Stochastic homogenization with space-time ergodic divergence-free drift

TL;DR

This paper proves that diffusion equations with space-time stationary, ergodic, divergence-free drift converge to a stochastic PDE with Stratonovich noise, revealing how certain drifts influence homogenization and noise emergence.

Contribution

It introduces a novel homogenization result for divergence-free drifts, including cases without spatial ergodicity, and characterizes the resulting stochastic limit.

Findings

Diffusion equations homogenize to a stochastic PDE with Stratonovich noise.

Partial absorption of drift into flux skew-symmetric part when spatial ergodicity is absent.

Convergence to Brownian noise with deterministic covariance for mildly decorrelating fields.

Abstract

We prove that diffusion equations with a space-time stationary and ergodic, divergence-free drift homogenize in law to a deterministic stochastic partial differential equation with Stratonovich transport noise. In the absence of spatial ergodicity, the drift is only partially absorbed into the skew-symmetric part of the flux through the use of an appropriately defined stream matrix. This leaves a time-dependent, spatially-homogenous transport which, for mildly decorrelating fields, converges to a Brownian noise with deterministic covariance in the homogenization limit. The results apply to uniformly elliptic, stationary and ergodic environments in which the drift admits a suitably defined stationary and $L^2$-integrable stream matrix.