Electronic Journal of Probability

Homogenization of Fractional Kinetic Equations with Random Initial Data

Gi-Ren Liu and Narn-Rueih Shieh

Full-text: Open access

Abstract

We present the small-scale limits for the homogenization of a class of spatial-temporal random fields; the field arises from the solution of a certain fractional kinetic equation and also from that of a related two-equation system, subject to given random initial data. The space-fractional derivative of the equation is characterized by the composition of the inverses of the Riesz potential and the Bessel potential. We discuss the small-scale (the micro) limits, opposite to the well-studied large-scale limits, of such spatial-temporal random field. Our scaling schemes involve both the Riesz and the Bessel parameters, and also involve the rescaling in the initial data; our results are completely new-type scaling limits for such random fields.

Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 32, 962-980.

Dates
Accepted: 14 April 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464820201

Digital Object Identifier
doi:10.1214/EJP.v16-896

Mathematical Reviews number (MathSciNet)
MR2801457

Zentralblatt MATH identifier
1231.60046

Subjects
Primary: 60G60: Random fields
Secondary: 60H05: Stochastic integrals 62M15: Spectral analysis

Keywords
Homogenization Small-scale limits Riesz-Bessel fractional equation and system Random initial data Hermite expansion Multiple It^{o}-Wiener integral Long-range dependence

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Liu, Gi-Ren; Shieh, Narn-Rueih. Homogenization of Fractional Kinetic Equations with Random Initial Data. Electron. J. Probab. 16 (2011), paper no. 32, 962--980. doi:10.1214/EJP.v16-896. https://projecteuclid.org/euclid.ejp/1464820201


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