Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

TL;DR

This paper establishes the existence and uniqueness of solutions to a biased infinity Laplacian equation using a novel biased tug-of-war game approach, extending previous methods to include exponential cone comparisons.

Contribution

It introduces the eta-biased \\eps-game and the concept of comparison with exponential cones, generalizing prior work on the infinity Laplacian.

Findings

Existence and uniqueness of viscosity solutions for the biased infinity Laplacian.

Convergence of the game value to the solution as \\eps \\to 0.

Equivalence between comparison with exponential cones and viscosity solutions.

Abstract

We prove that if U\subset\R^n is an open domain whose closure \overline{U} is compact in the path metric, and F is a Lipschitz function on \partial{U}, then for each \beta\in\R there exists a unique viscosity solution to the \beta-biased infinity Laplacian equation \beta |\nabla u| + \Delta_\infty u=0 on U that extends F, where \Delta_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}. In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the \beta-biased \eps-game as follows. The starting position is x_0 \in U. At the k^\text{th} step the two players toss a suitably biased coin (in our key example, player I wins with odds of \exp(\beta\eps) to 1), and the winner chooses x_k with d(x_k,x_{k-1}) < \eps. The game ends when x_k \in \partial{U}, and player II pays the amount F(x_k) to player I. We prove that the value u^{\eps}(x_0) of this game exists, and that \|u^\eps - u\|_\infty \to 0 as \eps \to 0, where u is the unique extension of F to \overline{U} that satisfies comparison with \beta-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with \beta-exponential cones if and only if it is a viscosity solution to the \beta-biased infinity Laplacian equation.