We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the mean-field limit the system is close to the solution of a kinetic PDE. Our aim is to include models widely studied in the literature such as the Cucker-Smale model, adding noise to the behavior of individuals. The difficulty, as compared to the classical case of globally Lipschitz potentials, is that in several models the interaction potential between particles is only locally Lipschitz, the local Lipschitz constant growing to infinity with the size of the region considered. With this in mind, we present an extension of the classical theory for globally Lipschitz interactions, which works for only locally Lipschitz ones.

We establish quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances. As an application, we provide some error bounds for particle simulations in a model mean field problem. The tools include coupling arguments, as well as regularity and moment estimates for solutions of certain diffusive partial differential equations.

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.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 [4]) have been studied by P. Cattiaux and the last two authors in particular [14], [19,20], 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 [11,12] (see also [26]), 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 [24], L. Wu [27] or D. Bakry, P. Cattiaux and the second author [2] 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 [16], [17] and [25, 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 [23] 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 [25, 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 , ...

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We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behavior of individuals with a fixed speed. We rigorously derive the kinetic mean-field partial differential equation satisfied when the number N of particles tends to infinity, quantifying the convergence of the law of one particle to the solution of the PDE. For this we adapt a classical coupling argument to the present case in which both the particle system and the PDE are defined on a surface rather than on the whole space. As part of the study we give existence and uniqueness results for both the particle system and the PDE

We describe conditions on non-gradient drift diffusion Fokker-Planck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, in any dimension, which to our knowledge is a novelty.

We study the long time asymptotics of a nonlinear, nonlocal equation used in the modelling of granular media. We prove a uniform exponential convergence to equilibrium for degenerately convex and non convex interaction or confinement potentials, improving in particular results by J. A. Carrillo, R. J. McCann and C. Villani. The method is based on studying the dissipation of the Wasserstein distance between a solution and the steady state.

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