Homogenization of nonlocal partial differential equations related to stochastic differential equations with Lévy noise

Abstract

We study the “periodic homogenization” for a class of nonlocal partial differential equations of parabolic-type with rapidly oscillating coefficients, related to stochastic differential equations driven by multiplicative isotropic α-stable Lévy noise ($1<\mathit{\alpha }<2$) which is nonlinear in the noise component. Our homogenization method is probabilistic. It turns out that, under suitable regularity assumptions, the limit of the solutions satisfies a nonlocal partial differential equation with constant coefficients, which are associated to a symmetric α-stable Lévy process.