Higher order homogenization for random non-autonomous parabolic operators

TL;DR

This paper investigates the asymptotic behavior of solutions to a class of parabolic operators with rapidly oscillating, mixed periodic and ergodic coefficients, focusing on non-diffusive scaling where spatial oscillations dominate.

Contribution

It introduces a higher order homogenization approach for non-autonomous parabolic operators with mixed periodic and ergodic coefficients under non-diffusive scaling.

Findings

Asymptotic behavior of the normalized difference is characterized.

Homogenized operator remains deterministic despite random coefficients.

New techniques for non-diffusive homogenization are developed.

Abstract

We consider Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variables and random stationary ergodic in time. As was proved in [24] and [12] in this case the homogenized operator is deterministic. The paper focuses on non-diffusive scaling, when the oscillation in spatial variables is faster than that in temporal variable. Our goal is to study the asymptotic behaviour of the normalized difference between solutions of the original and the homogenized problems.