The Annals of Probability

Homogenization of a singular random one-dimensional PDE with time-varying coefficients

Étienne Pardoux and Andrey Piatnitski

Full-text: Open access


In this paper we study the homogenization of a nonautonomous parabolic equation with a large random rapidly oscillating potential in the case of one-dimensional spatial variable. We show that if the potential is a statistically homogeneous rapidly oscillating function of both temporal and spatial variables, then, under proper mixing assumptions, the limit equation is deterministic, and convergence in probability holds. To the contrary, for the potential having a microstructure only in one of these variables, the limit problem is stochastic, and we only have convergence in law.

Article information

Ann. Probab., Volume 40, Number 3 (2012), 1316-1356.

First available in Project Euclid: 4 May 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 80M40: Homogenization 60H25: Random operators and equations [See also 47B80] 74Q10: Homogenization and oscillations in dynamical problems

Stochastic homogenization random operator large potential


Pardoux, Étienne; Piatnitski, Andrey. Homogenization of a singular random one-dimensional PDE with time-varying coefficients. Ann. Probab. 40 (2012), no. 3, 1316--1356. doi:10.1214/11-AOP650.

Export citation


  • [1] Bal, G. (2010). Homogenization with large spatial random potential. Multiscale Model. Simul. 8 1484–1510.
  • [2] Barlow, M. T. and Yor, M. (1982). Semimartingale inequalities via the Garsia–Rodemich–Rumsey lemma, and applications to local times. J. Funct. Anal. 49 198–229.
  • [3] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [4] Diop, M. A., Iftimie, B., Pardoux, E. and Piatnitski, A. L. (2006). Singular homogenization with stationary in time and periodic in space coefficients. J. Funct. Anal. 231 1–46.
  • [5] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [6] Ibragimov, I. A. and Has’minskiĭ, R. Z. (1981). Statistical Estimation: Asymptotic Theory. Appl. Math. 16. Springer, New York.
  • [7] Iftimie, B., Pardoux, É. and Piatnitski, A. (2008). Homogenization of a singular random one-dimensional PDE. Ann. Inst. Henri Poincaré Probab. Stat. 44 519–543.
  • [8] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • [9] Nualart, D. and Pardoux, É. (1988). Stochastic calculus with anticipating integrands. Probab. Theory Related Fields 78 535–581.
  • [10] Pardoux, E. and Piatnitski, A. (2006). Homogenization of a singular random one dimensional PDE. In Multi Scale Problems and Asymptotic Analysis. GAKUTO Internat. Ser. Math. Sci. Appl. 24 291–303. Gakkōtosho, Tokyo.
  • [11] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.