Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Strong convergence to the homogenized limit of parabolic equations with random coefficients

Authors: Joseph G. Conlon and Arash Fahim
Journal: Trans. Amer. Math. Soc. 367 (2015), 3041-3093
MSC (2010): Primary 81T08, 82B20, 35R60, 60J75
Published electronically: December 10, 2014
MathSciNet review: 3314801
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients and their convergence to solutions of a homogenized equation. It has previously been shown that if the random environment is translational invariant and ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized parabolic PDE. In this paper point-wise estimates are obtained on the difference between the averaged solution to the random equation and the solution to the homogenized equation for certain random environments which are strongly mixing.

References [Enhancements On Off] (What's this?)

  • [1] C. Boldrighini, R. A. Minlos, and A. Pellegrinotti, Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive, Probab. Theory Related Fields 129 (2004), no. 1, 133-156. MR 2052866 (2005e:60048),
  • [2] Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366-389. MR 0450480 (56 #8774)
  • [3] Leo Breiman, Probability, Classics in Applied Mathematics, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. Corrected reprint of the 1968 original. MR 1163370 (93d:60001)
  • [4] Luis A. Caffarelli and Panagiotis E. Souganidis, Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media, Invent. Math. 180 (2010), no. 2, 301-360. MR 2609244 (2011c:35041),
  • [5] A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139. MR 0052553 (14,637f)
  • [6] G. A. Chechkin, A. L. Piatnitski, and A. S. Shamaev, Homogenization, Methods and applications, Translations of Mathematical Monographs, vol. 234, American Mathematical Society, Providence, RI, 2007. Translated from the 2007 Russian original by Tamara Rozhkovskaya. MR 2337848 (2008j:35013)
  • [7] Joseph G. Conlon, PDE with random coefficients and Euclidean field theory, J. Statist. Phys. 116 (2004), no. 1-4, 933-958. MR 2082201 (2005e:81132),
  • [8] Joseph G. Conlon, Green's functions for elliptic and parabolic equations with random coefficients. II, Trans. Amer. Math. Soc. 356 (2004), no. 10, 4085-4142 (electronic). MR 2058840 (2005f:35341),
  • [9] Joseph G. Conlon and Ali Naddaf, On homogenization of elliptic equations with random coefficients, Electron. J. Probab. 5 (2000), no. 9, 58 pp. (electronic). MR 1768843 (2002j:35328),
  • [10] Joseph G. Conlon and Thomas Spencer, Strong convergence to the homogenized limit of elliptic equations with random coefficients, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1257-1288. MR 3145731,
  • [11] T. Delmotte and J.-D. Deuschel, On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to $ \nabla \phi $ interface model, Probab. Theory Related Fields 133 (2005), no. 3, 358-390. MR 2198017 (2007a:60057),
  • [12] Dmitry Dolgopyat, Gerhard Keller, and Carlangelo Liverani, Random walk in Markovian environment, Ann. Probab. 36 (2008), no. 5, 1676-1710. MR 2440920 (2009f:60124),
  • [13] Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR 1609153 (98m:60001)
  • [14] T. Funaki and H. Spohn, Motion by mean curvature from the Ginzburg-Landau $ \nabla \phi $ interface model, Comm. Math. Phys. 185 (1997), no. 1, 1-36. MR 1463032 (98f:60206),
  • [15] Antoine Gloria and Felix Otto, An optimal variance estimate in stochastic homogenization of discrete elliptic equations, Ann. Probab. 39 (2011), no. 3, 779-856. MR 2789576 (2012j:35018),
  • [16] Mathieu Gourcy and Liming Wu, Logarithmic Sobolev inequalities of diffusions for the $ L^2$ metric, Potential Anal. 25 (2006), no. 1, 77-102. MR 2238937 (2008h:60214),
  • [17] Bernard Helffer and Johannes Sjöstrand, On the correlation for Kac-like models in the convex case, J. Statist. Phys. 74 (1994), no. 1-2, 349-409. MR 1257821 (95g:82022),
  • [18] B. Frank Jones Jr., A class of singular integrals, Amer. J. Math. 86 (1964), 441-462. MR 0161099 (28 #4308)
  • [19] S. M. Kozlov, Averaging of random structures, Dokl. Akad. Nauk SSSR 241 (1978), no. 5, 1016-1019 (Russian). MR 510894 (80e:60078)
  • [20] S. M. Kozlov, The averaging method and walks in inhomogeneous environments, Uspekhi Mat. Nauk 40 (1985), no. 2(242), 61-120, 238 (Russian). MR 786087 (87b:60104)
  • [21] C. Landim, S. Olla, and H. T. Yau, Convection-diffusion equation with space-time ergodic random flow, Probab. Theory Related Fields 112 (1998), no. 2, 203-220. MR 1653837 (99j:35084),
  • [22] Per-Gunnar Martinsson and Gregory J. Rodin, Asymptotic expansions of lattice Green's functions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2027, 2609-2622. MR 1942800 (2003j:82019),
  • [23] Jean-Christophe Mourrat, Kantorovich distance in the martingale CLT and quantitative homogenization of parabolic equations with random coefficients, Probab. Theory Related Fields 160 (2014), no. 1-2, 279-314. MR 3256815,
  • [24] A. Naddaf and T. Spencer,
    Estimates on the variance of some homogenization problems,
    1998 preprint.
  • [25] David Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233 (2006j:60004)
  • [26] G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random fields, Vol. I, II (Esztergom, 1979) Colloq. Math. Soc. János Bolyai, vol. 27, North-Holland, Amsterdam, 1981, pp. 835-873. MR 712714 (84k:58233)
  • [27] William Parry, Topics in ergodic theory, Cambridge Tracts in Mathematics, vol. 75, Cambridge University Press, Cambridge, 2004. Reprint of the 1981 original. MR 2140546 (2005m:37003)
  • [28] Rémi Rhodes, On homogenization of space-time dependent and degenerate random flows, Stochastic Process. Appl. 117 (2007), no. 10, 1561-1585. MR 2353040 (2008j:60195),
  • [29] Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975. MR 0493420 (58 #12429b)
  • [30] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44 #7280)
  • [31] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972 (46 #4102)
  • [32] V. V. Yurinskiĭ, Averaging of symmetric diffusion in a random medium, Sibirsk. Mat. Zh. 27 (1986), no. 4, 167-180, 215 (Russian). MR 867870 (88e:35190)
  • [33] V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosifyan]. MR 1329546 (96h:35003b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 81T08, 82B20, 35R60, 60J75

Retrieve articles in all journals with MSC (2010): 81T08, 82B20, 35R60, 60J75

Additional Information

Joseph G. Conlon
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Arash Fahim
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
Address at time of publication: Department of Mathematics, Florida State University, Tallahassee, Florida 32306

Keywords: Euclidean field theory, PDE with random coefficients, homogenization
Received by editor(s): March 23, 2012
Received by editor(s) in revised form: November 1, 2012
Published electronically: December 10, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society