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

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Fix a bounded domain Ω⊂Rd, a continuous function F:∂Ω→R, and constants ε>0 and 1<p,q<∞ with p−1+q−1=1. For each x∈Ω, let uε(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∈B̲(0,ε) 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 ε tends to zero, the functions uε converge uniformly to the unique p-harmonic extension of F. Using a modified game (in which ε 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 [15], [16] (p=∞) 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

Abstract Fix a bounded domain Ω⊂Rd, a continuous function F:∂Ω→R, and constants ε>0 and 1<p,q<∞ with p−1+q−1=1. For each x∈Ω, let uε(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∈B̲(0,ε) 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…

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