Generalized BSDEs, weak convergence, and homogenization of semilinear PDEs with the Wentzell-type boundary condition
Aboubakary Diakhaby, Youssef Ouknine · Stochastic Analysis and Applications · 2016
We study the homogenization property of semilinear second-order PDEs on the domain according to the Wentzell-type boundary condition, periodic coefficients, highly oscillating drift, and nonlinear term. Our method builds on earlier work on homogenization of Tanaka [22 Tanaka, H. 1984. Homogenization of diffusion processes with boundary conditions. In Stochastic Analysis and Applications. Advanced Probability and Related Topics, 7. New York: Dekker, 411–437. [Google Scholar]], which deals with of diffusion with Wentzell boundary and for semilinear PDEs, Pardoux [13 Pardoux, É. 1999. Homogenization of linear and semilinear second order parabolic {PDEs} with periodic coefficients: A probabilistic approach. Journal of Functional Analysis 167(2):498–520.[Crossref], [Web of Science ®] , [Google Scholar]], who provides a probabilistic approch in whole space, and Ouknine-Pardoux [11 Ouknine, Y., Pardoux, É. 2002. Homogenization of PDEs with non linear boundary condition. In Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999). Progress in Probability, 52. Basel: Birkhauser, 229–242.[Crossref] , [Google Scholar]], who is devoted to the domain D with the Neumann boundary condition. In particular, we use the weak convergence of reflected BSDEs and then obtain pointwise convergence of PDEs but not functional convergence as with the analytical method.