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

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 interpretation 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. Introduction and main results We are interested in the long time behaviour and in a particle approximation of a distribution ft(x, v) in the space of positions x ∈ Rd and velocities v ∈ Rd (with d > 1) evolving according to the VlasovFokker-Planck equation ∂ft ∂t + v · ∇xft − C ∗x ρ[ft](x) · ∇vft = ∆vft +∇v · ((A(v) +B(x))ft), t > 0, x, v ∈ R (1) where ρ[ft](x) = ∫ Rd ft(x, v) dv is the macroscopic density in the space of positions x ∈ Rd (or the space marginal of ft). Here a · b denotes the scalar product of two vectors a and b in Rd and ∗x stands for the convolution with respect to x ∈ Rd : C ∗x ρ[ft](x) = ∫ Rd C(x− y) ρ[ft](y) dy = ∫ R2d C(x− y) ft(y, v) dy dv. Moreover ∇x stands for the gradient with respect to the position variable x ∈ Rd whereas ∇v, ∇v· and ∆v respectively stand for the gradient, divergence and Laplace operators with respect to the velocity variable v ∈ Rd. 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 Rd. This equation is used in the modelling of the distribution ft(x, v) of diffusive, confined and interacting stellar or charged matter when C respectively derives from the Newton and Coulomb potential (see [10] for instance). It has the following natural probabilistic interpretation: if f0 is a density function, the solution ft of (1) is the density of the law at time t of the R2d-valued process (Xt, Vt)t>0 evolving according to the mean field stochastic differential equation (diffusive Newton’s equations) { dXt = Vt dt dVt = −A(Vt) dt−B(Xt) dt − C ∗x ρ[ft](Xt) dt + √ 2 dWt. (2)