Harnack's Inequality forp-Harmonic Functions via Stochastic Games

Abstract

We give a proof of Lipschitz continuity of p-harmonious functions, that are tug-of-war game analogies of ordinary p-harmonic functions. This result is used to obtain a new proof of Harnack's inequality for p-harmonic functions in the case p > 2 that avoids classical techniques like Moser iteration, but instead relies on suitable choices of strategies for the stochastic tug-of-war game.

Uniform measure density condition and game regularity for tug-of-war games

“…We have the following estimate for the probability that the cylinder walk exits the cylinder through its bottom; the proof is in the appendix of the paper [LPS13].…”

“…The cancellation strategy was introduced in the paper [LPS13] to prove Harnack's inequality for p-harmonic functions via tug-of-war games. In addition, the cancellation strategy can be used to prove regularity properties for viscosity solutions of the inhomogeneous p-Laplace equation (see [Ruo16]).…”

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

“…The fixed starting point x 0 and the strategies S I and S II determine a unique probability measure P x 0 S I ,S II on the product σ-algebra, see e.g. [15]. The expected total payoff, when starting from x 0 and using strategies S I and S II , is obtained as a sum of final payoff and running payoff…”

A priori Hölder and Lipschitz regularity for generalizedp-harmonious functions in metric measure spaces

“…Furthermore, {u ε } → u uniformly in Ω as ε → 0, where u is the unique p-harmonic function solving the Dirichlet problem in Ω with boundary data f . See also [15] for an analytic approach, still in the constant radius case.…”

“…In the constant radius case, the local Lipschitz regularity of p-harmonious functions for p ≥ 2 was obtained in [16]. As for the case α = 1 (or p = ∞), not much is known.…”