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

DOI:10.1051/m2an/2010045 www.esaim-m2an.org

TREND TO EQUILIBRIUM AND PARTICLE APPROXIMATION

FOR A WEAKLY SELFCONSISTENT VLASOV-FOKKER-PLANCK EQUATION

Franc ¸ois Bolley

, Arnaud Guillin

and Florent Malrieu

Abstract. We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpre- tation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates on the approximation of the solution by the particle system. Finally, a transportation inequality for the distribution of the particle system leads to quantitative deviation bounds on the approximation of the equilibrium solution of the equation by an empirical mean of the particles at given time.

Mathematics Subject Classification. 65C35, 35K55, 65C05, 82C22, 26D10, 60E15.

Received June 7, 2009. Revised January 18, 2010.

Published online August 26, 2010.

1. Introduction and main results

We are interested in the long time behaviour and in a particle approximation of a distributionft(x, v) in the space of positions x∈R^d and velocities v ∈R^d (with d1) evolving according to the Vlasov-Fokker-Planck equation

∂ft

∂t +v· ∇xft−C∗xρ[ft](x)· ∇vft= Δ_vft+∇v·((A(v) +B(x))ft), t >0, x, v∈R^d (1.1) where

ρ[ft](x) =

R^dft(x, v) dv

Keywords and phrases. Vlasov-Fokker-Planck equation, particular approximation, concentration inequalities, transportation inequalities.

1 Ceremade, UMR CNRS 7534, Universit´e Paris-Dauphine, Place du Mar´echal De Lattre De Tassigny, 75775 Paris Cedex 16, France. bolley@ceremade.dauphine.fr

2UMR CNRS 6620, Laboratoire de Math´ematiques, Universit´e Blaise Pascal, avenue des Landais, 63177 Aubi`ere Cedex, France.

guillin@math.univ-bpclermont.fr

3 UMR CNRS 6625, Institut de Recherche Math´ematique de Rennes (IRMAR), Universit´e de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France. florent.malrieu@univ-rennes1.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2010

is the macroscopic density in the space of positionsx∈R^d (or the space marginal offt). Herea·bdenotes the scalar product of two vectorsaandbin R^d and∗xstands for the convolution with respect tox∈R^d:

C∗xρ[ft](x) =

R^dC(x−y)ρ[ft](y) dy=

R^2dC(x−y)ft(y, v) dydv.

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

The three maps A, B andC areR^d to R^d maps. TheA(v) term models the friction, theB(x) term models an exterior confinement and the C(x−y) term in the convolution models the symmetric interaction between positionsxandy in the underlying physical system. For that reason we assume thatC is an odd map onR^d. This equation is used in the modelling of the distributionft(x, v) of diffusive, confined and interacting stellar or charged matter when C respectively derives from the Newton and Coulomb potential (see [9] for instance).

We shall consider measure solutions, which have the following natural probabilistic interpretation: if f0 is a density function, the solutionftof (1.1) is the density of the law at timetof theR^2d-valued process (Xt, Vt)_t0 evolving according to the mean field stochastic differential equation (diffusive Newton’s equations)

dXt = Vtdt

dVt = −A(Vt) dt−B(Xt) dt−C∗xρ[ft](Xt) dt+√

2 dWt. (1.2)

Here (Wt)_t0 is a Brownian motion in the velocity spaceR^d andftis the law of (Xt, Vt) inR^2d, so thatρ[ft] is the law ofXt inR^d.

Space homogeneous models of diffusive and interacting granular media (see [4]) have been studied by Cattiaux et al. in particular [13], [24,25] (see also [30] for non-gradient equations), by means of a stochastic interpretation analogous to (1.2) and a particle approximation analogous to (1.4) below. These models were interpreted as gra- dient flows in the space of probability measures by Carrilloet al.[11,12] (see also [32]), both approaches leading to explicit exponential (or algebraic for non uniformly convex potentials) rates of convergence to equilibrium.

Time-uniform propagation of chaos was also proven for the associated particle system.

