Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization

TL;DR

This paper extends regularity results for degenerate elliptic equations with random coefficients, providing probabilistic bounds and applications to stochastic homogenization, including growth of correctors and two-scale expansions.

Contribution

It introduces stretched exponential moment bounds for the minimal regularity radius and generalizes stochastic homogenization results to degenerate coefficient settings.

Findings

Established probabilistic bounds for the minimal radius $r_*$.

Generalized growth and decay estimates for correctors in degenerate settings.

Extended two-scale expansion techniques to degenerate elliptic operators.

Abstract

We consider degenerate elliptic equations of second order in divergence form with a symmetric random coefficient field $a$. Extending the work of the first author, Fehrman, and Otto [Ann. Appl. Probab. 28 (2018), no. 3, 1379-1422], who established the large-scale $C^{1,\alpha}$ regularity of $a$-harmonic functions in a degenerate situation, we provide stretched exponential moments for the minimal radius $r_*$ describing the minimal scale for this $C^{1,\alpha}$ regularity. As an application to stochastic homogenization, we partially generalize results by Gloria, Neukamm, and Otto [Anal. PDE 14 (2021), no. 8, 2497-2537] on the growth of the corrector, the decay of its gradient, and a quantitative two-scale expansion to the degenerate setting. On a technical level, we demand the ensemble of coefficient fields to be stationary and subject to a spectral gap inequality, and we impose moment bounds on $a$ and $a^{-1}$. We also introduce the ellipticity radius $r_e$ which encodes the minimal scale where these moments are close to their positive expectation value.