On near optimal trajectories for a game associated with the \infty-Laplacian

TL;DR

This paper explores the connection between a two-player stochastic differential game and solutions to the -Laplacian equation, identifying the limiting behavior of the game as a diffusion process under optimal play.

Contribution

It provides a novel stochastic game representation for -Laplacian solutions and characterizes the limiting diffusion process explicitly in terms of the solution's derivatives.

Findings

The game representation links to solutions of the -Laplacian equation.

The limiting process is a diffusion with explicitly given coefficients.

Under smoothness assumptions, the limit law is identified precisely.

Abstract

A two-player stochastic differential game representation has recently been obtained for solutions of the equation -\Delta_\infty u=h in a \calC^2 domain with Dirichlet boundary condition, where h is continuous and takes values in \RR\setminus\{0\}. Under appropriate assumptions, including smoothness of u, the vanishing \delta limit law of the state process, when both players play \delta-optimally, is identified as a diffusion process with coefficients given explicitly in terms of derivatives of the function u.