Homogenization with fractional random fields

TL;DR

This paper develops a homogenization theory for differential equations in long-range dependent random environments, resulting in effective dynamics described by rough differential equations involving mixed stochastic integrals.

Contribution

It introduces a homogenization framework for systems in fractional random fields, utilizing rough path theory to handle complex stochastic limits and including the negatively correlated case in one dimension.

Findings

Proves a homogenization theorem with multiple scaling constants.

Derives effective dynamics as a rough differential equation with mixed integrals.

Provides error bounds for the second order fractional Brownian motion model.

Abstract

We consider a system of differential equations in a fast long range dependent random environment and prove a homogenization theorem involving multiple scaling constants. The effective dynamics solves a rough differential equation, which is `equivalent' to a stochastic equation driven by mixed It\^o integrals and Young integrals with respect to Wiener processes and Hermite processes. Lacking other tools we use the rough path theory for proving the convergence, our main technical endeavour is on obtaining an enhanced scaling limit theorem for path integrals (Functional CLT and non-CLT's) in a strong topology, the rough path topology, which is given by a H\"older distance for stochastic processes and their lifts. In dimension one we also include the negatively correlated case, for the second order / kinetic fractional BM model we also bound the error.