Weak KAM Theory topics in the stationary ergodic setting

TL;DR

This paper develops a stochastic version of Weak KAM Theory for stationary ergodic Hamiltonians, introducing random Aubry sets and representation formulas to analyze correctors in a non-compact, random setting.

Contribution

It introduces a new framework for stochastic Weak KAM Theory, defining random Aubry sets and deriving representation formulas in the ergodic setting.

Findings

Defined a notion of random Aubry set.

Established conditions for the existence of correctors.

Developed a Lax-type formula in the stochastic environment.

Abstract

We perform a qualitative analysis of the critical equation associated with a stationary ergodic Hamiltonian through a stochastic version of the metric method, where the notion of closed random stationary set, issued from stochastic geometry, plays a major role. Our purpose is to give an appropriate notion of random Aubry set, to single out characterizing conditions for the existence of exact or approximate correctors, and write down representation formulae for them. For the last task, we make use of a Lax--type formula, adapted to the stochastic environment. This material can be regarded as a first step of a long--term project to develop a random analog of Weak KAM Theory, generalizing what done in the periodic case or, more generally, when the underlying space is a compact manifold.