Tug-of-war with noise: A game-theoretic view of the p-Laplacian

Abstract

Fix a bounded domain $\Omega \subset {\mathbb{R}}^d$, a continuous function $F:\partial \Omega \to \mathbb{R}$, and constants $\epsilon >0$ and $1<p,q<\infty$ with $p^{-1}+q^{-1}=1$. For each $x\in \Omega$, let $u^{\epsilon }(x)$ be the value for player I of the following two-player, zero-sum game. The initial game position is $x$. At each stage, a fair coin is tossed, and the player who wins the toss chooses a vector $v\in \stackrel{̲}{B}(0,\epsilon )$ to add to the game position, after which a random noise vector with mean zero and variance $(q/p)|v{|}^2$ in each orthogonal direction is also added. The game ends when the game position reaches some $y\in \partial \Omega$, and player I's payoff is $F(y)$.

We show that (for sufficiently regular $\Omega$) as $\epsilon$ tends to zero, the functions $u^{\epsilon }$ converge uniformly to the unique $p$-harmonic extension of $F$. Using a modified game (in which $\epsilon$ gets smaller as the game position approaches $\partial \Omega$), we prove similar statements for general bounded domains $\Omega$ and resolutive functions $F$.

These games and their variants interpolate between the tug-of-war games studied by Peres, Schramm, Sheffield, and Wilson [15], [16] ($p=\infty$) and the motion-by-curvature games introduced by Spencer [17] and studied by Kohn and Serfaty [9] ($p=1$). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about $p$-capacity and $p$-harmonic measure