Gradient Estimates for Nonlinear Diffusion Semigroups by Coupling Methods

Our purpose is to obtain gradient estimates for certain nonlinear partial differential equations by coupling methods. First we derive uniform gradient estimates for a certain semi-linear PDEs based on the coupling method introduced in Wang (2011) and the theory of backward SDEs. Then we generalize Wang's coupling to the $G$-expectation space and obtain gradient estimates for nonlinear diffusion semigroups, which correspond to the solutions of a certain fully nonlinear PDEs.