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

Abstract

We prove that if ${U\subset \mathbb {R}^n}$ is an open domain whose closure ${\overline U}$ is compact in the path metric, and F is a Lipschitz function on ∂U, then for each ${\beta \in \mathbb {R}}$ there exists a unique viscosity solution to the β-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 β-biased ${\epsilon}$-game as follows. The starting position is ${x_0 \in U}$. At the kth step the two players toss a suitably biased coin (in our key example, player I wins with odds of ${\exp(\beta\epsilon)}$ to 1), and the winner chooses x _k with ${d(x_k,x_{k-1}) < \epsilon}$. 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 ${u^{\epsilon}(x_0)}$ of this game exists, and that ${\|u^\epsilon - u\|_\infty \to 0}$ as ${\epsilon \to 0}$, where u is the unique extension of F to ${\overline{U}}$ that satisfies comparison with β-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 β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation.