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

Abstract

AbstractWe consider degenerate elliptic equations of second order in divergence form with a symmetric random coefficient field a. Extending the work of Bella et al. (Ann Appl Probab 28(3):1379–1422, 2018), who established the large-scale $$C^{1,\alpha }$$ C 1 , α regularity of a-harmonic functions in a degenerate situation, we provide stretched exponential moments for the minimal radius $$r_*$$ r ∗ describing the minimal scale for this $$C^{1,\alpha }$$ C 1 , α regularity. As an application to stochastic homogenization, we partially generalize results by Gloria et al. (Anal PDE 14(8):2497–2537, 2021) 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 $$a^{-1}$$ a - 1 . We also introduce the ellipticity radius $$r_e$$ r e which encodes the minimal scale where these moments are close to their positive expectation value.