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We consider degenerate elliptic equations of second order in divergence formwith a symmetric random coefficient field $a$. Extending the work of the firstauthor, Fehrman, and Otto [Ann. Appl. Probab. 28 (2018), no. 3, 1379-1422], whoestablished the large-scale $C^{1,\alpha}$ regularity of $a$-harmonic functionsin a degenerate situation, we provide stretched exponential moments for theminimal radius $r_*$ describing the minimal scale for this $C^{1,\alpha}$regularity. As an application to stochastic homogenization, we partiallygeneralize 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 aquantitative two-scale expansion to the degenerate setting. On a technicallevel, we demand the ensemble of coefficient fields to be stationary andsubject 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 theminimal scale where these moments are close to their positive expectationvalue.

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