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

Abstract

Fix a bounded domain Omega in R^d, a continuous function F on the boundary of Omega, and constants epsilon>0, p>1, and q>1 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 of length at most 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 on the boundary of 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 the boundary), 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 (p=infinity) and the motion-by-curvature games introduced by Spencer and studied by Kohn and Serfaty (p=1). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about p-capacity and p-harmonic measure.