Electronic Journal of Probability

On generalized Gaussian free fields and stochastic homogenization

Yu Gu and Jean-Christophe Mourrat

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Gaussian free field Markov property stochastic homogenization corrector

Creative Commons Attribution 4.0 International License.


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

Export citation


  • [1] N. Berger and M. Biskup. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (1-2), 83–120 (2007).
  • [2] M. Biskup, M. Salvi, and T. Wolff. A central limit theorem for the effective conductance: linear boundary data and small ellipticity contrasts. Comm. Math. Phys., 328(2):701–731, 2014.
  • [3] P. Courrège. Sur la forme intégro-différentielle des opérateurs de $C^\infty _k$ dans $C$ satisfaisant au principe du maximum. In Séminaire de Théorie du Potentiel, Dirigé par M. Brelot, G. Choquet et J. Deny, 1965/66, tome 10, exposé 2, 1–38. Secrétariat mathématique, Paris, 1966.
  • [4] E. B. Dynkin. Markov processes and random fields. Bull. Amer. Math. Soc. (N.S.), 3(3):975–999, 1980.
  • [5] M. Fukushima, Y. Ōshima, and M. Takeda. Dirichlet forms and symmetric Markov processes, volume 19 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1994.
  • [6] G. Giacomin, S. Olla, and H. Spohn. Equilibrium fluctuations for $\nabla \varphi $ interface model. Ann. Probab. 29 (3), 1138–1172 (2001).
  • [7] A. Gloria and F. Otto. An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab., 39(3):779–856, 2011.
  • [8] Y. Gu and J.-C. Mourrat. Scaling limit of fluctuations in stochastic homogenization. Multiscale Model. Simul., 14(1):452–481, 2016.
  • [9] G. Kallianpur and V. Mandrekar. The Markov property for generalized Gaussian random fields. Ann. Inst. Fourier (Grenoble), 24(2):vi, 143–167, 1974. Colloque International sur les Processus Gaussiens et les Distributions Aléatoires (Colloque Internat. du CNRS, No. 222, Strasbourg, 1973).
  • [10] D. Koller and N. Friedman. Probabilistic graphical models. Adaptive Computation and Machine Learning. MIT Press, Cambridge, MA, 2009.
  • [11] T. Kolsrud. On the Markov property for certain Gaussian random fields. Probab. Theory Related Fields, 74(3):393–402, 1987.
  • [12] R. Künnemann. The diffusion limit for reversible jump processes on $\mathbb{Z} ^d$ with ergodic random bond conductivities. Comm. Math. Phys., 90(1):27–68, 1983.
  • [13] H. Künsch. Gaussian Markov random fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 26(1):53–73, 1979.
  • [14] P. Lévy. Processus Stochastiques et Mouvement Brownien. Suivi d’une note de M. Loève. Gauthier-Villars, Paris, 1948.
  • [15] H. P. McKean, Jr. Brownian motion with a several-dimensional time. Teor. Verojatnost. i Primenen., 8:357–378, 1963. Engl. transl. Theor. Probability Appl. 8:335–354, 1963.
  • [16] J. Miller. Fluctuations for the Ginzburg-Landau $\nabla \phi $ interface model on a bounded domain. Comm. Math. Phys., 308(3):591–639, 2011.
  • [17] G. M. Molčan. Some problems connected with the Brownian motion of Lévy. Teor. Verojatnost. i Primenen., 12:747–755, 1967. Engl. transl. Theor. Probability Appl. 12:682–690, 1967.
  • [18] J.-C. Mourrat and J. Nolen. Scaling limit of the corrector in stochastic homogenization. Preprint, arXiv:1502.07440 (2015).
  • [19] J.-C. Mourrat and F. Otto. Correlation structure of the corrector in stochastic homogenization. Ann. Probab., 44(5):3207–3233, 2016.
  • [20] A. Naddaf and T. Spencer. On homogenization and scaling limit of some gradient perturbations of a massless free field. Comm. Math. Phys. 183 (1), 55–84 (1997).
  • [21] G. C. Papanicolaou and S. R. S. Varadhan. Boundary value problems with rapidly oscillating random coefficients. In Random fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, pages 835–873. North-Holland, Amsterdam, 1981.
  • [22] L. D. Pitt. A Markov property for Gaussian processes with a multidimensional parameter. Arch. Rational Mech. Anal., 43:367–391, 1971.
  • [23] D. Preiss and R. Kotecký. Markoff property of generalized random fields. In Seventh Winter School on Abstract Analysis, pages 61–66. Czechoslovak Academy of Sciences, Mathematical Institute, Prague, 1979.
  • [24] M. Röckner. Generalized Markov fields and Dirichlet forms. Acta Appl. Math., 3(3):285–311, 1985.
  • [25] S. Sheffield. Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 (3-4), 521–541 (2007).