An Asymptotic Mean Value Characterization for a Class of Nonlinear Parabolic Equations Related to Tug-of-War Games

Abstract

We characterize solutions to the homogeneous parabolic p-Laplace equation $u_{t}=|\nabla u|^{2-p}\Delta_{p}u=(p-2)\Delta_{\infty}u+\Delta u$ in terms of an asymptotic mean value property. The results are connected with the analysis of tug-of-war games with noise in which the number of rounds is bounded. The value functions for these games approximate a solution to the PDE above when the parameter that controls the size of the possible steps goes to zero.