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 previous results so far limited to dimension , and to the continuum setting with strong correlations recent results in the discrete i.i.d. case.
Funding Statement
Financial support of M. Duerinckx is acknowledged from the CNRS-Momentum program. Financial support of A. Gloria is acknowledged from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM 335410).
Acknowledgments
The authors thank Ivan Nourdin and Felix Otto for inspiring discussions.
M. Duerinckx is also affiliated with Laboratoire de Mathématique d’Orsay, CNRS, Université Paris-Saclay.
A. Gloria is also affiliated with Département de Mathématique, Université Libre de Bruxelles (ULB), and CNRS, Université de Paris.
Citation
Mitia Duerinckx. Julian Fischer. Antoine Gloria. "Scaling limit of the homogenization commutator for Gaussian coefficient fields." Ann. Appl. Probab. 32 (2) 1179 - 1209, April 2022. https://doi.org/10.1214/21-AAP1705
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