A Course on Tug-of-War Games with Random Noise

TL;DR

This paper explores the connection between linear potential theory and random walks, extending concepts to nonlinear cases through tug-of-war games with random noise, and introduces a discrete ball walk process to analyze harmonic functions.

Contribution

It introduces a nonlinear extension of potential theory via tug-of-war games with noise, linking stochastic processes to harmonic functions in a novel way.

Findings

The ball walk process converges to harmonic functions with boundary data.

Harmonic functions can be characterized through stochastic game-based processes.

Extension of linear potential theory concepts to nonlinear settings.

Abstract

This is a preprint of Chapter 2 in the following work: Marta Lewicka, A Course on Tug-of-War Games with Random Noise, 2020, Springer, reproduced with permission of Springer Nature Switzerland AG. We present the basic relation between the linear potential theory and random walks. This fundamental connection, developed by Ito, Doob, Levy and others, relies on the observation that harmonic functions and martingales share a common cancellation property, expressed via mean value properties. It turns out that, with appropriate modifications, a similar observation and approach can be applied also in the nonlinear case, which is of main interest in our Course Notes. Thus, the present Chapter serves as a stepping stone towards gaining familiarity with more complex nonlinear constructions. After recalling the equivalent defining properties of harmonic functions, we introduce the ball walk. This is an infinite horizon discrete process, in which at each step the particle, initially placed at some point $x_0$ in the open, bounded domain $\mathcal{D}\subset\mathbb{R}^N$, is randomly advanced to a new position, uniformly distributed within the following open ball: centered at the current placement, and with radius equal to the minimum of the parameter $\epsilon$ and the distance from the boundary $\partial\mathcal{D}$. With probability one, such process accumulates on $\partial\mathcal{D}$ and $u^\epsilon(x_0)$ is then defined as the expected value of the given boundary data $F$ at the process limiting position. Each function $u^\epsilon$ is harmonic, and if $\partial\mathcal{D}$ is regular, then each $u^\epsilon$ coincides with the unique harmonic extension of $F$ in $\mathcal{D}$. One sufficient condition for regularity is the exterior cone condition.