BSDEs, weak convergence and homogenization of semilinear PDEs

Abstract

In these lectures, we present the theory of backward stochastic differential equations, and its connection with solutions of semilinear second order partial differential equations of parabolic and elliptic type. This connection provides a probabilistic tool for studying solutions of semilinear PDEs. We apply our results to the proof of the homogenization result for such PDEs, both with periodic and random coefficients. For that purpose, we need to present the theory of weak limits of solutions of backward stochastic differential equations. We also present a complete probabilistic proof, under apparently minimal assumptions, of the homogenization result of linear second order PDEs.