A game interpretation for the weighted $p$-Laplace equation

In this paper, we obtain a stochastic approximation that converges to the viscosity solution of the weighted $p$-Laplace equation. We consider a stochastic two-player zero-sum game controlled by a random walk, two player’s choices, and the gradient of the weight function. The proof is based on the boundary conditions in the viscosity sense and the comparison principle. These results extend previous findings for the non-weighted $p$-Laplace equation [Manfredi, Parviainen, Rossi, 2012]. In addition, we study the limiting behavior of the viscosity solution of the weighted $p$-Laplace equation as $p\to \mathrm{\infty }$.