Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

Scott N. Armstrong, Charles K. Smart

Published 2012-08-22, updated 2014-10-28Version 4

We prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form. The equation is required to be strictly elliptic, but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite $d$th moment, where $d$ is the dimension. In the general stationary-ergodic framework, we show that the equation homogenizes to a deterministic, uniformly elliptic equation, and we obtain an explicit estimate of the effective ellipticity, which is new even in the uniformly elliptic context. Showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory. We prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity, which are controlled in the macroscopic limit by the ergodic theorem. We show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite $p$th moment, for every $p<d$, but for which regularity and homogenization break down. In probabilistic terms, the homogenization results correspond to quenched invariance principles for diffusion processes in random media, including linear diffusions as well as diffusions controlled by one controller or two competing players.