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.

Acknowledgment

The authors would like to thanks the anonymous referee for his valuable suggestions and remarks. Youssef Ouknine was supported by Moroccan Program PARS MI 37.