### Abstract

This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well known that the corrector to homogenization, i.e., the difference between the random solution and the homogenized solution, converges in distribution to a Gaussian process when the correlations in the random medium are sufficiently short-range. Moreover, the limiting process may be written as a stochastic integral with respect to standard Brownian motion. We generalize the result to a large class of processes with long-range correlations. In this setting, the corrector also converges to a Gaussian random process, which has an interpretation as a stochastic integral with respect to fractional Brownian motion. Moreover, we show that the longer the range of the correlations, the larger is the amplitude of the corrector. Derivations are based on a careful analysis of random oscillatory integrals of processes with long-range correlations. We also make use of the explicit expressions for the solutions to the one-dimensional elliptic equation.

Original language | English (US) |
---|---|

Pages (from-to) | 1-26 |

Number of pages | 26 |

Journal | Asymptotic Analysis |

Volume | 59 |

Issue number | 1-2 |

DOIs | |

State | Published - 2008 |

Externally published | Yes |

### Fingerprint

### Keywords

- Central limit
- Gaussian processes
- Homogenization
- Long-range memory effects
- Partial differential equations with random coefficients

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Asymptotic Analysis*,

*59*(1-2), 1-26. https://doi.org/10.3233/ASY-2008-0890

**Random integrals and correctors in homogenization.** / Bal, Guillaume; Garnier, Josselin; Motsch, Sebastien; Perrier, Vincent.

Research output: Contribution to journal › Article

*Asymptotic Analysis*, vol. 59, no. 1-2, pp. 1-26. https://doi.org/10.3233/ASY-2008-0890

}

TY - JOUR

T1 - Random integrals and correctors in homogenization

AU - Bal, Guillaume

AU - Garnier, Josselin

AU - Motsch, Sebastien

AU - Perrier, Vincent

PY - 2008

Y1 - 2008

N2 - This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well known that the corrector to homogenization, i.e., the difference between the random solution and the homogenized solution, converges in distribution to a Gaussian process when the correlations in the random medium are sufficiently short-range. Moreover, the limiting process may be written as a stochastic integral with respect to standard Brownian motion. We generalize the result to a large class of processes with long-range correlations. In this setting, the corrector also converges to a Gaussian random process, which has an interpretation as a stochastic integral with respect to fractional Brownian motion. Moreover, we show that the longer the range of the correlations, the larger is the amplitude of the corrector. Derivations are based on a careful analysis of random oscillatory integrals of processes with long-range correlations. We also make use of the explicit expressions for the solutions to the one-dimensional elliptic equation.

AB - This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well known that the corrector to homogenization, i.e., the difference between the random solution and the homogenized solution, converges in distribution to a Gaussian process when the correlations in the random medium are sufficiently short-range. Moreover, the limiting process may be written as a stochastic integral with respect to standard Brownian motion. We generalize the result to a large class of processes with long-range correlations. In this setting, the corrector also converges to a Gaussian random process, which has an interpretation as a stochastic integral with respect to fractional Brownian motion. Moreover, we show that the longer the range of the correlations, the larger is the amplitude of the corrector. Derivations are based on a careful analysis of random oscillatory integrals of processes with long-range correlations. We also make use of the explicit expressions for the solutions to the one-dimensional elliptic equation.

KW - Central limit

KW - Gaussian processes

KW - Homogenization

KW - Long-range memory effects

KW - Partial differential equations with random coefficients

UR - http://www.scopus.com/inward/record.url?scp=50249087368&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50249087368&partnerID=8YFLogxK

U2 - 10.3233/ASY-2008-0890

DO - 10.3233/ASY-2008-0890

M3 - Article

AN - SCOPUS:50249087368

VL - 59

SP - 1

EP - 26

JO - Asymptotic Analysis

JF - Asymptotic Analysis

SN - 0921-7134

IS - 1-2

ER -