a way that (2) is dissipative for this metric.

Remark 7. Let us also point out that if we do not suppose a confinement/convexity assumption as in (A) but only Lipschitz regularity on the drift fields

If Q is a positive quadratic form on R 2d and µ and ν are two probability measures in P 2 we let

d Q (µ, ν) 2 = inf (X,V ), (Y,W ) E(Q((X, V ) -(Y, W )))
where again the infimum runs over all the couples (X, V ) and (Y, W ) of random variables on R 2d with respective laws µ and ν; so that

d Q = d if Q is the squared Euclidean norm on R 2d .
The key step in the proof is the following Proposition 8. Under the assumptions of Theorem 1, there exist a positive constant C and a positive quadratic form Q on R 2d such that

d Q (f t , ft ) e -Ct d Q (f 0 , f0 ), t 0
for all solutions (f t ) t 0 and ( ft ) t 0 to (1) with respective initial data f 0 and f0 in P 2 .

Proof of Proposition 8. Let (f t ) t 0 and ( ft ) t 0 be two solutions to [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF] with initial data f 0 and f0 in P 2 . Let also (X 0 , V 0 ) and ( X0 , V0 ) with respectively law f 0 and f0 , evolving into (X t , V t ) and ( Xt , Vt ) according to (2), both with the same Brownian motion (W t ) t 0 in R d . Then, by difference, (x t , v t ) = (X t -Xt , V t -Vt ) evolves according to

dx t = v t dt dv t = -A(V t ) -A( Vt ) + β x t + D(X t ) -D( Xt ) dt -C * x ρ[f t ](X t ) -C * x ρ[ ft ]( Xt ) dt.
Then, if a and b are positive constants to be chosen later on,

d dt (a|x t | 2 + 2 x t • v t + b|v t | 2 ) = 2 a x t • v t + 2|v t | 2 -2 x t • A(V t ) -A( Vt ) + βx t + D(X t ) -D( Xt ) -2 b v t • A(V t ) -A( Vt ) + βx t + D(X t ) -D( Xt ) -2 (x t + b v t ) • C * x ρ[f t ](X t ) -C * x ρ[ ft ]( Xt ) .
By the Cauchy-Schwarz inequality and assumptions on A and D, the first four terms are bounded by above by 2(abβ)

x t • v t + 2(α + bδ)|x t | |v t | -2 (bα ′ -1)|v t | 2 -2 (β -δ) |x t | 2 .
Let now π t be the law of (X t , V t ; Xt , Vt ) on R 2d × R 2d : then its marginals on R 2d are the respective distributions f t and ft of (X t , V t ) and ( Xt , Vt ), so that, since moreover C is odd:

-2 E x t • C * x ρ[f t ](X t ) -C * x ρ[ ft ]( Xt ) = -2 R 8d (Y -Ȳ ) • C(Y -y) -C( Ȳ -ȳ) dπ t (y, w; ȳ, w) dπ t (Y, W ; Ȳ , W ) = - R 8d (Y -y) -( Ȳ -ȳ) • C(Y -y) -C( Ȳ -ȳ) dπ t (y, w; ȳ, w) dπ t (Y, W ; Ȳ , W ) γ R 8d (Y -y) -( Ȳ -ȳ) 2 dπ t (y, w; ȳ, w) dπ t (Y, W ; Ȳ , W ) = 2 γ R 4d
|y -ȳ| 2 dπ t (y, w; ȳ, w) -R 4d

(yȳ)dπ t (y, w; ȳ, w)

