Probabilistic approach for nonlinear PDEs and SPDEs with Neumann boundary conditions

Abstract:  We study nonlinear PDEs and stochastic PDEs with Neumann boundary conditions on domains satisfying conditions put forward by P.L. Lions--A. Sznitman and Saisho. We apply the penalization convergence of reflected SDEs, and prove the existence and comparison principle of viscosity solutions for HJB equations with Neumann conditions associated with optimal control problems. We also prove for nonlinear SPDEs and backward SPDEs with Neumann conditions representations of the stochastic viscosity solutions.