Regularity of degenerate elliptic equations in nondivergence form on large scales

I will present a regularity result for degenerate elliptic equations in nondivergence form. In joint work with Charlie Smart, we prove a Harnack inequality for equations with possibly unbounded ellipticity-- provided that the ellipticity belongs to $L^d$ (where $d$ is the dimension). As an application we obtain a stochastic homogenization result for such equations (and a new estimate for the effective coefficients) as well as an invariance principle for random diffusions in random environments.