10 research outputs found
Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization
We consider degenerate elliptic equations of second order in divergence form
with a symmetric random coefficient field . Extending the work of the first
author, Fehrman, and Otto [Ann. Appl. Probab. 28 (2018), no. 3, 1379-1422], who
established the large-scale regularity of -harmonic functions
in a degenerate situation, we provide stretched exponential moments for the
minimal radius describing the minimal scale for this
regularity. As an application to stochastic homogenization, we partially
generalize results by Gloria, Neukamm, and Otto [Anal. PDE 14 (2021), no. 8,
2497-2537] on the growth of the corrector, the decay of its gradient, and a
quantitative two-scale expansion to the degenerate setting. On a technical
level, we demand the ensemble of coefficient fields to be stationary and
subject to a spectral gap inequality, and we impose moment bounds on and
. We also introduce the ellipticity radius which encodes the
minimal scale where these moments are close to their positive expectation
value
Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations
We consider the corrector equation from the stochastic homogenization of
uniformly elliptic finite-difference equations with random, possibly
non-symmetric coefficients. Under the assumption that the coefficients are
stationary and ergodic in the quantitative form of a Logarithmic Sobolev
inequality (LSI), we obtain optimal bounds on the corrector and its gradient in
dimensions . Similar estimates have recently been obtained in the
special case of diagonal coefficients making extensive use of the maximum
principle and scalar techniques. Our new method only invokes arguments that are
also available for elliptic systems and does not use the maximum principle. In
particular, our proof relies on the LSI to quantify ergodicity and on
regularity estimates on the derivative of the discrete Green's function in
weighted spaces.Comment: added applications, e.g. two-scale expansion, variance estimate of
RV
Lipschitz regularity for elliptic equations with random coefficients
We develop a higher regularity theory for general quasilinear elliptic
equations and systems in divergence form with random coefficients. The main
result is a large-scale -type estimate for the gradient of a
solution. The estimate is proved with optimal stochastic integrability under a
one-parameter family of mixing assumptions, allowing for very weak mixing with
non-integrable correlations to very strong mixing (e.g., finite range of
dependence). We also prove a quenched estimate for the error in
homogenization of Dirichlet problems. The approach is based on subadditive
arguments which rely on a variational formulation of general quasilinear
divergence-form equations.Comment: 85 pages, minor revisio
The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations
We consider uniformly elliptic coefficient fields that are randomly
distributed according to a stationary ensemble of a finite range of dependence.
We show that the gradient and flux of the
corrector , when spatially averaged over a scale decay like the
CLT scaling . We establish this optimal rate on the level of
sub-Gaussian bounds in terms of the stochastic integrability, and also
establish a suboptimal rate on the level of optimal Gaussian bounds in terms of
the stochastic integrability. The proof unravels and exploits the
self-averaging property of the associated semi-group, which provides a natural
and convenient disintegration of scales, and culminates in a propagator
estimate with strong stochastic integrability. As an application, we
characterize the fluctuations of the homogenization commutator, and prove sharp
bounds on the spatial growth of the corrector, a quantitative two-scale
expansion, and several other estimates of interest in homogenization.Comment: 114 pages. Revised version with some new results: optimal scaling
with nearly-optimal stochastic integrability on top of nearly-optimal scaling
with optimal stochastic integrability, CLT for the homogenization commutator,
and several estimates on growth of the extended corrector, semi-group
estimates, and systematic error