A Stochastic Analog of Aubry-Mather Theory

Details A Stochastic Analog of Aubry-Mather Theory A Stochastic Analog of Aubry-Mather Theory

Publishing date

2001-04-24

URL

arxiv.org/abs/math/0104230v1

Science

Mathematics

Field

Analysis of PDEs

Type

Scientific paper

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.

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