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 $\Omega \subset {\mathbb R}^d$, a continuous function $F:\partial \Omega \rightarrow {\mathbb R}$, and constants $\epsilon >0$ and $1 \lt p,q \lt \infty$ with $p^{-1} + q^{-1} = 1$. For each $x \in \Omega$, 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 $v \in \overline{B}(0,\epsilon)$ 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 $y \in\partial \Omega$, and player I's payoff is $F(y)$.

We show that (for sufficiently regular $\Omega$) as $\epsilon$ tends to zero, the functions $u^\epsilon$ converge uniformly to the unique $p$-harmonic extension of $F$. Using a modified game (in which $\epsilon$ gets smaller as the game position approaches $\partial \Omega$), 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] ($p=\infty$) 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. http://projecteuclid.org/euclid.dmj/1221656864.

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