Notes on tug-of-war games and the p-Laplace equation

TL;DR

This paper explores the relationship between tug-of-war games with noise and the p-Laplace equation, highlighting their connection, recent insights, and regularity theory in both elliptic and parabolic cases.

Contribution

It provides an accessible introduction to the stochastic game connection with the p-Laplace equation and discusses recent developments and regularity results.

Findings

Connection between tug-of-war with noise and p-Laplace equation established

Introduction of parabolic case alongside elliptic case

Discussion of regularity theory in the context of these equations

Abstract

The objective is the interplay between stochastic processes and partial differential equations. To be more precise, we focus on the connection between the nonlinear p-Laplace equation, and the stochastic game called tug-of-war with noise. The connection in this context was discovered roughly 15 years ago, and has provided novel insight and approaches ever since. These lecture notes provide a short introduction to the topic and to more research oriented literature. We also introduce the parabolic case side by side with the elliptic one, and cover some parts of the regularity theory.