A Stochastic Analog of Aubry-Mather Theory

TL;DR

This paper extends Aubry-Mather theory to stochastic control problems involving controlled diffusions, establishing existence, regularity, and asymptotic properties of Mather measures and solutions.

Contribution

It introduces a stochastic analog of Aubry-Mather theory, proving existence of Mather measures and analyzing their properties using viscosity solutions.

Findings

Existence of a Mather measure in the stochastic setting

Regularity estimates for viscosity solutions of Hamilton-Jacobi equations

Asymptotic behavior of controlled diffusion trajectories as diffusion vanishes

Abstract

In this paper we discuss a stochastic analog of Aubry-Mather theory in which a deterministic control problem is replaced by a controlled diffusion. We prove the existence of a minimizing measure (Mather measure) and discuss its main properties using viscosity solutions of Hamilton-Jacobi equations. Then we prove regularity estimates on viscosity solutions of Hamilton-Jacobi equation using the Mather measure. Finally we apply these results to prove asymptotic estimates on the trajectories of controlled diffusions and study the convergence of Mather measures as the rate of diffusion vanishes.