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 -integrable stream matrix.
Funding Statement
The author acknowledges financial support from the Engineering and Physical Sciences Research Council of the United Kingdom through the EPSRC Early Career Fellowship EP/V027824/1.
Acknowledgments
The author would like to thank the two anonymous referees for their excellent comments which have substantially improved the quality of the paper.
Citation
Benjamin Fehrman. "Stochastic homogenization with space-time ergodic divergence-free drift." Ann. Probab. 52 (1) 350 - 380, January 2024. https://doi.org/10.1214/23-AOP1663
Information