Scaling limit of the homogenization commutator for Gaussian coefficient 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 $\mathit{d}\ge 1$ previous results so far limited to dimension $\mathit{d}=1$, and to the continuum setting with strong correlations recent results in the discrete i.i.d. case.