2 2 γ E|x t | 2 .
In the same way, and by Young's inequality,

-2 Ev t • C * x ρ[f t ](X t ) -C * x ρ[ ft ]( Xt ) = -2 R 8d (W -W ) • C(Y -y) -C( Ȳ -ȳ) dπ t (y, w; ȳ, w) dπ t (Y, W ; Ȳ , W ) = - R 8d (W -W ) -(w -w) • C(Y -y) -C( Ȳ -ȳ) dπ t (y, w; ȳ, w) dπ t (Y, W ; Ȳ , W ) γ 2 R 8d (W -W ) -(w -w) 2 + (Y -Ȳ -(y -ȳ) 2 dπ t (y, w; ȳ, w) dπ t (Y, W ; Ȳ , W ) γ E[|x t | 2 + |v t | 2 ].
Collecting all terms leads to the bound

d dt E a|x t | 2 + 2 x t • v t + b|v t | 2 ) 2(a -bβ) E x t • v t + 2(α + bδ) E|x t | |v t | -2 β -δ -γ - γ b 2 E|x t | 2 -2 α ′ b -1 - γ b 2 E|v t | 2
for every positive ε, and then (with a = bβ) to

d dt E bβ |x t | 2 + 2 x t • v t + b|v t | 2 ) -2β -2η -ε -ηb E|x t | 2 -(2α ′ -η)b -2 - α 2 ε E|v t | 2
by Young's inequality, where η = γ + δ.

If 4 -4βb 2 < 0, that is, if b > 1/ √ β, then Q : (x, v) → bβ|x| 2 + 2x • v + b|v| 2
is a positive quadratic form on R 2d . Then we look for b and ε such that

2β -2η -ε -ηb > 0 and (2α ′ -η)b -2 - α 2 ε > 0, (6)
in such a way that

d dt E Q(x t , v t ) -C E |x t | 2 + |v t | 2
holds for a positive constant C. Necessarily η < 2α ′ , which is assumed in the sequel. Then, for instance for ε = β, the conditions (6) are equivalent to

2 + α 2 /β 2α ′ -η < b < β -2η η , η < 2α ′ .
We look for η such that

2 + α 2 /β 2α ′ -η < β -2η η , that is, 2η 2 -η(2 + α 2 β + β + 4α ′ ) + 2α ′ β0
. This polynomial takes negative values at η = 2α ′ , so it is positive on an interval [0, η 0 [ for some η 0 < 2α ′ .

We further notice that η 0 < β √ β 1 + 2 √ β , so that there exists b with all the above conditions for any 0 η < η 0 .

Hence there exists a constant η 0 , depending only on α, α ′ and β, such that, if γ + δ < η 0 , then there exist a positive quadratic form Q on R 2d and a constant depending only on α, α ′ , β, γ and δ such that

d dt E Q(x t , v t ) -C E |x t | 2 + |v t | 2
for all t 0. In turn, since Q(x, v) and |x| 2 + |v| 2 are equivalent on R 2d , this is bounded by

-C E Q(x t , v t ) for a new constant C, so that E Q (X t , V t ) -( Xt , Vt ) e -Ct E Q (X 0 , V 0 ) -( X0 , V0 )
for all t 0 by integration. We finally optimize over (X 0 , V 0 ) and ( X0 , V0 ) with respective laws f 0 and f0 and use the relation

d Q (f t , ft ) E Q((X t , V t ) -( Xt , Vt )) to deduce d Q (f t , ft ) e -Ct d Q (f 0 , f0 ).
This concludes the argument.

Remark 9. This coupling argument can also be performed for the (space homogeneous) granular media equation, for which it exactly recovers the contraction property in Wasserstein distance given in [START_REF]Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF]Theorem 5], whence the statements which follow on the trend to equilibrium.

We now turn to the

Proof of Theorem 1. First of all, the positive quadratic form Q(x, v) on R 2d given by Proposition 8 is equivalent to |x| 2 + |v| 2 , so there exist positive constants C ′′ and C ′ such that

d(f t , ft ) C ′′ d Q (f t , ft ) C ′′ e -Ct d Q (f 0 , f0 ) C ′ e -Ct d(f 0 , f0 ), t 0
for all solutions (f t ) t 0 and ( ft ) t 0 to (1) by the contraction property of Proposition 8: this proves the first assertion (3) of Theorem 1.

