Tug-of-war and the infinity Laplacian

We prove that every bounded Lipschitz function F on a subset Y of a length space X admits a tautest extension to X , i.e., a unique Lipschitz extension u:X rightarrow {R} for which operatorname{Lip}_U u =operatorname{Lip}_{partial U} u for all open U subset Xsetminus Y . This was previously known only for bounded domains in {R}^n , in which case u is infinity harmonic; that is, a viscosity solution to Δ_infty u = 0 , where Δ_infty u = \vertnabla u\vert^{-2} sum_{i,j} u_{x_i} u_{x_ix_j} u_{x_j}. We also prove the first general uniqueness results for Δ_{infty} u = g on bounded subsets of {R}^n (when g is uniformly continuous and bounded away from 0) and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u . Let u^\varepsilon(x) be the value of the following two-player zero-sum game, called tug-of-war: fix x_0=xin X setminus Y . At the k{^{{th}}} turn, the players toss a coin and the winner chooses an x_k with d(x_k, x_{k-1})< \varepsilon . The game ends when x_k in Y , and player I's payoff is F(x_k) - frac{\varepsilon^2}{2}sum_{i=0}^{k-1} g(x_i) . We show that Vert u^\varepsilon- uVert _{infty} to 0 . Even for bounded domains in {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 -neighborhood of a Cantor set on the unit circle.