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.
Highlights
The purpose of this paper is to present new-type spatial-temporal scaling limits for the homogenization theory associated with a certain fractional kinetic equation and a related system of reaction-diffusion type formed from it:
We aim to study the small-scale limit of spatial-temporal random field arising from the solution of the fractional Riesz-Bessel equation and its associated system (1) with μ1 = μ2 = μ > 0, subject to given the random initial data u0 and v0, of which are independent and each one has a certain long-range dependence in its random structure
The opposite large-scale limits have been explored intensively in the above cited literatures; a notable point in such large-scale limits as in [1, 2] is that the Bessel parameter does not play its role; while in our small-scale limits, both the Riesz and the Bessel parameters play their roles, and we need the rescaling on the initial data
Summary
The purpose of this paper is to present new-type spatial-temporal scaling limits for the homogenization theory associated with a certain fractional kinetic equation and a related system of reaction-diffusion type formed from it:. We aim to study the small-scale limit of spatial-temporal random field arising from the solution of the fractional Riesz-Bessel equation and its associated system (1) with μ1 = μ2 = μ > 0, subject to given the random initial data u0 and v0, of which are independent and each one has a certain long-range dependence in its random structure. The opposite large-scale limits have been explored intensively in the above cited literatures; a notable point in such large-scale limits as in [1, 2] is that the Bessel parameter does not play its role; while in our small-scale limits, both the Riesz and the Bessel parameters play their roles, and we need the rescaling on the initial data Such small-scale spatial-temporal scaling limit is a completely new result in the homogenization theory of random fields, to our best knowledge. The proofs of main results are give in the final Section 5
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