Regularity of non-uniformly elliptic equations in nondivergence form at unit scale

I will present recent joint work with Charles Smart, in which we prove a Harnack inequality for fully nonlinear elliptic equations in dimension $d$ with possibly unbounded ellipticity– provided the ellipticity belongs to $L^d$. This yields a stochastic homogenization result for such equations, with applications to random diffusions in (strictly elliptic) random environments.