Random Tug of War games for the ${\mathbf p}$-Laplacian: ${\mathbf{1<p<{\boldsymbol \infty}}}$

TL;DR

This paper introduces a novel finite difference method based on Tug of War games with noise to approximate solutions of the p-Laplacian, demonstrating convergence and regularity properties.

Contribution

It develops a new game-theoretic approximation for the p-Laplacian involving stochastic and deterministic averages, establishing convergence to the viscosity solution.

Findings

Solutions are continuous for continuous boundary data.

The game value coincides with the unique solution of the Dirichlet problem.

Domains with exterior corkscrew condition ensure uniform convergence.

Abstract

We propose a new finite difference approximation to the Dirichlet problem for the homogeneous $\mathbf{p}$-Laplace equation posed on an $N$-dimensional domain, in connection with the Tug of War games with noise. Our game and the related mean-value expansion that we develop, superposes the ``deterministic averages'' ``$\frac{1}{2}(\inf +\sup)$'' taken over balls, with the ``stochastic averages'' ``$\fint$'', taken over $N$-dimensional ellipsoids whose aspect ratio depends on $N,\mathbf{p}$ and whose orientations span all directions while determining $\inf / \sup$. We show that the unique solutions $u_\epsilon$ of the related dynamic programming principle are automatically continuous for continuous boundary data, and coincide with the well-defined game values. Our game has thus the min-max property: the order of supremizing the outcomes over strategies of one player and infimizing over strategies of their opponent, is immaterial. We further show that domains satisfying the exterior corkscrew condition are game regular in this context, i.e. the family $\{u_\epsilon\}_{\epsilon\to 0}$ converges uniformly to the unique viscosity solution of the Dirichlet problem.