Duke Mathematical Journal

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

Yuval Peres and Scott Sheffield

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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

Article information

Duke Math. J., Volume 145, Number 1 (2008), 91-120.

First available in Project Euclid: 17 September 2008

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J55 91A24: Positional games (pursuit and evasion, etc.) [See also 49N75]
Secondary: 91A15: Stochastic games 49N70: Differential games 31C15: Potentials and capacities


Peres, Yuval; Sheffield, Scott. Tug-of-war with noise: A game-theoretic view of the $p$ -Laplacian. Duke Math. J. 145 (2008), no. 1, 91--120. doi:10.1215/00127094-2008-048. https://projecteuclid.org/euclid.dmj/1221656864

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