Obtaining rates of convergence to equilibrium for (1.1) is much more complex, as the equation simultane- ously presents Hamiltonian and gradient flows aspects. Much attention has recently been called to the linear noninteracting case of (1.1), whenC= 0, also known as the kinetic Fokker-Planck equation. On the one hand a probabilistic approach based on Lyapunov functionals, and thus easy to check conditions, has led Talay [29], Wu [33] or Bakryet al. [3] to exponential or subexponential convergence to equilibrium in total variation dis- tance. On the other hand the case when A(v) = v and B(x) = ∇Ψ(x), and when the equilibrium solution is explicitly given by f∞(x, v) = e^{−Ψ(x)−|v|2/2} was studied in [18,21,22] and [31], Chapter 7: hypocoerciv- ity analytic techniques are developed which, applied to this situation, give sufficient conditions, in terms of Poincar´e or logarithmic Sobolev inequalities for the measure e^−Ψ, to ensure L² or entropic convergence with an explicit exponential rate. We also refer to [20] for the evolution of two species, modelled by two coupled Vlasov-Fokker-Planck equations.

Villani’s approach extends to the selfconsistent situation whenCderives from a nonzero potentialU (see [31], Chap. 17): replacing the confinement forceB(x) by a periodic boundary condition, and for small and smooth potentialU, 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 wholeR^d, with linear-like frictionA(v) and confinementB(x) forces, and small Lipschitz interactionC(x): in the whole paper we make the following:

Assumption. We say that Assumption (A) is fulfilled 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 allx, y, v, winR^d.

One may also consider the case of non-constant diffusion coefficients: then our method would require smallness assumptions on their Lipschitz seminorms with respect toα, α andβ.

Convergence of solutions will be measured in terms of Wasserstein distances: let P2 be the space of Borel probability measuresμonR^2dwith finite second moment, that is, such that the integral

R^2d

(|x|²+|v|²) dμ(x, v) be finite. The spaceP2 is equipped with the (Monge-Kantorovich) Wasserstein distancedof 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 onR^2d with respective laws μandν.Convergence in this metric is equivalent to narrow convergence plus convergence of the second moment (see [32], Chap. 6, for instance).

The coefficients A, B and C being Lipschitz, existence and uniqueness for equation (1.2) with square- integrable initial data are ensured by [26]. It follows that, for all initial dataf0in P2, equation (1.1) admits a unique measure solution inP2, that is, continuous on [0,+∞[ with values inP2.

Assumption (A) is made in the whole paper. Under an additional assumption on the smallness ofγ and δ, we shall prove a quantitative exponential convergence of all solutions to a unique equilibrium:

Theorem 1.1. Under Assumption(A), for all positiveα, α andβ there exists a positive constant c such that, if 0γ, δ < c, then there exist positive constants C andC such that

d(ft,f¯t)Ce^−Ctd(f0,f¯0), t0 (1.3) for all measure solutions (ft)_t0 and( ¯ft)_t0 to (1.1)with respective initial data f0 andf¯0 inP2.

Moreover (1.1)admits a unique stationary solutionμ∞ and all solutions(ft)_t0 converge towards it, with d(ft, μ∞)Ce^−Ctd(f0, μ∞), t0.

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 whenγ=δ= 0,the convergence rate is given byC= 1/3, and for instance forγ andδwithγ+δ= 0.1 we obtainC∼0.27.

Compared to Villani’s results, convergence is here proven in the (weak) Wasserstein distance, not inL¹norm, or relative entropy – the latter being a stronger convergence since, in this specific situation, the equilibrium measure e^−|v|2/2satisfies a logarithmic Sobolev inequality, hence a transportation inequality. We refer to [1,23]

or [32], 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 cases. 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).

Theorem1.1provides the convergence to equilibrium for a large class of initial data, but also a stability result of all solutions. Let us finally note that it is based on the natural stochastic interpretation (1.2) and a simple

coupling argument, and does not need any regularity property of the solutions that could be deduced from the hypoellipticity of the model.

