Quantitative results in stochastic homogenization for divergence-form equations

Scott Armstrong (Paris-Dauphine)

I will summarize some recent progress in the study of linear, divergence form elliptic equations with random coefficients. Several years ago, Gloria and Otto and collaborators used an idea from statistical mechanics (concentration of measure) to establish strong quantitative bounds on the sizes of fluctuations of solutions, and then Nolen, Mourrat and others proved central limit theorems for Dirichlet forms and obtained the scaling limits of solutions. In all of these results, concentration inequalities were the essential tool for transferring ergodic information from the coefficients (which satisfy an iid assumption) to the solutions themselves, and this requires some restrictive assumptions. In this talk, I will describe an alternative approach developed in the last several years with Smart and more recently with Kuusi and Mourrat. We obtain optimal quantitative estimates, central limit theorems, and the scaling limit of the correctors to a variant of the Gaussian Free Field– under fewer assumptions, with much stronger stochastic integrability, and without using abstract concentration of measure. Rather, our approach is to “linearize the randomness around the homogenized limit”: that is, we use renormalization arguments which reveal that certain Dirichlet forms are essentially additive quantities. The complicated nonlinear structure of the randomness is reduced to a linear one (essentially a sum of iid random variables) and a complete quantitative theory can then be easily read off.

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