The Annals of Probability

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

Étienne Pardoux and Andrey Piatnitski

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Abstract

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

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

Dates
First available in Project Euclid: 4 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1336136065

Digital Object Identifier
doi:10.1214/11-AOP650

Mathematical Reviews number (MathSciNet)
MR2962093

Zentralblatt MATH identifier
1255.60108

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

Keywords
Stochastic homogenization random operator large potential

Citation

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. https://projecteuclid.org/euclid.aop/1336136065


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References

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