The particle approximation of solutions to (1.1) consists in the introduction of a large number N of R^2d-valued processes (X_t^i,N, V_t^i,N)_t0with 1iN, evolving according to the force fieldC∗xρ[ˆμ^N_t ] generated by the empirical measure ˆμ^N_t = 1

N N i=1

δ_(X^i,N

t ,V_t^i,N) of the system: if (W_·ⁱ)_i1 with i1 are independent stan- dard Brownian motions onR^dand (X₀ⁱ, V₀ⁱ) withi1 are independent random vectors onR^2dwith lawf0inP2

and independent of (W_·ⁱ)_i1, we let (Xt^(N), Vt^(N))_t0 = (Xt^1,N, . . . , Xt^N,N, Vt^1,N, . . . , Vt^N,N)_t0 be the solution of the following stochastic differential equation in (R^2d)^N:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

dX_t^i,N =V_t^i,Ndt

dV_t^i,N =−A(V_t^i,N) dt−B(X_t^i,N) dt− 1 N

N j=1

C(X_t^i,N−X_t^j,N) dt+√

2 dW_tⁱ, 1iN (X₀^i,N, V₀^i,N) = (X₀ⁱ, V₀ⁱ).

(1.4)

The mean field forceC∗xρ[ft] in (1.2) is replaced by the sum of the pairwise actions 1

NC(X_t^i,N −X_t^j,N) of particlejon particlei.Since this interaction is of order 1/N, it may be reasonable that two of these interacting particles (or a fixed numberkof them) become less and less correlated asN gets large.

In order to state thispropagation of chaosproperty we let, for eachi1, ( ¯X_tⁱ,V¯_tⁱ)_t0be the solution of the kinetic McKean-Vlasov type equation onR^2d

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

d ¯X_tⁱ= ¯V_tⁱdt

d ¯V_tⁱ=−A( ¯V_tⁱ) dt−B( ¯X_tⁱ) dt−C∗xρ[νt]( ¯X_tⁱ) dt+√ 2 dW_tⁱ, ( ¯X₀ⁱ,V¯₀ⁱ) = (X₀ⁱ, V₀ⁱ)

(1.5)

where νtis the distribution of ( ¯Xtⁱ,V¯tⁱ).The processes ( ¯Xtⁱ,V¯tⁱ)_t0 withi1 are independent since the initial conditions and driving Brownian motions are independent. Moreover they are identically distributed and their common law evolves according to (1.1), so at time t is the solution ft of (1.1) with initial datum f0. In this notation, and as N gets large, the N processes (X_t^i,N, V_t^i,N)_t0 look more and more like the N independent processes ( ¯X_tⁱ,V¯_tⁱ)_t0:

Theorem 1.2 (time-uniform propagation of chaos). Let (X₀ⁱ, V₀ⁱ)for 1iN be N independent R^2d-valued random variables with lawf0inP2(R^2d).Let also(X_t^i,N, V_t^i,N)_t0,1iN be the solution to(1.2)and( ¯X_tⁱ,V¯_tⁱ)_t0 the solution to (1.5)with initial datum(X₀ⁱ, V₀ⁱ)for1iN.Under Assumption(A), for all positiveα, α and β there exists a positive constantcsuch that, if0γ, δ < c, then there exists a positive constantC, independent of N, such that fori= 1, . . . , N

supt0 EXt^i,N−X¯tⁱ²+Vt^i,N−V¯tⁱ²

C

N·

Here the constantCdepends only on the coefficients of the equation and the second moment off0.

Remark 1.3. In particular the lawf_t^(1,N) at timetof any (X_t^i,N, V_t^i,N) (by symmetry) converges toft asN goes to infinity, according to

d(ft^(1,N), ft)²EXt^i,N −X¯tⁱ²+Vt^i,N−V¯tⁱ²

C

N·

Theorem 1.2also ensures propagation of chaos for the particle system as it is defined in the famous review by Sznitman [28]: let k 1 be fixed, and let ϕ : (R^2d)^k →R be l-Lipschitz with respect to each argument.

