### 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.

Original language | English |
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Pages (from-to) | 962-980 |

Number of pages | 19 |

Journal | Electronic Journal of Probability |

Volume | 16 |

DOIs | |

Publication status | Published - 2011 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Electronic Journal of Probability*,

*16*, 962-980. https://doi.org/10.1214/EJP.v16-896

}

*Electronic Journal of Probability*, vol. 16, pp. 962-980. https://doi.org/10.1214/EJP.v16-896

**Homogenization of fractional kinetic equations with random initial data.** / Liu, Gi-Ren; Shieh, Narn Rueih.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Homogenization of fractional kinetic equations with random initial data

AU - Liu, Gi-Ren

AU - Shieh, Narn Rueih

PY - 2011/1/1

Y1 - 2011/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=79957469482&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957469482&partnerID=8YFLogxK

U2 - 10.1214/EJP.v16-896

DO - 10.1214/EJP.v16-896

M3 - Article

VL - 16

SP - 962

EP - 980

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -