Tug-of-war games and the infinity Laplacian with spatial dependence

Abstract

In this paper we look for PDEs that arise as limits of values of Tug-of-War games when the possible movements of the game are taken in a family of sets that are not necessarily euclidean balls. In this way we find existence of viscosity solutions to the Dirichlet problem for an equation of the form -(x) = 0, that is, an infinity Laplacian with spatial dependence. Here J_{x}(Dv(x)) is a vector that depends on the the spatial location and the gradient of the solution.