Then, for allk-uple (i1, . . . , ik) of [1, N], and lettingZ_t^i,N = (X_t^i,N, V_t^i,N) and ¯Z_tⁱ= ( ¯X_tⁱ,V¯_tⁱ), the quantity Eϕ(Z_t^i1,N, . . . , Z_t^ik,N)−

R^2kdϕdf_t^⊗k=Eϕ(Z_t^i1,N, . . . , Z_t^ik,N)−Eϕ( ¯Z_tⁱ¹, . . . ,Z¯_t^ik) is bounded by

Eϕ(Z_t^i1,N, Z_t^i2,N, . . . , Z_t^ik,N)−ϕ( ¯Z_tⁱ¹, Z_t^i2,N, . . . , Z_t^ik,N)+. . .+Eϕ( ¯Z_tⁱ¹, . . . ,Z¯_t^ik−1, Z_t^ik,N)

−ϕ( ¯Z_tⁱ¹, . . . ,Z¯_t^ik−1,Z¯_t^ik), whence by

l k j=1

E|Z_t^ij,N−Z¯_t^ij|Cl√k N whereC is independent ofN: hence it tends to 0 as N tends to infinity.

Propagation of chaos at the level of the trajectories, and not only of the time-marginals, is studied in [7,26]

for a broad class of equations, but with time dependent constants.

We finally turn to the approximation of the equilibrium solution of the Vlasov-Fokker-Planck equation (as given by Thm. 1.1) by the particle system at a given timeT.

Since all solutions (ft)_t0 to (1.1) with initial data in P2 converge to the equilibrium solutionμ∞, we let (x0, v0) in R^2d be given and we consider the Dirac mass δ(x0,v0) at (x0, v0) as the initial datumf0. We shall give precise bounds on the approximation of μ∞ by the empirical measure of the particles (X_t^i,N, V_t^i,N) for 1iN, all of them initially at (x0, v0).

In the space homogeneous case of the granular media equation, this was performed by the third author [24,25]

by proving a logarithmic Sobolev inequality for the joint law ft^(N) of theN particles at time t. In turn this inequality was proved by a Bakry-Emery curvature criterion (see [2]). The argument does not work here as the particle system has−∞curvature, and we shall only prove a (Talagrand)T2 transportation inequality for the joint law of the particles.

Remark 1.4. At this stage we have to point out that, for instance when the force fieldsA, BandCare 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 carr´e du champ|∇xf|²+|∇vf|²; however it does not satisfy a logarithmic Sobolev inequality with carr´e du champ |∇vf|², corresponding to the diffusion term Δ_vf, which would at once lead to exponential entropic convergence to equilibrium for the particle system (see [2]).

Let us recall that a probability measure μ onR^2d is said to satisfy aT2 transportation inequality if there exists a constantD such that

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

H(ν|μ) =

log dν

dμ

dν ifνμ, and +∞otherwise, is the relative entropy ofν with respect toμ.

Theorem 1.5. Under Assumption(A), for all positiveα, α andβ there exists a positive constantcsuch that if 0γ, δ < c, then the joint law of theN particles(X_T^(i,N), V_T^(i,N))at given timeT, all with deterministic starting

points(x0, v0)∈R^2d, satisfies aT2 inequality with a constant D independent of the number N of particles, of timeT and of the point(x0, v0).

It follows that there exists a constantD such that

P

1 N

N i=1

h(X_T^i,N, V_T^i,N)−

R^2dhdμ∞(h)≥r+D 1

√N + e^−CT

≤exp

−N r² 2D

for allN, T, r0and all 1-Lipschitz observables hon R^2d.

Here the constantChas been obtained in Theorem1.1and the constantDdepends only on the point (x0, v0) and the coefficients of the equation.

Remark 1.6. Such single observable deviation inequalities were obtained in [24] for the space homogeneous granular media equation; they were upgraded in [8] 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.

Remark 1.7. 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 fieldsA, B andC, then Theorems 1.1,1.2and1.5still hold but with constants growing exponentially fast with timeT.

