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

Fix a bounded domain Ω ⊂ Rd, a continuous function F : ∂Ω → R, and constants ǫ > 0 and 1 < p, q < ∞ with p−1 + q−1 = 1. For each x ∈ Ω, let uǫ(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 ∈ B(0, ǫ) 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 ∈ ∂Ω, and player I’s payoff is F (y). We show that (for sufficiently regular Ω) as ǫ tends to zero the functions uǫ converge uniformly to the unique p-harmonic extension of F . Using a modified game (in which ǫ gets smaller as the game position approaches ∂Ω), we prove similar statements for general bounded domains Ω and resolutive functions F . These games and their variants interpolate between the tug of war games studied by Peres, Schramm, Sheffield, and Wilson (p = ∞) 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.