Analysis & PDE
- Anal. PDE
- Volume 11, Number 8 (2018), 1945-2014.
Quantitative stochastic homogenization and regularity theory of parabolic equations
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 -type estimate and a Liouville theorem of every finite order.
Anal. PDE, Volume 11, Number 8 (2018), 1945-2014.
Received: 22 May 2017
Revised: 15 February 2018
Accepted: 9 April 2018
First available in Project Euclid: 15 January 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 35B45: A priori estimates
Secondary: 60K37: Processes in random environments 60F05: Central limit and other weak theorems
Armstrong, Scott; Bordas, Alexandre; Mourrat, Jean-Christophe. Quantitative stochastic homogenization and regularity theory of parabolic equations. Anal. PDE 11 (2018), no. 8, 1945--2014. doi:10.2140/apde.2018.11.1945. https://projecteuclid.org/euclid.apde/1547521444