As a conclusion to this introduction, let us point out that our model belongs to the huge class of mean field interacting particle models. Another important family of such processes consists in the Nambu and Moran type interacting jump processes and their associated Feynmac-Kac semigroups, which play a role in filtering problems or in the study of Schr¨odinger operators for example (see [14]). For this kind of models, asymptotic stability, uniform propagation of chaos and sharp concentration bounds have been obtained in [14–17] or [27]

for instance.

Sections2–4 are respectively devoted to the proofs of Theorems1.1,1.2and1.5.

2. Long time behaviour for the Vlasov-Fokker-Planck equation

This section is devoted to the proof of Theorem 1.1, which is based on the stochastic interpretation (1.2) of (1.1) and a coupling argument. The main idea, also present in [29,31], is to perturb the Euclidean metric onR^2d in such a way that (1.2) is dissipative for this metric.

IfQis a positive definite quadratic form onR^2d andμ andν are two probability measures inP₂ we let dQ(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 onR^2d with respective laws μ and ν; so that dQ = d if Q is the squared Euclidean norm on R^2d. The key step in the proof is the following:

Proposition 2.1. Under the assumptions of Theorem 1.1, there exist a positive constant C and a positive definite quadratic formQ onR^2d such that

dQ(ft,f¯t)e^−CtdQ(f0,f¯0), t0

for all solutions (ft)_t0 and( ¯ft)_t0 to (1.1)with respective initial data f0 andf¯0 inP₂.

Proof of Proposition 2.1. Let (ft)_t0and ( ¯ft)_t0be two solutions to (1.1) with initial dataf0and ¯f0inP2.Let also (X0, V0) and ( ¯X0,V¯0) with respectively lawf0and ¯f0, evolving into (Xt, Vt) and ( ¯Xt,V¯t) according to (1.2),

both with the same Brownian motion (Wt)_t0 inR^d.Then, by difference, (xt, vt) = (Xt−X¯t, Vt−V¯t) evolves according to

dxt=vtdt dvt=−

A(Vt)−A( ¯Vt) +βxt+D(Xt)−D( ¯Xt) dt−

C∗xρ[ft](Xt)−C∗xρ[ ¯ft]( ¯Xt) dt.

Then, ifaandbare positive constants to be chosen later on, d

dt(a|xt|²+ 2xt·vt+b|vt|²) = 2a xt·vt+ 2|vt|²−2xt·

A(Vt)−A( ¯Vt) +βxt+D(Xt)−D( ¯Xt)

−2b vt·

A(Vt)−A( ¯Vt) +βxt+D(Xt)−D( ¯Xt)

−2 (xt+b vt)·

C∗xρ[ft](Xt)−C∗xρ[ ¯ft]( ¯Xt) .

By the Cauchy-Schwarz inequality and assumptions on A and D, the first three terms are bounded from above by

2(a−bβ)xt·vt+ 2(α+bδ)|xt| |vt| −2 (bα−1)|vt|²−2 (β−δ)|xt|².

Let nowπtbe the law of (Xt, Vt; ¯Xt,V¯t) onR^2d×R^2d: then its marginals onR^2dare the respective distributions ftand ¯ftof (Xt, Vt) and ( ¯Xt,V¯t), so that, since moreoverC is odd:

−2Ext·

C∗xρ[ft](Xt)−C∗xρ[ ¯ft]( ¯Xt)

= −2

R^8d(Y −Y¯)·

C(Y −y)−C( ¯Y −y¯)

×dπt(y, w; ¯y,w¯) dπt(Y, W; ¯Y ,W¯)

= −

R^8d

(Y −y)−( ¯Y −y¯)

·

C(Y −y)−C( ¯Y −y¯)

dπt(y, w; ¯y,w¯) dπt(Y, W; ¯Y ,W¯)

γ

R^8d

(Y −y)−( ¯Y −y¯)²dπt(y, w; ¯y,w¯) dπt(Y, W; ¯Y ,W¯)

= 2γ

R^4d|y−y|¯²dπt(y, w; ¯y,w¯)−

R^4d(y−y¯)dπt(y, w; ¯y,w¯)² 2γE|x_t|².