Now, if Q is the positive quadratic form on R 2d given by Proposition 8, then √ Q is a norm on R 2d so that the space (P 2 , d Q ) is a complete metric space (see [START_REF] Bolley | Separability and completeness for the Wasserstein distance, Sém. de probabilités XLI[END_REF] or [START_REF]Optimal transport, old and new[END_REF]Chapter 6] for instance).

Then Lemma 10 below (see [START_REF] Carrillo | Contractive probability metrics and asymptotic behavior of dissipative kinetic equations[END_REF]Lemma 7.3] for instance) and the contraction property of Proposition 8 ensure the existence of a unique stationary solution µ ∞ in P 2 to (1): Lemma 10. Let (S, dist) be a complete metric space and (T (t)) t 0 be a continuous semigroup on (S, dist) for which for all positive t there exists L(t) ∈]0, 1[ such that dist(T (t)(x), T (t)(y)) L(t) dist(x, y) for all positive t and x, y in S. Then there exists a unique stationary point x ∞ in S, that is, such that T (t)(x ∞ ) = x ∞ for all positive t. Moreover all solutions (f t ) t 0 with initial data f 0 in P 2 converge to this stationary solution µ ∞ ,

with d Q (f t , µ ∞ ) e -Ct d Q (f 0 , µ ∞ ), t 0.
Finally, with f0 = µ ∞ , (3) specifies into

d(f t , µ ∞ ) C ′ e -Ct d(f 0 , µ ∞ ), t 0
which concludes the proof of Theorem 1.

Particle approximation

The time-uniform propagation of chaos in Theorem 2 requires a time-uniform bound of the second moment of the solutions to (1).

Lemma 11. Under Assumption (A), for all positive α, α ′ and β there exists a positive constant c such that sup

t 0 R 2d (|x| 2 + |v| 2 f t (x, v) dx dv is finite for all solutions (f t ) t 0 to (1) with initial datum f 0 in P 2 and γ, δ in [0, c).
Proof of Lemma 11. Let (f t ) t 0 be a solution to [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF] with initial datum f 0 in P 2 , and let a and b be positive numbers to be chosen later on. Then

d dt R 2d a |x| 2 + 2 x • v + b |v| 2 df t (x, v) = 2bd + 2 R 2d v • (ax + v) -(x + bv) • A(v) + B(x) + C * x ρ[f t ](x) df t (x, v)
where, by Young's inequality and assumption on A, B and C,

-2 x • A(v) = -2 x • A(v) -A(0) -2 x • A(0) 2 α |x| |v| -2x • A(0), -2 x • B(x) = -2 x • β x + D(x) -D(0) + D(0) -(2β -2δ)|x| 2 -2 x • D(0), -2 b v • A(v) = -2 b v • (A(v) -A(0) + A(0) -2 b α ′ |v| 2 -2 b v • A(0), -2 b v • B(x) = -2 b v • β x + D(x) -D(0) + D(0) -2bβ v • x + 2bδ |v| |x| -2b v • D(0), -2 R 2d x • C * x ρ[f t ](x) df t (x, v) = - R 4d (x -y) • C(x -y) df t (x, v) df t (y, w) γ R 4d |x -y| 2 df t (x, v) df t (y, w) 2 γ R 2d |x| 2 df t (x, v)
and

-2 b R 2d v • C * x ρ[f t ](x) df t (x, v) = -b R 4d (v -w) • C(x -y) df t (x, v) df t (y, w) b γ R 4d |v -w| |x -y| df t (x, v) df t (y, w) bγ 2 R 4d |v -w| 2 + |x -y| 2 df t (x, v) df t (y, w) b γ R 2d (|x| 2 + |v| 2 )df t (x, v).
Collecting all terms and using Young's inequality we obtain, with a = βb and η = γ + δ,

