Journal
ANNALS OF PROBABILITY
Volume 42, Issue 6, Pages 2558-2594Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/13-AOP833
Keywords
Stochastic homogenization; quenched invariance principle; regularity; effective ellipticity; random diffusions in random environments; fully nonlinear equations
Categories
Funding
- NSF [DMS-10-04645, DMS-10-04595]
- Chaire Junior of la Fondation Sciences Mathematiques de Paris
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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 dth 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 pth 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.
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