In the same way, and by Young’s inequality,

−2Evt·

C∗xρ[ft](Xt)−C∗xρ[ ¯ft]( ¯Xt)

= −2

R^8d(W −W¯)·

C(Y −y)−C( ¯Y −y¯)

×dπt(y, w; ¯y,w¯) dπt(Y, W; ¯Y ,W¯)

= −

R^8d

(W −W¯)−(w−w¯)

·

C(Y −y)−C( ¯Y −y¯)

dπt(y, w; ¯y,w¯) dπt(Y, W; ¯Y ,W¯)

γ

2

R^8d

(W−W¯)−(w−w¯)²+(Y −Y¯)−(y−y¯)²dπt(y, w; ¯y,w¯) dπt(Y, W; ¯Y ,W¯) γE[|xt|²+|vt|²].

Collecting all terms leads to the bound d

dtE

a|xt|²+ 2xt·vt+b|vt|²) 2(a−bβ)Ext·vt+ 2(α+bδ)E|xt| |vt|

−2

β−δ−γ−γ b 2

E|xt|²−2

αb−1−γ b 2

E|vt|²

and then (witha=bβ) to d

dtE

bβ|xt|²+ 2xt·vt+b|vt|²)−

2β−2η−ε−ηb

E|xt|²−

(2α−η)b−2−α² ε

E|vt|²

for every positiveε: here we have used by Young’s inequality and letη=γ+δ. If 4−4βb²<0, that is, ifb >1/√

β, then Q: (x, v)→bβ|x|²+ 2x·v+b|v|² is a positive definite quadratic form onR^2d.Then we look forb andεsuch that

2β−2η−ε−ηb >0 and (2α−η)b−2−α²

ε >0, (2.1)

in such a way that

d

dtEQ(xt, vt)−CE

|xt|²+|vt|² holds for a positive constantC.

Necessarilyη <2α, which is assumed in the sequel. Then, for instance forε=β, the conditions (2.1) are equivalent to

2 +α²/β

2α−η < b < β−2η

η , η <2α. We look forη such that 2 +α²/β

2α−η <β−2η

η ,that is, 2η²−η(2 +α²

β +β+ 4α) + 2αβ >0.This polynomial takes a positive value at η= 0 and a negative value atη= 2α, so it is positive on an interval [0, η0[ for some η0 <2α. We further notice that η0 < β√

β 1 + 2√

β, so that there exists b with all the above conditions for any 0η < η0.

Hence there exists a constantη0, depending only onα, α andβ, such that, ifγ+δ < η0, then there exist a positive definite quadratic formQonR^2d and a constantC, depending only onα, α, β, γ andδ such that

d

dtEQ(xt, vt)−CE

|xt|²+|vt|²

for allt0. In turn, sinceQ(x, v) and|x|²+|v|² are equivalent onR^2d, this is bounded by −CEQ(xt, vt) for a new constantC, so that

EQ

(Xt, Vt)−( ¯Xt,V¯t)

e^−CtEQ

(X0, V0)−( ¯X0,V¯0)

for allt0 by integration. We finally optimize over (X0, V0) and ( ¯X0,V¯0) with respective lawsf0 and ¯f0 and use the relationdQ(ft,f¯t)EQ((Xt, Vt)−( ¯Xt,V¯t)) to deduce

dQ(ft,f¯t)e^−CtdQ(f0,f¯0).

This concludes the argument.

Remark 2.2. This coupling argument can also be performed for the (space homogeneous) granular media equa- tion; in this case it exactly recovers the contraction property in Wasserstein distance given in [12], Theorem 5, and then the statements which follow on the trend to equilibrium.

We now turn to the:

Proof of Theorem 1.1. First of all, the positive definite quadratic formQ(x, v) onR^2dgiven by Proposition2.1 is equivalent to|x|²+|v|², so there exist positive constantsC andC such that

d(ft,f¯t)CdQ(ft,f¯t)Ce^−CtdQ(f0,f¯0)Ce^−Ctd(f0,f¯0), t0

for all solutions (ft)_t0 and ( ¯ft)_t0 to (1.1) by the contraction property of Proposition2.1: this proves the first assertion (1.3) of Theorem1.1.

