#### Vol. 11, No. 8, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
Quantitative stochastic homogenization and regularity theory of parabolic equations

### Scott Armstrong, Alexandre Bordas and Jean-Christophe Mourrat

Vol. 11 (2018), No. 8, 1945–2014
DOI: 10.2140/apde.2018.11.1945
##### Abstract

We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain subadditive quantities derived from a variational interpretation of parabolic equations. These subadditive quantities are intimately connected to spatial averages of the fluxes and gradients of solutions. We implement a renormalization-type scheme to obtain an algebraic rate for their convergence, which is essentially a quantification of the weak convergence of the gradients and fluxes of solutions to their homogenized limits. As a consequence, we obtain estimates of the homogenization error for the Cauchy–Dirichlet problem which are optimal in stochastic integrability. We also develop a higher regularity theory for solutions of the heterogeneous equation, including a uniform ${C}^{0,1}$-type estimate and a Liouville theorem of every finite order.

##### Keywords
stochastic homogenization, parabolic equation, large-scale regularity, variational methods
##### Mathematical Subject Classification 2010
Primary: 35B27, 35B45
Secondary: 60K37, 60F05