Quantitative stochastic homogenization and regularity theory of parabolic equations

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.

69 pages, minor revision

References in corpus (2)

• An optimal error estimate in stochastic homogenization of discrete elliptic equations
• An optimal variance estimate in stochastic homogenization of discrete elliptic equations

Cited by in corpus (13)

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• Homogenization of locally periodic parabolic operators with non-self-similar scales
• Convergence rates in homogenization of higher order parabolic systems
• Optimal Hardy weights on the Euclidean lattice
• Weighted heat kernel estimates: rate of convergence in Kolmogorov distance
• Coarsen entropy inequalities and its applications in quantitative stochastic homogenization
• Weighted Meyers estimates in perforated domains
• Regularity theory of elliptic systems in $\epsilon$-scale flat domains