Viscous Aubry-Mather theory and the Vlasov equation

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.