Duke Mathematical Journal
- Duke Math. J.
- Volume 145, Number 1 (2008), 91-120.
Tug-of-war with noise: A game-theoretic view of the -Laplacian
Fix a bounded domain , a continuous function , and constants and with . For each , let be the value for player I of the following two-player, zero-sum game. The initial game position is . At each stage, a fair coin is tossed, and the player who wins the toss chooses a vector to add to the game position, after which a random noise vector with mean zero and variance in each orthogonal direction is also added. The game ends when the game position reaches some , and player I's payoff is .
We show that (for sufficiently regular ) as tends to zero, the functions converge uniformly to the unique -harmonic extension of . Using a modified game (in which gets smaller as the game position approaches ), we prove similar statements for general bounded domains and resolutive functions .
These games and their variants interpolate between the tug-of-war games studied by Peres, Schramm, Sheffield, and Wilson ,  () and the motion-by-curvature games introduced by Spencer  and studied by Kohn and Serfaty  (). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about -capacity and -harmonic measure
Duke Math. J., Volume 145, Number 1 (2008), 91-120.
First available in Project Euclid: 17 September 2008
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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