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

We 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 \(C^{1,\alpha }\) $C^{1,\alpha }$ regularity of a-harmonic functions in a degenerate situation, we provide stretched exponential moments for the minimal radius \(r_*\) $r_{\ast }$ describing the minimal scale for this \(C^{1,\alpha }\) $C^{1,\alpha }$ 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.