Abstract
We study a generalization of the notion of Gaussian free field (GFF). Although the extension seems minor, we first show that a generalized GFF does not satisfy the spatial Markov property, unless it is a classical GFF. In stochastic homogenization, the scaling limit of the corrector is a possibly generalized GFF described in terms of an “effective fluctuation tensor” that we denote by $\mathsf{Q} $. We prove an expansion of $\mathsf{Q} $ in the regime of small ellipticity ratio. This expansion shows that the scaling limit of the corrector is not necessarily a classical GFF, and in particular does not necessarily satisfy the Markov property.
Highlights
We study a generalization of the notion of Gaussian free field (GFF)
The minimizer in H01(U ) of the mapping v → U (ξ + ∇v) · a(ξ + ∇v), where U is a large domain, offers a finite-volume approximation of the corrector. This minimization of a random quadratic functional suggests a parallel with gradient Gibbs measures, where a determinimistic functional of quadratic type is used to produce a probability measure via the Gibbs principle
Such gradient Gibbs measures are known to rescale to Gaussian free fields (GFFs), see [20, 6, 16]
Summary
We show that a generalized GFF satisfies the Markov property if and only if it is a classical GFF. For random fields on discrete graphs, the Markov property is equivalent to the locality of the “energy function” (in the Gaussian case, this is the Dirichlet form, and more generally, we mean the logarithm of the probability density, up to a constant). The results of [3] can be understood as follows: if the operator satisfies the maximum principle, it is the generator of a Lévy process This Lévy process must have the same scale invariance as Brownian motion, in view of the scaling properties of L. 1.1 Proof of Theorem 1.1 To sum up: using some mild regularity property of the field Φ defined by (1.1), we show that the Markov property implies that L is a local operator.
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