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

Fix a bounded domain Omega subset of R-d, a continuous function F : partial derivative Omega --> R, and constants epsilon > 0 and 1 < p, q < infinity with p(-1) + q(-1) = 1. For each x is an element of 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 upsilon is an element of (B) over bar (0, epsilon) to add to the game position, after which a random noise vector with mean zero and variance (q/p)vertical bar upsilon vertical bar(2) in each orthogonal direction is also added. The game ends when the game position reaches some y is an element of partial derivative Omega, and player I's payoff is F(y). We show that (for sufficiently regular Omega) as e tends to zero, the functions u(epsilon) converge uniformly to the unique p-harmonic extension of F. Using a modified game (in which c gets smaller as the game position approaches a partial derivative 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 = infinity) and the motion-bycurvature 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.