A game interpretation for the weighted $p$-Laplace equation

TL;DR

This paper introduces a stochastic game approach to approximate solutions of the weighted p-Laplace equation, extending previous methods and analyzing the behavior as p approaches infinity.

Contribution

It develops a stochastic approximation scheme for the weighted p-Laplace equation and explores the solution's limiting behavior as p tends to infinity.

Findings

Converges to viscosity solutions of the weighted p-Laplace equation.

Extends previous results from the non-weighted case.

Analyzes the solution behavior as p approaches infinity.

Abstract

In this paper, we obtain a stochastic approximation that converges to the viscosity solution of the weighted $p$-Laplace equation. We consider a stochastic two-player zero-sum game controlled by a random walk, two player's choices, and the gradient of the weight function. The proof is based on the boundary conditions in the viscosity sense and the comparison principle. These results extend previous findings for the non-weighted $p$-Laplace equation [Manfredi, Parviainen, Rossi, 2012]. In addition, we study the limiting behavior of the viscosity solution of the weighted $p$-Laplace equation as $p\rightarrow\infty$.