Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the exit time and stochastic homogenization of isotropic diffusions in large domains

Benjamin Fehrman

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Stochastic homogenization is achieved for a class of elliptic and parabolic equations describing the lifetime, in large domains, of stationary diffusion processes in random environment which are small, statistically isotropic perturbations of Brownian motion in dimension at least three. Furthermore, the homogenization is shown to occur with an algebraic rate. Such processes were first considered in the continuous setting by Sznitman and Zeitouni (Invent. Math. 164 (2006) 455–567), upon whose results the present work relies strongly.


On effectue l’homogénéisation stochastique d’une certaine classe d’équations elliptiques et paraboliques. Ces équations décrivent la durée de vie, dans des domaines grands, de processus de diffusion stationnaire en environnement aléatoire qui sont des petites perturbations statistiquement isotropes du mouvement brownien, en dimension au moins trois. On démontre que l’homogénéisation a lieu à vitesse algébrique. De tels processus ont été étudiés dans un cadre continu en premier lieu par Snitzman et Zeitouni (Invent. Math. 164 (2006) 455–567), sur les résultats desquels le présent travail s’appuie fortement.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 720-755.

Received: 27 January 2017
Revised: 1 February 2018
Accepted: 28 February 2018
First available in Project Euclid: 14 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 35J25: Boundary value problems for second-order elliptic equations 35K20: Initial-boundary value problems for second-order parabolic equations 60H25: Random operators and equations [See also 47B80] 60J60: Diffusion processes [See also 58J65] 60K37: Processes in random environments

Diffusion processes in random environment Stochastic homogenization Elliptic boundary-value problem Parabolic boundary-value problem


Fehrman, Benjamin. On the exit time and stochastic homogenization of isotropic diffusions in large domains. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 720--755. doi:10.1214/18-AIHP896.

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  • [1] M. A. Armstrong. Basic Topology. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1979 original.
  • [2] E. Baur. An invariance principle for a class of non-ballistic random walks in random environment. Probab. Theory Related Fields 166 (1–2) (2016) 463–514.
  • [3] E. Baur and E. Bolthausen. Exit laws from large balls of (an)isotropic random walks in random environment. Ann. Probab. 43 (6) (2015) 2859–2948.
  • [4] E. Bolthausen and O. Zeitouni. Multiscale analysis of exit distributions for random walks in random environments. Probab. Theory Related Fields 138 (3–4) (2007) 581–645.
  • [5] J. Bricmont and A. Kupiainen. Random walks in asymmetric random environments. Comm. Math. Phys. 142 (2) (1991) 345–420.
  • [6] A. De Masi, P. A. Ferrari, S. Goldstein and W. D. Wick. An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55 (3–4) (1989) 787–855.
  • [7] B. Fehrman. Isotropic diffusions in random environment. Ph.D. thesis, Univ. Chicago, ProQuest LLC, Ann Arbor, MI, 2015.
  • [8] B. Fehrman. Exit laws of isotropic diffusions in random environment from large domains. Electron. J. Probab. 22 (63 (2017) 37.
  • [9] B. Fehrman. On the existence of an invariant measure for isotropic diffusions in random environment. Probab. Theory Related Fields 168 (1–2) (2017) 409–453.
  • [10] A. Friedman. Partial Differential Equations of Parabolic Type. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.
  • [11] S. M. Kozlov. The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 (2(242)) (1985) 61–120.
  • [12] J. W. Milnor. Topology from the Differentiable Viewpoint. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997. Based on notes by David W. Weaver, Revised reprint of the 1965 original.
  • [13] B. Øksendal. Stochastic Differential Equations: An Introduction with Applications, 6th edition. Universitext. Springer-Verlag, Berlin, 2003.
  • [14] S. Olla. Homogenization of diffusion processes in random fields. In C.M.A.P.-École Polytechnique, Palaiseau, 1994.
  • [15] H. Osada. Homogenization of diffusion processes with random stationary coefficients. In Probability Theory and Mathematical Statistics (Tbilisi, 1982) 507–517. Lecture Notes in Math. 1021. Springer, Berlin, 1983.
  • [16] G. C. Papanicolaou and S. R. S. Varadhan. Boundary value problems with rapidly oscillating random coefficients. In Random Fields, Vol. I, II (Esztergom, 1979) 835–873. Colloq. Math. Soc. János Bolyai 27. North-Holland, Amsterdam, 1981.
  • [17] G. C. Papanicolaou and S. R. S. Varadhan. Diffusions with random coefficients. In Statistics and Probability: Essays in Honor of C. R. Rao 547–552. North-Holland, Amsterdam, 1982.
  • [18] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer-Verlag, Berlin, 1999.
  • [19] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Classics in Mathematics. Springer-Verlag, Berlin, 2006. Reprint of the 1997 edition.
  • [20] A. S. Sznitman and O. Zeitouni. An invariance principle for isotropic diffusions in random environment. Invent. Math. 164 (3) (2006) 455–567.