Scaling limit of the homogenization commutator for Gaussian coefficient fields

TL;DR

This paper studies the fluctuations of solutions to elliptic PDEs with Gaussian random coefficients, showing that the key homogenization commutator converges to a fractional Gaussian field, extending previous results to higher dimensions and correlated fields.

Contribution

It establishes the scaling limit of the homogenization commutator for Gaussian coefficient fields with correlations, generalizing prior one-dimensional and iid results to higher dimensions.

Findings

Rescaled commutator converges to a fractional Gaussian field.

The limit's degeneracy depends on the correlation decay.

Results extend to all dimensions $d\\ge1$ and correlated Gaussian fields.

Abstract

Consider a linear elliptic partial differential equation in divergence form with a random coefficient field. The solution operator displays fluctuations around its expectation. The recently developed pathwise theory of fluctuations in stochastic homogenization reduces the characterization of these fluctuations to those of the so-called standard homogenization commutator. In this contribution, we investigate the scaling limit of this key quantity: starting from a Gaussian-like coefficient field with possibly strong correlations, we establish the convergence of the rescaled commutator to a fractional Gaussian field, depending on the decay of correlations of the coefficient field, and we investigate the (non)degeneracy of the limit. This extends to general dimension $d\ge1$ previous results so far limited to dimension $d=1$, and to the continuum setting with strong correlations recent results in the discrete iid case.