d dt R 2d (β b |x| 2 + 2 x • v + b |v| 2 ) df t (x, v) 2bd + (2α + 2bδ) |x| |v| df t (x, v) + bγ + 2γ -2β + 2δ |x| 2 df t (x, v) + 2 + γb -2α ′ b |v| 2 df t (x, v) -2 A(0) + D(0) • x df t (x, v) + b v df t (x, v) 2bd + 2 ε + b 2 ε 2α 2 |A(0) + D(0)| 2 -2β -2η -ε -ηb |x| 2 df t (x, v) -(2α ′ -η)b -2 - 4α 2 ε |v| 2 df t (x, v)
for all positive ε. Now, as in the proof of Proposition 8, with α replaced by 2α, we get the existence of a positive constant η 0 , depending only on α, α ′ and β, such that for all 0 γ + δ < η 0 there exist b (and

ε) such that Q(x, v) = β b |x| 2 + 2 x • v + b |v| 2 be a positive quadratic form on R 2d and such that d dt R 2d Q(x, v) f t (x, v) dx dv C 1 -C 2 R 2d (|x| 2 +|v| 2 ) f t (x, v) dx dv C 1 -C 3 R 2d Q(x, v) f t (x, v) dx dv for positive constants C i . It follows that sup t 0 R 2d Q(x, v) f t (x, v) dx dv < +∞ if initially R 2d Q(x, v) f 0 (x, v) dx dv < +∞, that is, sup t 0 R 2d (|x| 2 + |v| 2 ) f t (x, v) dx dv < +∞
if initially f 0 belongs to P 2 . This concludes the argument.

We now turn to the Proof of Theorem 2. For each 1 i N the law f t of ( Xi t , V i t ) is the solution to (1) with f 0 as initial datum and the processes ( Xi t , V i t ) t 0 and (X i,N t , V i,N t ) t 0 are driven by the same Brownian motion.

In particular the differences

x i t = X i,N t -Xi t and v i t = V i,N t -V i t evolve according to      dx i t = v i t dt dv i t = -A(V i,N t ) -A( V i t ) + β x i t + D(X i,N t ) -D( Xi t ) dt - 1 N N j=1 C(X i,N t -X j,N t ) -C * x ρ[f t ]( Xi t ) dt
with (x i 0 , v i 0 ) = (0, 0). Then, if a and b are positive constants to be chosen later on,

d dt (a|x i t | 2 + 2 x i t • v i t + b|v i t | 2 ) = 2 a x i t • v i t + 2|v i t | 2 -2 x i t • A(V i,N t ) -A( V i t ) + βx i t + D(X i,N t ) -D( Xi t ) -2 b v i t • A(V i,N t ) -A( V i t ) + βx i t + D(X i,N t ) -D( Xi t ) - 2 N N j=1 (x i t + b v i t ) • C(X i,N t -X j,N t ) -C * x ρ[f t ]( Xi t ) .
By the Young inequality and assumptions on A and D, for all positive ε the third and fourth terms are bounded by above according to

-2 x i t • (A(V i,N t ) -A( V i t )) 2|x i t | |A(V i,N t ) -A( V i t )| 2 α |x i t | |v i t | ε 2 |x i t | 2 + 2α 2 ε |v i t | 2 , -2 x i t • (D(X i,N t ) -D( Xi t )) = -2b(X i t -Xi t ) • (D(X i,N t ) -D( Xi t )) 2 δ |x i t | 2 , -2 b v i t • (A(V i,N t ) -A( V i t )) = -2 b(V i,N t -V i t ) • (A(V i,N t ) -A( V i t )) -2 b α|v i t | 2 and -2 b v i t • (D(X i,N t ) -D( Xi t )) 2 b δ|v i t | |x i t | b δ(|x i t | 2 + |v i t | 2 ). Hence, with a = β b, d dt (β b|x i t | 2 + 2 x i t • v i t + b|v i t | 2 ) ε 2 -2β + 2δ + δb |x i t | 2 + 2 + 2α 2 ε -2α ′ b + δb |v i t | 2 - 2 N N j=1 (x i t + b v i t ) • C(X i,N t -X j,N t ) -C * x ρ[f t ]( Xi t ) .
Moreover, by symmetry, E|x i t | 2 , Ex i t • v i t , ... are independent of i = 1, . . . , N , so that, by averaging on i,

