Electronic Journal of Probability

On generalized Gaussian free fields and stochastic homogenization

Yu Gu and Jean-Christophe Mourrat

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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.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 28, 21 pp.

Dates
Received: 24 January 2016
Accepted: 20 March 2017
First available in Project Euclid: 24 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1490320844

Digital Object Identifier
doi:10.1214/17-EJP51

Mathematical Reviews number (MathSciNet)
MR3629872

Zentralblatt MATH identifier
1359.60064

Subjects
Primary: 60G60: Random fields 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]

Keywords
Gaussian free field Markov property stochastic homogenization corrector

Rights
Creative Commons Attribution 4.0 International License.

Citation

Gu, Yu; Mourrat, Jean-Christophe. On generalized Gaussian free fields and stochastic homogenization. Electron. J. Probab. 22 (2017), paper no. 28, 21 pp. doi:10.1214/17-EJP51. https://projecteuclid.org/euclid.ejp/1490320844


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