Homogenization of Multivalued Partial Differential Equations via Reflected Backward Stochastic Differential Equations

Abstract

We study the limit of the solution of multivalued semi-linear Partial Differential Equations (PDEs for short) involving a second order differential operator of parabolic type where the non-linear term is a function of the solution, not of its gradient. Our basic tool is the approach given by Pardoux [Pardoux, E. Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. In Stochastic Analysis and Related Topics: The Geilo Workshop, 1996; Decreusefond, L., Gjerde, J., Oksendal, B., Ustüunel, A.S., Eds.; Birkhäuser, 1998; 79–127] and Ouknine [Ouknine, Y. Reflected BSDE with jumps. Stoch. Stoch. Reports 65, 111–125]. In particular, we use the weak convergence of an associated reflected Backward Stochastic Differential Equation (BSDE for short) involving the subdifferential operator of a lower semi-continuous, proper and convex function. An homogenization property for solutions of semi-linear PDEs in Sobolev spaces is also proved.