Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

TL;DR

This paper establishes regularity and stochastic homogenization for certain degenerate elliptic equations with oscillating ellipticity, providing explicit estimates and demonstrating sharpness of moment conditions in a stationary-ergodic framework.

Contribution

It introduces a novel approach to regularity and homogenization for degenerate elliptic equations with finite moment ellipticity, extending results to nonuniformly elliptic contexts.

Findings

Homogenization to a deterministic, uniformly elliptic equation under finite moment conditions.

Explicit estimates of effective ellipticity depending on the distribution of ellipticity.

Sharpness of moment condition demonstrated through explicit counterexamples.

Abstract

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.