Tug-of-war and the infinity Laplacian

Abstract

We prove that every bounded Lipschitz function F F on a subset Y Y of a length space X X admits a tautest extension to X X , i.e., a unique Lipschitz extension u : X → R u:X \rightarrow \mathbb {R} for which Lip U ⁡ u = Lip ∂ U ⁡ u \operatorname {Lip}_U u =\operatorname {Lip}_{\partial U} u for all open U ⊂ X ∖ Y U \subset X\smallsetminus Y . This was previously known only for bounded domains in R n \mathbb {R}^n , in which case u u is infinity harmonic; that is, a viscosity solution to Δ ∞ u = 0 \Delta _\infty u = 0 , where \[ Δ ∞ u = | ∇ u | − 2 ∑ i , j u x i u x i x j u x j . \Delta _\infty u = |\nabla u|^{-2} \sum _{i,j} u_{x_i} u_{x_ix_j} u_{x_j}. \] We also prove the first general uniqueness results for Δ ∞ u = g \Delta _{\infty } u = g on bounded subsets of R n \mathbb {R}^n (when g g is uniformly continuous and bounded away from 0 0 ) and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u u . Let u ε ( x ) u^\varepsilon (x) be the value of the following two-player zero-sum game, called tug-of-war: fix x 0 = x ∈ X ∖ Y x_0=x\in X \smallsetminus Y . At the k t h k^{\mathrm {th}} turn, the players toss a coin and the winner chooses an x k x_k with d ( x k , x k − 1 ) > ε d(x_k, x_{k-1})> \varepsilon . The game ends when x k ∈ Y x_k \in Y , and player I’s payoff is F ( x k ) − ε 2 2 ∑ i = 0 k − 1 g ( x i ) F(x_k) - \frac {\varepsilon ^2}{2}\sum _{i=0}^{k-1} g(x_i) . We show that ‖ u ε − u ‖ ∞ → 0 \|u^\varepsilon - u\|_{\infty } \to 0 . Even for bounded domains in R n \mathbb {R}^n , the game theoretic description of infinity harmonic functions yields new intuition and estimates; for instance, we prove power law bounds for infinity harmonic functions in the unit disk with boundary values supported in a δ \delta -neighborhood of a Cantor set on the unit circle.