d dt E β b|x 1 t | 2 + 2 x 1 t • v 1 t + b|v 1 t | 2 -2β -2δ - ε 2 -δb E|x 1 t | 2 -(2α ′ -δ)b -2 - 2α 2 ε E|v 1 t | 2 - 2 N 2 N i,j=1 E (x i t + b v i t ) • C(X i,N t -X j,N t ) -C * x ρ[f t ]( Xi t ) . (7)
We decompose the last term in (7) according to

C(X i,N t -X j,N t ) -C * x ρ[f t ]( Xi t ) = C(X i,N t -X j,N t ) -C( Xi t -Xj t ) + C( Xi t -Xj t ) -C * x ρ[f t ]( Xi t )
which leads to estimating four terms:

1. By symmetry and assumption on C,

- N i,j=1 E x i t • C(X i,N t -X j,N t ) -C( Xi t -Xj t ) = - N i,j=1 E (X i,N t -Xi t ) • C(X i,N t -X j,N t ) -C( Xi t -Xj t ) = - 1 2 N i,j=1 E (X i,N t -X j,N t ) -( Xi t -Xj t ) • C(X i,N t -X j,N t ) -C( Xi t -Xj t ) γ 2 N i,j=1 E (X i,N t -X j,N t ) -( Xi t -Xj t ) 2 = γ 2 N i,j=1 E (X i,N t -Xi t ) -( Xj,N t -Xj t ) 2 = γ N i,j=1 E|x i t | 2 -γE N i=1 (X i,N t -Xi t ) 2 γN 2 |x 1 t | 2 .

2.

By assumption on C and the Young inequality,

- N i,j=1 E v i t • C(X i,N t -X j,N t ) -C( Xi t -Xj t ) = - 1 2 N i,j=1 E (v i t -v j t ) • C(X i,N t -X j,N t ) -C( Xi t -Xj t ) γ 2 N i,j=1 E |v i t -v j t | (X i,N t -X j,N t ) -( Xi t -Xj t ) γ 2 N i,j=1 E 1 2 |v i t -v j t | 2 + 1 2 (X i,N t -Xi t ) -(X j,N t -Xj t ) 2 γ 2 N 2 E(|v 1 t | 2 + |x 1 t | 2 ).

3.

For each i = 1, . . . , N , and again by the Young inequality

-2 E x i t • N j=1 C( Xi t -Xj t )-C * x ρ[f t ]( Xi t ) LN E |x i t | 2 + 1 LN E N j=1 C( Xi t -Xj t )-C * x ρ[f t ]( Xi t ) 2
for any positive constant L, where the last expectation is

N j=1 E C( Xi t -Xj t )-C * x ρ[f t ]( Xi t ) 2 + j =k E C( Xi t -Xj t )-C * x ρ[f t ]( Xi t ) • C( Xi t -Xk t )-C * x ρ[f t ]( Xi t ) .
First of all, C is odd, so C(0) = 0 and hence |C(z)| γ |z|. Then, for each j = 1, . . . , N, Then, for all j = k,

E C( Xi t -Xj t ) -C * x ρ[f t ]( Xi t ) 2 2 E C( Xi t -Xj t ) 2 + 2 E C * x ρ[f t ]( Xi t ) 2 2 γ 2 E Xi t -Xj t 2 + R 4d |y -x| 2 f t (x, v) f t (
E C( Xi t -Xj t ) -C * x ρ[f t ]( Xi t ) • C( Xi t -Xk t ) -C * x ρ[f t ]( Xi t ) = E Xi t E Xj t C( Xi t -Xj t ) -C * x ρ[f t ]( Xi t ) • E Xk t C( Xi t -Xk t ) -C * x ρ[f t ]( Xi t ) = E Xi t 0 =
E x i t • C( Xi t -Xj t ) -C * x ρ[f t ]( Xi t ) L N 2 E|x 1 t | 2 + M L N.
4. In the same way for any positive L ′ we obtain the bound

