Tug-of-war games associated with boundary value problems involving derivatives

TL;DR

This paper explores a noisy tug-of-war game linked to boundary value problems for the normalized p-Laplace equation with 1<p<2, focusing on boundary regularity and convergence to viscosity solutions.

Contribution

It introduces a game-theoretic interpretation of the boundary value problem for the normalized p-Laplace equation and analyzes the regularity and convergence properties of the associated value function.

Findings

Proves boundary regularity of the value function.

Shows convergence to viscosity solutions.

Connects game theory with PDE boundary problems.

Abstract

In this paper, we study a certain type of noisy tug-of-war game which can be regarded as an interpretation of a certain type of boundary value problem for the normalized $p$-Laplace equation, where $1<p<2$. More precisely, we will investigate the boundary regularity of the value function to the game and its convergence to a viscosity solution of the model problem.