Asymptotic decomposition of solutions to random parabolic operators with oscillating coefficients

TL;DR

This paper investigates the asymptotic behavior of solutions to parabolic equations with rapidly oscillating, periodic-in-space, and ergodic-in-time coefficients, revealing deterministic leading terms and stochastic limits.

Contribution

It provides the leading asymptotic expansion of solutions and shows convergence to an SPDE for the renormalized difference, advancing understanding of homogenization in stochastic PDEs.

Findings

Deterministic leading terms in asymptotic expansion

Convergence to SPDE for renormalized difference

Asymptotic behavior characterized for oscillating coefficients

Abstract

We consider Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variable and random stationary ergodic in time. As was proved in [25] and [13] in this case the homogenized operator is deterministic. We obtain the leading terms of the asymptotic expansion of the solution , these terms being deterministic functions, and show that a properly renormalized difference between the solution and the said leading terms converges to a solution of some SPDE.