Vol. 11, No. 8, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 C0,1-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
Milestones
Received: 22 May 2017
Revised: 15 February 2018
Accepted: 9 April 2018
Published: 6 June 2018
Authors
Scott Armstrong
Courant Institute of Mathematical Sciences
New York University
New York, NY
United States
Alexandre Bordas
Ecole normale supérieure de Lyon
Lyon
France
Jean-Christophe Mourrat
Ecole normale supérieure de Lyon
CNRS
Lyon
France