Homogenization of non-autonomous evolution problems for convolution type operators in randomly evolving media

Abstract

We study homogenization problem for non-autonomous parabolic equations of the form ${\partial }_{t}u=L\left(t\right)u$ with an integral convolution type operator $L\left(t\right)$ that has a non-symmetric jump kernel which is periodic in spatial variables and stationary random in time. We show that asymptotically the spatial and temporal evolutions of the solutions are getting decoupled and can be described separately, and, under additional mixing conditions on the coefficient, the homogenized equation is a SPDE with a finite dimensional multiplicative noise.