Now, ifQis the positive definite quadratic form onR^2d given by Proposition2.1, then√

Qis a norm onR^2d so that the space (P2, dQ) is a complete metric space (see [6] or [32], Chap. 6, for instance).

Then Lemma 2.3 below (see [10], Lem. 7.3, for instance) and the contraction property of Proposition 2.1 ensure the existence of a unique stationary solutionμ∞in P2 to (1.1):

Lemma 2.3. Let (S,dist)be a complete metric space and(T(t))_t0 be a continuous semigroup on(S,dist) for which for all positivet there existsL(t)∈]0,1[such that

dist(T(t)(x), T(t)(y))L(t) dist(x, y)

for all positivetandx, yinS.Then there exists a unique stationary pointx∞inS, that is, such thatT(t)(x∞) = x∞ for all positivet.

Finally, with ¯f0=μ∞, (1.3) specifies into

d(ft, μ∞)Ce^−Ctd(f0, μ∞), t0

which concludes the proof of Theorem1.1.

3. Particle approximation

The time-uniform propagation of chaos in Theorem1.2requires a time-uniform bound of the second moment of the solutions to (1.1).

Lemma 3.1. Under Assumption (A), for all positive α, α andβ there exists a positive constant c such that supt0

R^2d(|x|²+|v|²) dft(x, v) is finite for all solutions (ft)_t0 to (1.1) with initial datum f0 in P2 and γ, δ in [0, c).

Lemma 3.1, as suggested by a referee, can be seen as consequence of Theorem 1.1 since, for all solutions (ft)_t0 to (1.1) with initial datumf0in P₂,

R^2d(|x|²+|v|²) dft(x, v) =d(ft, δ0)²2d(ft, μ∞)²+ 2d(μ∞, δ0)²2Ce^−Ctd(f0, μ∞)²+ 2d(μ∞, δ0)² is bounded in t.

However we prefer to give an elementary proof of it, to stress on the fact that our particle approximation estimate is independent of the long time behaviour of the solutions.

Proof of Lemma 3.1. Let (ft)_t0be a solution to (1.1) with initial datumf0 inP₂, and letaandbbe positive numbers to be chosen later on. Then

d dt

R^2d

a|x|²+ 2x·v+b|v|²

dft(x, v) = 2bd+ 2

R^2d

v·(ax+v)−(x+bv)·

A(v) +B(x)

+C∗xρ[ft](x)

dft(x, v)

where, by Young’s inequality and assumption onA, Band C,

−2x·A(v) =−2x·

A(v)−A(0)

−2x·A(0)2α|x| |v| −2x·A(0),

−2x·B(x) =−2x·

β x+D(x)−D(0) +D(0)

−(2β−2δ)|x|²−2x·D(0),

−2b v·A(v) =−2b v·(A(v)−A(0) +A(0)

−2b α|v|²−2b v·A(0),

−2b v·B(x) =−2b v·

β x+D(x)−D(0) +D(0)

−2bβ v·x+ 2bδ|v| |x| −2b v·D(0),

−2

R^2dx·C∗xρ[ft](x) dft(x, v) =−

R^4d

(x−y)·C(x−y) dft(x, v) dft(y, w)

γ

R^4d|x−y|²dft(x, v) dft(y, w) 2γ

R^2d|x|²dft(x, v) and

−2b

R^2dv·C∗xρ[ft](x) dft(x, v) =−b

R^4d(v−w)·C(x−y) dft(x, v) dft(y, w) b γ

R^4d|v−w| |x−y|dft(x, v) dft(y, w)

bγ

2

R^4d

|v−w|²+|x−y|²

dft(x, v) dft(y, w) b γ

R^2d(|x|²+|v|²)dft(x, v).

