Electronic Journal of Probability

Homogenization of Fractional Kinetic Equations with Random Initial Data

Gi-Ren Liu and Narn-Rueih Shieh

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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