# American Institute of Mathematical Sciences

February  2014, 34(2): 379-420. doi: 10.3934/dcds.2014.34.379

## Viscous Aubry-Mather theory and the Vlasov equation

 1 Dip. di Matematica, Università di Roma Tre, L.go S. Leonardo Murialdo 1, 00146, Roma, Italy

Received  November 2012 Revised  May 2013 Published  August 2013

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 $T^p$ and that the interaction potential $W$ is smooth. We are going to perturb this equation by a Brownian motion on $T^p$; 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.
Citation: Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
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