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Mathematics 2015, 3(2), 131-152; doi:10.3390/math3020131

Fractional Diffusion in Gaussian Noisy Environment

Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA
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Academic Editor: Hari M. Srivastava
Received: 16 February 2015 / Accepted: 24 March 2015 / Published: 31 March 2015
(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)
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Abstract

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: \(D_t^{(\alpha)} u(t, x)=\textit{B}u+u\cdot \dot W^H\), where \(D_t^{(\alpha)}\) is the Caputo fractional derivative of order \(\alpha\in (0,1)\) with respect to the time variable \(t\), \(\textit{B}\) is a second order elliptic operator with respect to the space variable \(x\in\mathbb{R}^d\) and \(\dot W^H\) a time homogeneous fractional Gaussian noise of Hurst parameter \(H=(H_1, \cdots, H_d)\). We obtain conditions satisfied by \(\alpha\) and \(H\), so that the square integrable solution \(u\) exists uniquely. View Full-Text
Keywords: fractional derivative; fractional order stochastic heat equation; mild solution; time homogeneous fractional Gaussian noise; stochastic integral of the Itô type; multiple integral of the Itô type; chaos expansion; Fox's H-function; Green's functions fractional derivative; fractional order stochastic heat equation; mild solution; time homogeneous fractional Gaussian noise; stochastic integral of the Itô type; multiple integral of the Itô type; chaos expansion; Fox's H-function; Green's functions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Hu, G.; Hu, Y. Fractional Diffusion in Gaussian Noisy Environment. Mathematics 2015, 3, 131-152.

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