A probabilistic approach to the homogenization of divergence‐form operators in periodic media

Article type: Research Article

Authors: Lejay, Antoine

Affiliations: Projet OMEGA, INRIA Lorraine, IECN, Campus Scientifique, BP 239, 54506 Vandœuvre‐lès‐Nancy cedex, France E‐mail: [email protected]

Note: [] Projet SYSDYS (INRIA/LATP, Marseille).

Abstract: We prove here using stochastic analysis the homogenization property of second‐order divergence‐form operators with lower‐order differential terms (possibly highly‐oscillating) in periodic media. The coefficients are not assumed to have any regularity, so the stochastic calculus theory for processes associated to Dirichlet forms is used. The Girsanov theorem and the Feynman–Kac formula are used to work on the probabilistic representation of the solutions of some PDEs.

Keywords: divergence‐form operators, Dirichlet forms, homogenization, Feynman–Kac formula, Girsanov theorem

Journal: Asymptotic Analysis, vol. 28, no. 2, pp. 151-162, 2001