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

TL;DR

This paper investigates the homogenization of non-autonomous parabolic equations with convolution operators in random media, revealing that the effective equation is a stochastic PDE with finite-dimensional noise.

Contribution

It introduces a homogenization framework for non-autonomous convolution-type operators with non-symmetric kernels in random media, leading to a stochastic PDE as the homogenized limit.

Findings

Homogenized equation is a stochastic PDE with finite-dimensional multiplicative noise.

The kernel's periodicity in space and stationarity in time are crucial assumptions.

The approach extends homogenization theory to non-autonomous, non-symmetric convolution operators.

Abstract

We study homogenization problem for non-autonomous parabolic equations of the form $\partial_t u=L(t)u$ with an integral convolution type operator $L(t)$ that has a non-symmetric jump kernel which is periodic in spatial variables and stationary random in time. We show that the homogenized equation is a SPDE with a finite dimensional multiplicative noise.