-2 N i,j=1 E v i t • C( Xi t -Xj t ) -C * x ρ[f t ]( Xi t ) L ′ N 2 E|v 1 t | 2 + M L ′ N.
Collecting all terms and letting for instance L = ε 2 and L ′ = 2α 2 bε , it follows from ( 7) that there exists a positive constant c such that for all γ, δ in [0, c) there exists a constant M such that

d dt E β b|x 1 t | 2 + 2 x 1 t • v 1 t + b|v 1 t | 2 -(2β -ε -2η -ηb)E|x 1 t | 2 -(2α ′ -η)b -2 - 4α 2 ε E|v 1 t | 2 + M N 2 ε + εb 2 2α 2 •
for all positive t, b and ε, where η = γ + δ. Now, as in the proof of Proposition 8, with α replaced by 2α, we get the existence of a positive constant η 0 , depending only on α, α ′ and β, such that for all 0 γ + δ < η 0 there exist b (and ε) such that

Q(x, v) = β b |x| 2 + 2 x • v + b |v| 2 be a positive quadratic form on R 2d and such that d dt E Q(x 1 t , v 1 t ) -C 1 E |x 1 t | 2 + |v 1 t | 2 + C 2 N
for all t 0 and for positive constants C 1 and C 2 , also depending on f 0 through its second moment, but not on N. In turn this is bounded by

-C 3 E Q(x 1 t , v 1 t ) + C 2 N , so that sup t 0 E Q(x 1 t , v 1 t ) C 4 N
and finally

sup t 0 E |X 1,N t -X1 t | 2 + |V 1,N t -V 1 t | 2 C N
where the constant C depends on the parameters of the equation and on the second moment of f 0 , but not on N. This concludes the proof of Theorem 2.

by proving the existence of a positive constant D, independent of N and T , such that We now turn to the deviation inequality in Theorem 5. First of all, if h is a 1-Lipschitz function on R 2d , then Hence for all r 0. This concludes the proof of Theorem 5.

E Q a|x t | 2 + 2x t • v t + b|v t | 2 ≤ -D t 0 E Q |x s | 2 + |v s | 2 ds + t 0 E Q ∇Q(x s , v s ) • σ (N ) (X (N ) s , V (N )
1 N N i=1 h(X i,N T , V i,N T ) - R 2dN 1 N N i=1 h(x i , v i ) df (N ) T (x 1 , . . . , v N ) = 1 N N i=1 h(X i,N T , V i,N T ) -hdµ ∞ + h dµ ∞ -h df T + h df T -h df (1,N ) T ≥ 1 N N i=1 h(X i,N T , V i,N T ) -h dµ ∞ -d(f T , µ ∞ ) -d(f T , f (1,
P 1 N N i=1 h(X i,N T , V i,N T ) - R 2d h dµ ∞ ≥ r + D ′ 1 √ N + e -CT ≤ P 1 N N i=1 h(X i,N T , V i,N T ) - R 2dN 1 N N i=1 h(x i , v i ) df

  A, B and C, then Theorems 1, 2 and 5 still hold but with constants growing exponentially fast with time T .Sections 1, 2 and 3 are respectively devoted to the proofs of Theorems 1, 2 and 5.

s )β s ds in the notation x t = X. 0 Eε t 0 E

 00 Then we bound the last term byε t Q |x s | 2 + |v s | 2 ds + 1 Q |β s | 2 dsand the transportation inequality for the trajectory law P follows again by Gronwall's lemma, with a new constant D independent of T.The transportation inequality for the law f (N ) T at time T finally follows by projection at time T.

