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.

A Liouville theorem for stationary and ergodic ensembles of parabolic systems

“…It is an immediately consequence of the ergodic theorem [11,Theorems 2,3] and the fact that the flux q is stationary with finite energy that, for · -a.e. a, for each i ∈ {1, .…”

“…. , d}, the Poincaré inequality and the ergodic theorem [11,Theorems 2,3] together with the Rellich-Kondrachov embedding theorem imply that the family…”

“…The quantitative homogenization of such ensembles has only recently been considered, and the preprint [3] contains, to our knowledge, the first results in this direction. In particular, in [3, Theorem 1.2], a full hierarchy of Liouville theorems is obtained for ensembles satisfying a finite-range dependence in space and time.…”

“…Since the ergodic theorem [11,Theorems 2,3] implies that, for · -a.e. a, it follows from (81) that, for · -a.e.…”

Homogenization of Parabolic Equations with Non-self-similar Scales

“…for any 2 < q < ∞. By the Meyers-type estimates for parabolic systems [2,Appendix], there exist some q > 2 and C > 0, depending on d and µ, such that…”