Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 1 (2016), 193-228.

Limiting distribution of elliptic homogenization error with periodic diffusion and random potential

Wenjia Jing

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Abstract

We study the limiting probability distribution of the homogenization error for second order elliptic equations in divergence form with highly oscillatory periodic conductivity coefficients and highly oscillatory stochastic potential. The effective conductivity coefficients are the same as those of the standard periodic homogenization, and the effective potential is given by the mean. We show that the limiting distribution of the random part of the homogenization error, as random elements in proper Hilbert spaces, is Gaussian and can be characterized by the homogenized Green’s function, the homogenized solution and the statistics of the random potential. This generalizes previous results in the setting with slowly varying diffusion coefficients, and the current setting with fast oscillations in the differential operator requires new methods to prove compactness of the probability distributions of the random fluctuation.

Article information

Source
Anal. PDE, Volume 9, Number 1 (2016), 193-228.

Dates
Received: 8 June 2015
Revised: 8 October 2015
Accepted: 28 October 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843207

Digital Object Identifier
doi:10.2140/apde.2016.9.193

Mathematical Reviews number (MathSciNet)
MR3461305

Zentralblatt MATH identifier
1335.35313

Subjects
Primary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Keywords
periodic and stochastic homogenization random field probability measures on Hilbert space weak convergence of probability distributions

Citation

Jing, Wenjia. Limiting distribution of elliptic homogenization error with periodic diffusion and random potential. Anal. PDE 9 (2016), no. 1, 193--228. doi:10.2140/apde.2016.9.193. https://projecteuclid.org/euclid.apde/1510843207


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