This paper is about the homogenization of linear elliptic operators in divergence form
with stationary random coefficients that have only slowly decaying correlations. It
deduces optimal estimates of the homogenization error from optimal growth estimates of
the (extended) corrector. In line with the heuristics, there are transitions at dimension
, and for a
correlation-decay exponent
we capture the correct power of logarithms coming from these two sources of
criticality.
The decay of correlations is sharply encoded in terms of a multiscale
logarithmic Sobolev inequality (LSI) for the ensemble under consideration — the
results would fail if correlation decay were encoded in terms of an
-mixing
condition. Among other ensembles popular in modeling of random media, this class
includes coefficient fields that are local transformations of stationary Gaussian
fields.
The optimal growth of the corrector
is derived from bounding the size of spatial averages
of its
gradient. This in turn is done by a (deterministic) sensitivity estimate of
, that is, by estimating
the functional derivative
of
with respect to
the coefficient field
.
Appealing to the LSI in form of concentration of measure yields a stochastic estimate on
. The sensitivity
argument relies on a large-scale Schauder theory for the heterogeneous elliptic operator
. The treatment allows
for nonsymmetric
and for systems like linear elasticity.
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