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1 October 2008 Tug-of-war with noise: A game-theoretic view of the p-Laplacian
Yuval Peres, Scott Sheffield
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Duke Math. J. 145(1): 91-120 (1 October 2008). DOI: 10.1215/00127094-2008-048


Fix a bounded domain ΩRd, a continuous function F:ΩR, and constants ε>0 and 1<p,q< with p1+q1=1. For each xΩ, 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 vB̲(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 yΩ, 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


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Yuval Peres. Scott Sheffield. "Tug-of-war with noise: A game-theoretic view of the p-Laplacian." Duke Math. J. 145 (1) 91 - 120, 1 October 2008.


Published: 1 October 2008
First available in Project Euclid: 17 September 2008

zbMATH: 1206.35112
MathSciNet: MR2451291
Digital Object Identifier: 10.1215/00127094-2008-048

Primary: 35J55 , 91A24
Secondary: 31C15 , 49N70 , 91A15

Rights: Copyright © 2008 Duke University Press


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Vol.145 • No. 1 • 1 October 2008
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