Collecting all terms and using Young’s inequality we obtain, witha=βbandη=γ+δ, d

dt

R^2d(β b|x|²+ 2x·v+b|v|²) dft(x, v)2bd+ (2α+ 2bδ)

|x| |v|dft(x, v) +

bγ+ 2γ−2β+ 2δ

×

|x|²dft(x, v) +

2 +γb−2αb |v|²dft(x, v)−2

A(0) +D(0)

· xdft(x, v) +b

vdft(x, v)

2bd+

2 ε + b²ε

2α²

|A(0) +D(0)|²−

2β−2η−ε−ηb |x|²dft(x, v)−

(2α−η)b−2−4α² ε

×

|v|²dft(x, v) for all positiveε.

Now, as in the proof of Proposition2.1, withαreplaced by 2α, we get the existence of a positive constantη0, depending only on α, α and β, such that for all 0 γ+δ < η0 there exist b (and ε) such that Q(x, v) = β b|x|²+ 2x·v+b|v|² be a positive definite quadratic form onR^2dand such that

d dt

R^2dQ(x, v) dft(x, v)C1−C2

R^2d(|x|²+|v|²) dft(x, v)C1−C3

R^2dQ(x, v) dft(x, v)

for positive constants Ci.It follows that supt0

R^2dQ(x, v) dft(x, v)<+∞

if initially

R^2dQ(x, v) df0(x, v)<+∞, that is, supt0

R^2d(|x|²+|v|²) dft(x, v)<+∞

if initiallyf0belongs toP2.This concludes the argument.

We now turn to the:

Proof of Theorem 1.2. For each 1iN the lawftof ( ¯X_tⁱ,V¯_tⁱ) is the solution to (1.1) withf0as initial datum and the processes ( ¯X_tⁱ,V¯_tⁱ)_t0 and (X_t^i,N, V_t^i,N)_t0 are driven by the same Brownian motion. In particular the differencesxⁱ_t=X_t^i,N−X¯_tⁱ andv_tⁱ=V_t^i,N−V¯_tⁱ evolve according to

⎧⎪

⎪⎨

⎪⎪

⎩

dxⁱ_t=v_tⁱdt dv_tⁱ=−

A(V_t^i,N)−A( ¯V_tⁱ) +β xⁱ_t+D(X_t^i,N)−D( ¯X_tⁱ) dt− 1

N N j=1

C(X_t^i,N−X_t^j,N)−C∗xρ[ft]( ¯X_tⁱ) dt

with (xⁱ₀, v₀ⁱ) = (0,0).

Then, ifaandbare positive constants to be chosen later on, d

dt(a|xⁱ_t|²+ 2xⁱ_t·v_tⁱ+b|vⁱ_t|²) = 2a xⁱ_t·vⁱ_t+ 2|v_tⁱ|²−2xⁱ_t·

A(V_t^i,N)−A( ¯V_tⁱ) +βxⁱ_t+D(X_t^i,N)−D( ¯X_tⁱ)

−2b v_tⁱ·

A(V_t^i,N)−A( ¯V_tⁱ) +βxⁱ_t+D(X_t^i,N)−D( ¯X_tⁱ)

−2 N

N j=1

(xⁱ_t+b v_tⁱ)·

C(X_t^i,N −X_t^j,N)−C∗xρ[ft]( ¯X_tⁱ) .

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

−2xⁱt·(A(Vt^i,N)−A( ¯Vtⁱ))2|xⁱ_t| |A(Vt^i,N)−A( ¯Vtⁱ)|2α|xⁱ_t| |v_tⁱ| ε

2|xⁱ_t|²+2α² ε |vⁱ_t|²,

−2xⁱt·(D(Xt^i,N)−D( ¯Xtⁱ)) =−2(Xt^i,N −X¯tⁱ)·(D(Xt^i,N)−D( ¯Xtⁱ))2δ|xⁱ_t|²,

−2b vtⁱ·(A(Vt^i,N)−A( ¯Vtⁱ)) =−2b(Vt^i,N−V¯tⁱ)·(A(Vt^i,N)−A( ¯Vtⁱ))−2b α|vⁱ_t|² and

−2b v_tⁱ·(D(X_t^i,N)−D( ¯X_tⁱ))2b δ|vⁱ_t| |xⁱ_t|b δ(|xⁱ_t|²+|v_tⁱ|²).