Tug-of-war games with varying probabilities and the normalized $p(x)$-Laplacian

TL;DR

This paper analyzes a tug-of-war game with variable probabilities, demonstrating the regularity, existence, and uniqueness of the game's value, and connecting it to solutions of the normalized p(x)-Laplacian.

Contribution

It establishes the connection between the game and the normalized p(x)-Laplacian, proving convergence and regularity results for the value function.

Findings

Value function is locally Hölder continuous.

Existence and uniqueness of the game's value are proven.

Value function converges to the solution of the normalized p(x)-Laplacian.

Abstract

We study a tug-of-war game with varying probabilities. In particular, we show that the value of the game is locally asymptotically H\"{o}lder continuous. We also show the existence and uniqueness of values of the game. As an application, we prove that the value function of the game converges to a solution of the normalized $p(x)$-Laplacian.