A time-step approximation scheme for a viscous version of the Vlasov equation

TL;DR

This paper extends a time-step approximation scheme to a viscous Vlasov equation, connecting Aubry-Mather theory with probability measure spaces, and provides a new, more elementary proof of related theorems.

Contribution

It adapts a scheme for viscous Aubry-Mather theory to the Vlasov equation, bridging two areas and offering a simpler proof of existing results.

Findings

The scheme applies to viscous Vlasov equations.

A new proof of Feng and Nguyen's theorem is provided.

The approach links Aubry-Mather theory with probability measures.

Abstract

Gomes and Valdinoci have introduced a time-step approximation scheme for a viscous version of Aubry-Mather theory; this scheme is a variant of that of Jordan, Kinderlehrer and Otto. Gangbo and Tudorascu have shown that the Vlasov equation can be seen as an extension of Aubry-Mather theory, in which the configuration space is the space of probability measures, i. e. the different distributions of infinitely many particles on a manifold. Putting the two things together, we show that Gomes and Valdinoci's theorem carries over to a viscous version of the Vlasov equation. In this way, we shall recover a theorem of J. Feng and T. Nguyen, but by a different and more "elementary" proof.