Tug-of-war games related to oblique derivative boundary value problems with the normalized $p$-Laplacian

TL;DR

This paper establishes a game-theoretic framework for solving oblique derivative boundary value problems involving the normalized p-Laplacian, extending classical Robin problems with new stochastic tug-of-war games.

Contribution

It introduces novel tug-of-war game models for the normalized p-Laplacian with oblique boundary conditions, analyzing their properties and convergence.

Findings

Existence and uniqueness of game value functions

Regularity and convergence results for the game solutions

Extension of Robin boundary problem to normalized p-Laplacian context

Abstract

In this paper, we are concerned with game-theoretic interpretations to the following oblique derivative boundary value problem \begin{align*} \left\{ \begin{array}{ll} \Delta_{p}^{N}u=0 & \textrm{in $ \Omega$,}\\ \langle \beta , Du \rangle + \gamma u = \gamma G & \textrm{on $ \partial \Omega$,}\\ \end{array} \right. \end{align*} where $\Delta_{p}^{N}$ is the normalized $p$-Laplacian. This problem can be regarded as a generalized version of the Robin boundary value problem for the Laplace equations. We construct several types of stochastic games associated with this problem by using `shrinking tug-of-war'. For the value functions of such games, we investigate the properties such as existence, uniqueness, regularity and convergence.