The Annals of Probability

Periodic homogenization with an interface: The multi-dimensional case

Martin Hairer and Charles Manson

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Abstract

We consider a diffusion process with coefficients that are periodic outside of an “interface region” of finite thickness. The question investigated in this article is the limiting long time/large scale behavior of such a process under diffusive rescaling. It is clear that outside of the interface, the limiting process must behave like Brownian motion, with diffusion matrices given by the standard theory of homogenization. The interesting behavior therefore occurs on the interface. Our main result is that the limiting process is a semimartingale whose bounded variation part is proportional to the local time spent on the interface. The proportionality vector can have nonzero components parallel to the interface, so that the limiting diffusion is not necessarily reversible. We also exhibit an explicit way of identifying its parameters in terms of the coefficients of the original diffusion.

Similarly to the one-dimensional case, our method of proof relies on the framework provided by Freidlin and Wentzell [Ann. Probab. 21 (1993) 2215–2245] for diffusion processes on a graph in order to identify the generator of the limiting process.

Article information

Source
Ann. Probab. Volume 39, Number 2 (2011), 648-682.

Dates
First available in Project Euclid: 25 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1298669176

Digital Object Identifier
doi:10.1214/10-AOP564

Mathematical Reviews number (MathSciNet)
MR2789509

Zentralblatt MATH identifier
1217.60044

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Keywords
Periodic homogenization interface skew Brownian motion local time

Citation

Hairer, Martin; Manson, Charles. Periodic homogenization with an interface: The multi-dimensional case. Ann. Probab. 39 (2011), no. 2, 648--682. doi:10.1214/10-AOP564. https://projecteuclid.org/euclid.aop/1298669176


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References

  • [1] Allaire, G. and Amar, M. (1999). Boundary layer tails in periodic homogenization. ESAIM Control Optim. Calc. Var. 4 209–243 (electronic).
  • [2] Ben Arous, G. and Černý, J. (2007). Scaling limit for trap models on ℤd. Ann. Probab. 35 2356–2384.
  • [3] Bahlali, K., Elouaflin, A. and Pardoux, E. (2009). Homogenization of semilinear PDEs with discontinuous averaged coefficients. Electron. J. Probab. 14 477–499.
  • [4] Bensoussan, A., Lions, J.-L. and Papanicolaou, G. (1978). Asymptotic Analysis for Periodic Structures. Studies in Mathematics and Its Applications 5. North-Holland, Amsterdam.
  • [5] Benchérif-Madani, A. and Pardoux, É. (2005). Homogenization of a diffusion with locally periodic coefficients. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 363–392. Springer, Berlin.
  • [6] Bogachev, V. I. (2007). Measure Theory, Vol. I, II. Springer, Berlin.
  • [7] Bass, R. F. and Pardoux, É. (1987). Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Related Fields 76 557–572.
  • [8] Borodin, A. N. and Salminen, P. (1996). Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel.
  • [9] Dellacherie, C. and Meyer, P.-A. (1983). Probabilités et Potentiel. Chapitres IX à XI, Revised ed. Hermann, Paris.
  • [10] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229. Cambridge Univ. Press, Cambridge.
  • [11] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [12] Freidlin, M. I. and Wentzell, A. D. (1993). Diffusion processes on graphs and the averaging principle. Ann. Probab. 21 2215–2245.
  • [13] Freidlin, M. I. and Wentzell, A. D. (2006). Long-time behavior of weakly coupled oscillators. J. Stat. Phys. 123 1311–1337.
  • [14] Gŕard-Varet, D. and Masmoudi, N. (2008). Homogenization in polygonal domains. Preprint, Paris 7 and NYU.
  • [15] Hairer, M. (2009). Ergodic properties for a class of non-Markovian processes. In Trends in Stochastic Analysis. London Math. Soc. Lecture Note Ser. 353 65–98. Cambridge Univ. Press, Cambridge.
  • [16] Has’minskiĭ, R. Z. (1960). Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. Teor. Verojatnost. i Primenen. 5 196–214.
  • [17] Hairer, M. and Manson, C. (2010). Periodic homogenization with an interface: The one-dimensional case. Stochastic Process. Appl. 120 1589–1605.
  • [18] Khasminskii, R. and Krylov, N. (2001). On averaging principle for diffusion processes with null-recurrent fast component. Stochastic Process. Appl. 93 229–240.
  • [19] Lejay, A. (2006). On the constructions of the skew Brownian motion. Probab. Surv. 3 413–466 (electronic).
  • [20] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • [21] Olla, S. (1994). Lectures on Homogenization of Diffusion Processes in Random Fields. Publications de l’École Doctorale, École Polytechnique.
  • [22] Olla, S. and Siri, P. (2004). Homogenization of a bond diffusion in a locally ergodic random environment. Stochastic Process. Appl. 109 317–326.
  • [23] Pavliotis, G. A. and Stuart, A. M. (2008). Multiscale Methods: Averaging and Homogenization. Texts in Applied Mathematics 53. Springer, New York.
  • [24] Papanicolaou, G. C. and Varadhan, S. R. S. (1981). Boundary value problems with rapidly oscillating random coefficients. In Random Fields, Vol. I, II (Esztergom, 1979). Colloquia Mathematica Societatis János Bolyai 27 835–873. North-Holland, Amsterdam.
  • [25] Rhodes, R. (2009). Diffusion in a locally stationary random environment. Probab. Theory Related Fields 143 545–568.
  • [26] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [27] Seidler, J. (2001). A note on the strong Feller property. Unpublished lecture notes.
  • [28] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer, Berlin.