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Fractional Diffusion in Gaussian Noisy Environment
Hu, Guannan Hu, Yaozhong
Hu, Guannan
Hu, Yaozhong
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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(α)tu(t,x)=Bu+u⋅W˙H, where D(α)t is the Caputo fractional derivative of order α∈(0,1) with respect to the time variable t, B is a second order elliptic operator with respect to the space variable x∈Rd and W˙H a time homogeneous fractional Gaussian noise of Hurst parameter H=(H1,⋯,Hd). We obtain conditions satisfied by α and H, so that the square integrable solution u exists uniquely.
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2015-03-31
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MDPI
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Hu_2015.pdf
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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
Citation
Hu, G.; Hu, Y. Fractional Diffusion in Gaussian Noisy Environment. Mathematics 2015, 3, 131-152. https://doi.org/10.3390/math3020131