Viscous Aubry-Mather theory and the Vlasov equation

TL;DR

This paper investigates the viscous Vlasov equation on a torus, analyzing the existence of periodic solutions and the asymptotic behavior of the Hopf-Lax semigroup under stochastic perturbations.

Contribution

It extends Aubry-Mather theory to the viscous Vlasov equation with smooth interactions, providing new insights into periodic solutions and asymptotic analysis.

Findings

Existence of periodic solutions for the viscous Vlasov equation.

Asymptotic behavior of the Hopf-Lax semigroup in this context.

Application of methods from Gangbo, Nguyen, Tudorascu, and Gomes.

Abstract

The Vlasov equation models a group of particles moving under a potential $V$; moreover, each particle exerts a force, of potential $W$, on the other ones. We shall suppose that these particles move on the $p$-dimensional torus ${\bf T}^p$ and that the interaction potential $W$ is smooth. We are going to perturb this equation by a Brownian motion on ${\bf T}^p$; adapting to the viscous case methods of Gangbo, Nguyen, Tudorascu and Gomes, we study the existence of periodic solutions and the asymptotics of the Hopf-Lax semigroup.