The Annals of Probability

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

Scott N. Armstrong and Charles K. Smart

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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.

Article information

Source
Ann. Probab., Volume 42, Number 6 (2014), 2558-2594.

Dates
First available in Project Euclid: 30 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1412083632

Digital Object Identifier
doi:10.1214/13-AOP833

Mathematical Reviews number (MathSciNet)
MR3265174

Zentralblatt MATH identifier
1315.35019

Subjects
Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 35B45: A priori estimates 60K37: Processes in random environments 35J70: Degenerate elliptic equations 35D40: Viscosity solutions

Keywords
Stochastic homogenization quenched invariance principle regularity effective ellipticity random diffusions in random environments fully nonlinear equations

Citation

Armstrong, Scott N.; Smart, Charles K. Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity. Ann. Probab. 42 (2014), no. 6, 2558--2594. doi:10.1214/13-AOP833. https://projecteuclid.org/euclid.aop/1412083632


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