Probabilistic approach for nonlinear partial differential equations and stochastic partial differential equations with Neumann boundary conditions

Abstract

In this study, we consider nonlinear partial differential equations (PDEs) and stochastic PDEs (SPDEs) with Neumann boundary conditions on domains that satisfy the Lions–Sznitmann–Saisho conditions. Using the convergence result based on the penalization approximation for stochastic differential equations with normal reflections on nonsmooth and nonconvex domains, we establish the existence and comparison principle for the viscosity solutions of nonlinear PDEs with the Neumann conditions associated with the optimal control problem. We also obtain for nonlinear SPDEs and backward SPDEs with Neumann condition the representations of the stochastic viscosity solutions.