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 ¯nd existence of viscosity solutions to the Dirichlet problem for an equation of the form ¡hD 2 v ¢ Jx(Dv); Jx(Dv)i(x) = 0, that is, an in¯nity Laplacian with spatial dependence. Here Jx(Dv(x)) is a vector that depends on the the spatial location and the gradient of the solution.