A stochastic differential game for the inhomogeneous $\infty$-Laplace equation

TL;DR

This paper establishes a stochastic differential game framework to represent the unique viscosity solution of the inhomogeneous infinity Laplace equation with given boundary conditions in a smooth domain.

Contribution

It introduces a novel stochastic game representation for solutions of the inhomogeneous infinity Laplace equation, linking PDE theory with stochastic game models.

Findings

Representation of the solution as a stochastic game value

Extension of game-theoretic methods to inhomogeneous PDEs

Connection between viscosity solutions and stochastic differential games

Abstract

Given a bounded $\mathcaligr{C}^2$ domain $G\subset{\mathbb{R}}^m$, functions $g\in\mathcaligr{C}(\partial G,{\mathbb{R}})$ and $h\in\mathcaligr {C}(\bar{G},{\mathbb{R}}\setminus\{0\})$, let $u$ denote the unique viscosity solution to the equation $-2\Delta_{\infty}u=h$ in $G$ with boundary data $g$. We provide a representation for $u$ as the value of a two-player zero-sum stochastic differential game.