( 1

 1 N ) T) by exchangeability. But, by Theorem 1 with f 0 = δ (x 0 ,v 0 ) , there exist two constants C and C ′ , depending only on the coefficients of the equation, such thatd(f T , µ ∞ ) C ′ e -CT d(f 0 , µ ∞ )where f T is the solution at time T of Equation (1) with initial datum f 0 = δ (x 0 ,v 0 ) . Moreover, by Remark 3, there exists a constant C ′′ , depending only on the equation and on (x 0 , v 0 ), such that d(f T , f

T

  (x 1 , . . . , v N ) r where D ′ = max(C ′ d(f 0 , µ ∞ ), C ′′ ) depends on (x 0 , v 0 ). Now the law f (N ) Tsatisfies a T 2 inequality on R 2dN with constant D, hence a Gaussian deviation inequality for Lipschitz functions (see[START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF]); moreover the map (x 1 , . . . , v N ) on R 2dN , so the probability on the right-hand side is bounded by exp -N r 2 2D

  , provided γ and δ are small enough for the conclusion of Lemma 11 to hold. The constant M depends on the initial moment R 2d |x| 2 + |v| 2 f 0 (x, v) dx dv and the coefficients of the equation, but not on t or N.

			y, w) dx dv dy dw
	8 γ 2	R 2d	|x| 2 f

t (x, v) dx dv M for a constant M

Remark 12. One can prove a contraction property for the particle system, similar to Proposition 8 for the Vlasov-Fokker-Planck equation: if f 0 is an initial datum in P 2 we let f (1,N ) t be the common law at time t of any of the N particles (X i,N t , V i,N t

). Then there exists a positive constant c such that, if 0 γ, δ < c, then there exist a positive constant C and a positive quadratic form

for all t and all initial data f 0 and f0 in P 2 . Here the form Q and the constants c and C depend only on the coefficients of the equations, and not on N. From this and Remark 3, and following [START_REF] Cattiaux | Probabilistic approach for granular media equations in the non uniformly convex case[END_REF], one can recover the contraction property of Proposition 8, whence Theorem 1.

Transportation inequality and deviation result

This final section is devoted to the proof of Theorem 5. It is based on the idea, borrowed to [START_REF] Djellout | Transportation cost-information inequalities and applications to random dynamical systems and diffusions[END_REF], of proving a T 2 transportation inequality not only for the law f (N ) T at time T , but for the whole trajectory up to time T ; this transportation inequality will be proved by means of stochastic calculus, a coupling argument, a clever formulation of the relative entropy of two trajectory laws and a change of metric as in the previous sections; it will imply the announced transportation inequality by projection at time T.

We only sketch the proof, emphasizing the main steps and refering to the previous sections and to [START_REF] Djellout | Transportation cost-information inequalities and applications to random dynamical systems and diffusions[END_REF] for further details.

We equip the space C of R 2dN -valued continuous functions on [0, T ] with the L 2 norm and consider the space P(C) of Borel probability measures on C, equipped with the Wasserstein distance defined by the cost

We write Equation (4) on the particle system (X

)dt for some coefficients σ (N ) and b (N ) .

Let P ∈ P(C) be the law of the trajectory (X (N ) , V (N ) ) = (X

) 0 t T of the particles, all of them starting from the deterministic point (x 0 , v 0 ) ∈ R 2d .

The transportation inequality for P, which it is sufficient to prove for laws Q absolutely continuous with respect to P, will obtained in two steps.

Step 1. Following [15, Proof of Theorem 5.6], for every trajectory law Q ∈ P(C), there exists

β s ds is a Brownian motion under Q. We now build a coupling between Q and P by letting ( X(N) ,

whose law under Q is exactly P.

Step 2. In order to prove the T 2 inequality, and as in the previous sections, we change the metric induced on C by the L 2 norm and consider an equivalent positive quadratic form Q(x, v) = a|x| 2 + 2 x • v + b|v| 2 . We control the quantity


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