Local regularity results for value functions of tug-of-war with noise and running payoff

Abstract

Abstract. We prove local Lipschitz continuity and Harnack's inequality for value functions of the stochastic game tug-of-war with noise and running payoff. As a consequence, we obtain game-theoretic proofs for the same regularity properties for viscosity solutions of the inhomogeneous p-Laplace equation when p > 2.

C1, regularity for the normalized p-Poisson problem

“…In [37] Peres and Sheffield studied a connection between equation (1.1) and the game tug-of-war with noise and running pay-off. The game-theoretic interpretation led to new regularity proofs in the case f = 0 in [32], and later in the case of bounded and positive f in [39], see also [9] for a PDE approach. Regularity studies were extended to the parabolic version u t = Δ N p u in [35,4,21] and led to applications in image processing, see e.g.…”

A game-theoretic approach to dynamic boundary problems for level-set curvature flow equations and applications

“…We also refer to stochastic Tug-of-war games studied by Peres et al [39,40] for normalized p-Laplace equations with 1 < p ≤ ∞; see related results on the so-called asymptotic mean value properties in [30,36,37]. The game approximations turn out to be useful in understanding various properties of the associated nonlinear PDEs, as shown in [3,[33][34][35]38,41] etc.…”

Local regularity for time-dependent tug-of-war games with varying probabilities

“…Since the number of steps during the game is bounded, adding a bounded running payoff to the game would not cause any new difficulties. In the case of unlimited number of steps the situation is different, see [Ruo16].…”