In this article, we adapt the work of Jabin and Wang in [JW18] to show the first result of uniform in time propagation of chaos for a class of singular interaction kernels. In particular, our models contain the Biot-Savart kernel on the torus and thus the 2D vortex model.
We prove the existence of a contraction rate for Vlasov-Fokker-Planck equation in Wasserstein distance, provided the interaction potential is Lipschitz continuous and the confining potential is both (locally) Lipschitz continuous and greater than a quadratic function, thus requiring no convexity conditions. Our strategy relies on coupling methods suggested by A. Eberle [22] adapted to the kinetic setting enabling also to obtain uniform in time propagation of chaos in a non convex setting.
In this article, we are interested in the behavior of a fully connected
network of $N$ neurons, where $N$ tends to infinity. We assume that the neurons
follow the stochastic FitzHugh-Nagumo model, whose specificity is the
non-linearity with a cubic term. We prove a result of uniform in time
propagation of chaos of this model in a mean-field framework. We also exhibit
explicit bounds. We use a coupling method initially suggested by A. Eberle
(arXiv:1305.1233), and recently extended in (1805.11387), known as the
reflection coupling. We simultaneously construct a solution of the $N$-particle
system and $N$ independent copies of the non-linear McKean-Vlasov limit in such
a way that, considering an appropriate semi-metric that takes into account the
various possible behaviors of the processes, the two solutions tend to get
closer together as $N$ increases, uniformly in time. The reflection coupling
allows us to deal with the non-convexity of the underlying potential in the
dynamics of the quantities defining our network, and show independence at the
limit for the system in mean field interaction with sufficiently small
Lipschitz continuous interactions.
In this article, we prove the first quantitative uniform in time propagation of chaos for a class of systems of particles in singular repulsive interaction in dimension one that contains the Dyson Brownian motion. We start by establishing existence and uniqueness for the Riesz gases, before proving propagation of chaos with an original approach to the problem, namely coupling with a Cauchy sequence type argument. We also give a general argument to turn a result of weak propagation of chaos into a strong and uniform in time result using the long time behavior and some bounds on moments, in particular enabling us to get a uniform in time version of the result of Cépa-Lépingle [CL97].Résumé (Sur les systèmes de particules en interaction singulière répulsive en dimension 1 : loggaz et gaz de Riesz) Dans cet article, nous prouvons le premier résultat de propagation du chaos quantitative uniforme en temps pour une classe de systèmes de particules en interaction singulière répulsive en dimension 1 qui contient le mouvement brownien de Dyson. Nous commençons par établir l'existence et l'unicité des gaz de Riesz, avant de prouver la propagation du chaos par une approche originale du problème, à savoir un couplage avec un argument de type suite de Cauchy. Nous donnons également un argument général pour transformer un résultat faible de propagation du chaos en un résultat fort et uniforme en temps en utilisant le comportement en temps long et certaines bornes sur les moments, ce qui nous permet en particulier d'obtenir une version uniforme en temps du résultat de Cépa-Lépingle [CL97].
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