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

Abstract

Fix a bounded domain Ω ⊂ R d , a continuous function F : ∂Ω → R, and constants ǫ > 0 and 1 < p, q < ∞ with p −1 + q −1 = 1. For each x ∈ Ω, let u ǫ (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 ∈ B(0, ǫ) 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 ǫ tends to zero the functions u ǫ converge uniformly to the unique p-harmonic extension of F . Using a modified game (in which ǫ 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 (p = ∞) and the motion-by-curvature games introduced by Spencer and studied by Kohn and Serfaty (p = 1). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about p-capacity and p-harmonic measure.

An asymptotic mean value characterization for 𝑝-harmonic functions

“…This formal expansion was used by Peres and Sheffield in [13] (see also Peres et al [12]) to find p-harmonic functions as limits of values of Tug-of-War games.…”

Dynamic Programming Principle for tug-of-war games with noise

“…The players continue starting from the new point until the game position reaches a strip near the boundary, and then Player II pays Player I the amount given by a payoff function. This game was studied in a slightly different form by Peres and Sheffield in [9], and the tug-of-war game by Peres, Schramm, Sheffield, and Wilson in [8].…”

Waiting time effect for motion by positive second derivatives and applications

“…This approximation gives an interesting representation of the solutions to the equation. See also [17] for the deterministic game approach to general elliptic and parabolic equations and [22][23][24][25] for a stochastic tug-of-war game approach to the p-Laplace equation with p > 1. Related extensions of this new method to the Heisenberg group are recently addressed in [9,10].…”