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Quantitative estimates in stochastic homogenization
We consider the (steady-state) diffusion equation on the integer lattice with random coefficients. For stationary and ergodic coefficients a homogenized behavior emerges on large length scales; that is, in the macroscopic limit the random coefficient field can be effectively replaced by homogenized coefficients that are deterministic and constant in space. This is a great reduction of complexity!
The homogenized coefficients are defined via a formula that involves the solution to the so called "corrector problem". In contrast to periodic homogenization, in the stochastic setting the corrector problem is posed on an infinite-dimensional function space; and thus, its analysis becomes ricky.
In this talk we present quantitative estimates on the corrector equation. They rely on the observation that ergodicity can be quantified by combining a certain spectral gap property and estimates on the parabolic Green’s function.
This is joint work with Antoine Gloria, INRIA Lille, and Felix Otto, MPI Leipzig.