Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

From averaging to homogenization in cellular flows – An exact description of the transition

Martin Hairer, Leonid Koralov, and Zsolt Pajor-Gyulai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider a two-parameter averaging-homogenization type elliptic problem together with the stochastic representation of the solution. A limit theorem is derived for the corresponding diffusion process and a precise description of the two-parameter limit behavior for the solution of the PDE is obtained.


Nous considérons un problème elliptique de type moyennisation / homogénisation à deux paramètres, en combinaison avec la représentation stochastique de la solution. Nous obtenons un théorème limite pour le processus de diffusion correspondant ainsi qu’une description précise du comportement limite de la solution de l’EDP.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1592-1613.

Received: 4 July 2014
Revised: 20 April 2015
Accepted: 26 May 2015
First available in Project Euclid: 17 November 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]

Local time Diffusion on graphs Averaging Homogenization


Hairer, Martin; Koralov, Leonid; Pajor-Gyulai, Zsolt. From averaging to homogenization in cellular flows – An exact description of the transition. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1592--1613. doi:10.1214/15-AIHP690.

Export citation


  • [1] G. Ben Arous and J. Černý. Scaling limit for trap models on $\mathbb{Z}^{d}$. Ann. Probab. 35 (6) (2007) 2356–2384.
  • [2] Y. Bakhtin. Noisy heteroclinic networks. Probab. Theory Related Fields 150 (2011) 1–42.
  • [3] E. Bolthausen and I. Goldsheid. Lingering random walks in random environment on a strip. Comm. Math. Phys. 278 (1) (2008) 253–288.
  • [4] A. Bensoussan, J. Lions and G. Papanicolaou. Asymptotic Analysis for Periodic Structures. AMS Chelsea Publishing, Providence, RI, 2011. (Corrected reprint of the 1978 original Asymptotic Analysis for Periodic Structures. Studies in Mathematics and Its Applications 5. North-Holland Publishing Co., Amsterdam. MR0503330)
  • [5] D. Dolgopyat and L. Koralov. Averaging of Hamiltonian flows with an ergodic component. Ann. Probab. 36 (6) (2008) 1999–2049.
  • [6] M. Freidlin and S.-J. Sheu. Diffusion processes on graphs: Stochastic differential equations, large deviation principle. Probab. Theory Related Fields 116 (2) (2000) 181–220.
  • [7] M. I. Freidlin and A. D. Wentzell. Diffusion processes on graphs and the averaging principle. Ann. Probab. 21 (4) (1993) 2215–2245.
  • [8] M. Freidlin and A. Wentzell. Random Perturbations of Dynamical Systems, 3rd edition. Grundlehren der Mathematischen Wissenschaften 260. Springer, Heidelberg, 2012.
  • [9] G. Iyer, T. Komorowski, A. Novikov and L. Ryzhik. From homogenization to averaging in cellular flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (2014) 957–983.
  • [10] G. Iyer and K. C. Zygalakis. Numerical studies of homogenization under a fast cellular flow. Multiscale Model. Simul. 10 (3) (2012) 1046–1058.
  • [11] L. Koralov. Random perturbations of 2-dimensional Hamiltonian flows. Probab. Theory Related Fields 129 (1) (2004) 37–62.
  • [12] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 53. Springer, New York, 1991.
  • [13] G. Pavliotis and A. Stuart. Multiscale Methods: Averaging and Homogenization. Texts in Applied Mathematics. Springer, New York, 2008.
  • [14] G. Pavliotis, A. Stuart and K. Zygalakis. Calculating effective diffusivities in the limit of vanishing molecular diffusion. J. Comput. Phys. 228 (4) (2009) 1030–1055.
  • [15] V. Zhikov, S. Kozlov and O. Oleĭnik. Homogenization of Differential Operators and Integral Functionals. Springer, Berlin, 1994.