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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
Mathematics 2015, 3(2), 131-152; https://doi.org/10.3390/math3020131
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)
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
MDPI and ACS Style

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

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