Sharp error estimates in stochastic homogenization of parabolic systems with time-dependent coefficients

TL;DR

This paper establishes sharp error estimates in stochastic homogenization for parabolic systems with time-dependent coefficients, introducing new correctors and advanced analytical techniques.

Contribution

It proves the existence of stationary correctors under space-time spectral gap conditions and develops optimal homogenization error estimates without assuming small-scale coefficient smoothness.

Findings

Existence of stationary correctors with distinct properties from elliptic cases

Introduction of new flux correctors and fluctuation estimates

Derivation of optimal homogenization error bounds in strong and weak norms

Abstract

This article mainly proves the existence of stationary correctors under space-time spectral gap conditions, which exhibit different properties from those of elliptic operator correctors. Additionally, new flux correctors and their fluctuation estimates are introduced.Based on this, we obtain the optimal homogenization error in the sense of strong and weak norms on C1 cylinders by using the duality and distance-weighted arguments, in which the (weighted) annealed Calderon-Zygmund estimates coupled with a novel form of the minimal radius are developed. Throughout the paper, no small-scale smoothness of the coefficients is used.