Abstract
We consider a linear elliptic system in divergence form with random coefficients and study the random fluctuations of large-scale averages of the field and the flux of the solution operator. In the context of the random conductance model, we developed in a previous work a theory of fluctuations based on the notion of homogenization commutator: we proved that the two-scale expansion of this special quantity is accurate at leading order in the fluctuation scaling when averaged on large scales (as opposed to the two-scale expansion of the solution operator taken separately) and that the large-scale fluctuations of the field and the flux of the solution operator can be recovered from those of the commutator. This implies that the large-scale fluctuations of the commutator of the corrector drive all other large-scale fluctuations to leading order, which we refer to as the pathwise structure of fluctuations in stochastic homogenization. In the present contribution we extend this result in two directions: we treat continuum elliptic (possibly non-symmetric) systems and allow for strongly correlated coefficient fields (Gaussian-like with a covariance function that can display an arbitrarily slow algebraic decay at infinity). Our main result shows in this general setting that the two-scale expansion of the homogenization commutator is still accurate to leading order when averaged on large scales, which illustrates the robustness of the pathwise structure of fluctuations.
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Notes
In dimension \(d=1\), the homogenization commutator indeed simply takes the form \(\Xi (x)= {\bar{{\varvec{a}}}}(1 -\frac{{\bar{{\varvec{a}}}}}{{\varvec{a}}(x)})\), which is exactly local wrt \({\varvec{a}}\).
That is, \(\nabla \phi _i({\varvec{a}};\cdot +z)=\nabla \phi _i({\varvec{a}}(\cdot +z);\cdot )\) and \(\nabla \sigma _{ijk}({\varvec{a}};\cdot +z)=\nabla \sigma _{ijk}({\varvec{a}}(\cdot +z);\cdot )\) a.e. in \(\mathbb R^d\), for all shift vectors \(z\in {\mathbb {R}}^d\).
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Acknowledgements
The work of MD is supported by F.R.S.-FNRS through a Research Fellowship and by the CNRS-Momentum program. MD and AG acknowledge financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM 335410).
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Duerinckx, M., Gloria, A. & Otto, F. Robustness of the pathwise structure of fluctuations in stochastic homogenization. Probab. Theory Relat. Fields 178, 531–566 (2020). https://doi.org/10.1007/s00440-020-00983-w
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DOI: https://doi.org/10.1007/s00440-020-00983-w