Percolation games

TL;DR

This paper introduces a stochastic game model on integer lattices that serves as a simplified framework for studying the stochastic homogenization of Hamilton-Jacobi equations, providing convergence conditions and theoretical insights.

Contribution

It presents a new class of discrete-time stochastic games on $ abla^d$, linking game theory with homogenization of PDEs and establishing convergence criteria.

Findings

Conditions for the convergence of the n-stage game value.

Connections established between the game model and homogenization theory.

Discussion of implications for stochastic homogenization of Hamilton-Jacobi equations.

Abstract

This paper introduces a discrete-time stochastic game class on $\mathbb{Z}^d$, which plays the role of a toy model for the well-known problem of stochastic homogenization of Hamilton-Jacobi equations. Conditions are provided under which the $n$-stage game value converges as $n$ tends to infinity, and connections with homogenization theory is discussed.