Tug-of-war and the infinity Laplacian

Abstract

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 for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do not intersect Y. This was previously known only for bounded domains R^n, in which case u is infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also prove the first general uniqueness results for Delta_infty u = g on bounded subsets of R^n (when g is uniformly continuous and bounded away from zero), and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be the value of the following two-player zero-sum game, called tug-of-war: fix x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner chooses an x_k with d(x_k, x_{k-1})< epsilon. The game ends when x_k is in Y, and player one's payoff is F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i) We show that the u^\epsilon converge uniformly to u as epsilon tends to zero. 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 delta-neighborhood of a Cantor set on the unit circle.Comment: 44 pages, 4 figure

On near optimal trajectories for a game associated with the ∞-Laplacian

“…As a consequence of [8], that proves uniqueness (and existence) of viscosity solutions to (1.1) under the above assumptions on h, the function u of Assumption 1.1 is the unique classical solution of (1.1).…”

“…We refer the reader to [1,2,4,6,7,10] for background on the infinity-Laplacian and some related PDE theory. This paper is motivated by recent work of Peres et al [8], where a discrete time random turn game, referred to as Tugof-War, is developed in relation to (1.1). This game, parameterized by ε > 0, has the property that the vanishing-ε limit of the value function uniquely solves (1.1) in the viscosity sense (a result that is valid also in the homogenous case, h = 0, excluded from the current paper).…”

“…is suggested in [8] as the game's dynamics in the vanishing-ε limit. The relation is rigorously established in examples, but only heuristically justified in general.…”

Boundary behavior of large viscosity solutions to infinity Laplace equations

“…By using the viscosity solutions introduced by Crandall and Lions [22] (see, also, Crandall et al [19]), Jensen [27] proved that u ∈ C(Ω) is an absolute minimizing Lipschitz extension of g ∈ Lip(∂Ω) if and only if ∞ u = 0, in the viscosity sense, in Ω and u = g on ∂Ω. Since then, the infinity Laplace equation in turn is a very topical differential operator that appears in many contexts and has been extensively studied, see, for instance, [5,6,[8][9][10][11][12]15,20,21,24,[28][29][30]37,43,47,48] and the references therein.…”

Existence of Viscosity Solutions to a Parabolic Inhomogeneous Equation Associated with Infinity Laplacian

“…is related to game theory named tug-of-war [8]. In [9]- [12], Akagi, Suzuki, et al considered the following degenerate parabolic equation…”