Weak KAM theory for action minimizing random walks

TL;DR

This paper develops a weak KAM theory framework for action minimizing controlled random walks on a grid, connecting discrete stochastic processes with Hamilton-Jacobi equations and their continuous limits.

Contribution

It introduces a novel weak KAM theory for controlled random walks, bridging discrete stochastic models with classical PDE approaches.

Findings

Established a discrete weak KAM theory for controlled random walks.

Connected the discrete theory with classical weak KAM theory via hyperbolic scaling.

Provided insights into the structure of action minimizing paths in stochastic grid models.

Abstract

We introduce a class of controlled random walks on a grid in $\mathbb{T}^d$ and investigate global properties of action minimizing random walks for a certain action functional together with Hamilton-Jacobi equations on the grid. This yields an analogue of weak KAM theory, which recovers a part of original weak KAM theory through the hyperbolic scaling limit.