Fractional Diffusion in Gaussian Noisy Environment

Guannan Hu and Yaozhong Hu
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Guannan Hu: Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA
Yaozhong Hu: Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA

Mathematics, 2015, vol. 3, issue 2, 1-22

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.

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2015
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