Quantitative stochastic homogenization and regularity theory of parabolic equations

Scott Armstrong, Alexandre Bordas, Jean Christophe Mourrat

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1945-2014
Number of pages70
JournalAnalysis and PDE
Volume11
Issue number8
DOIs
StatePublished - Jan 1 2018

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Keywords

  • Large-scale regularity
  • Parabolic equation
  • Stochastic homogenization
  • Variational methods

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

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