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

TL;DR

This paper establishes local regularity properties for value functions of a stochastic game called tug-of-war with noise, and applies these results to viscosity solutions of the inhomogeneous p-Laplace equation, providing new game-theoretic proofs.

Contribution

It proves local Lipschitz continuity and Harnack's inequality for these value functions and connects game theory with PDE regularity results for the p-Laplace equation.

Findings

Proved local Lipschitz continuity of value functions

Established Harnack's inequality for the value functions

Provided game-theoretic proofs for regularity of p-Laplace solutions

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 $p>2$.