Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix

Rémi Rhodes

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Abstract

This paper deals with homogenization of second order divergence form parabolic operators with locally stationary coefficients. Roughly speaking, locally stationary coefficients have two evolution scales: both an almost constant microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims to give a macroscopic approximation that takes into account the microscopic heterogeneities. This paper follows [Probab. Theory Related Fields (2009)] and improves this latter work by considering possibly degenerate diffusion matrices.

Résumé

Nous étudions l’homogénéisation d’opérateurs paraboliques du second ordre sous forme divergence à coefficients localement stationnaires. Ces coefficients présentent deux échelles d’évolution: une évolution microscopique presque constante et une évolution macroscopique régulière. La théorie de l’homogénéisation consiste à donner une approximation macroscopique de l’opérateur initial qui tient compte des hétérogénéités microscopiques. Cet article fait suite à [Probab. Theory Related Fields (2009)] et généralise ce dernier en considérant des matrices de diffusion pouvant dégénérer.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 4 (2009), 981-1001.

Dates
First available in Project Euclid: 6 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1257529888

Digital Object Identifier
doi:10.1214/08-AIHP190

Mathematical Reviews number (MathSciNet)
MR2572160

Zentralblatt MATH identifier
1207.60029

Subjects
Primary: 60F17: Functional limit theorems; invariance principles

Keywords
homogenization random medium degenerate diffusion locally stationary environment

Citation

Rhodes, Rémi. Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 4, 981--1001. doi:10.1214/08-AIHP190. https://projecteuclid.org/euclid.aihp/1257529888


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References

  • [1] A. Benchérif-Madani and E. Pardoux. Homogenization of a diffusion with locally periodic coefficients. In Séminaire de Probabilités XXXVIII 363–392. Lecture Notes in Math. 1857. Springer, Berlin, 2005.
  • [2] A. Bensoussan, J. L. Lions and G. Papanicolaou. Asymptotic Methods in Periodic Media. North Holland, Amsterdam, 1978.
  • [3] F. Delarue and R. Rhodes. Stochastic homogenization of quasilinear PDEs with a spatial degeneracy. Asymptot. Anal. 61 (2009) 61–90.
  • [4] M. Fukushima. Dirichlet Forms and Markov Processes. North-Holland, Amsterdam, 1980.
  • [5] M. Hairer and E. Pardoux. Homogenization of periodic linear degenerate PDEs. J. Funct. Anal. 255 2462–2487.
  • [6] J. Jacod and A. N. Shiryaev.Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaft 288. Springer, Berlin, 1987.
  • [7] V. V. Jikov, S. M. Kozlov and O. A. Oleinik.Homogenization of Differential Operators and Integral Functionals. Springer, Berlin, 1994.
  • [8] N. V. Krylov.Controlled Diffusion Processes. Springer, New York, 1980.
  • [9] Z. M. Ma and M. Röckner.Introduction to the Theory of (Nonsymmetric) Dirichlet Forms. Universitext. Springer, Berlin, 1992.
  • [10] S. Olla. Homogenization of diffusion processes in Random Fields. Cours de l’école doctorale, Ecole polytechnique, 1994. Available at http://www.ceremade.dauphine.fr/~olla/pubolla.html.
  • [11] S. Olla and P. Siri. Homogenization of a bond diffusion in a locally ergodic random environment. Stochastic Process. Appl. 109 (2004) 317–326.
  • [12] R. Rhodes. On homogenization of space time dependent random flows. Stochastic Process. Appl. 117 (2007) 1561–1585.
  • [13] R. Rhodes. Diffusion in a locally stationary random environment. Probab. Theory Related Fields 143 (2009) 545–568.
  • [14] D. Stroock. Diffusion semi-groups corresponding to uniformly elliptic divergence form operators. In Séminaires de Probabilités XXII 316–347. Lecture Notes in Math. 1321. Springer, Berlin, 1988. (Section B 35 (1999) 121–141.)
  • [15] L. Wu. Forward–Backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 35 (1999) 121–141.