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    Fractional diffusion in Gaussian noisy environment

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    We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic partial equations of the following form: Dtαu(t,x)=Bu+uWHD_t^\alpha u(t, x)=\textit{B}u+u\cdot W^H, where DtαD_t^\alpha is the fractional derivative of order α\alpha with respect to the time variable tt, B\textit{B} is a second order elliptic operator with respect to the space variable xRdx\in\mathbb{R}^d, and WHW^H a fractional Gaussian noise of Hurst parameter H=(H1,,Hd)H=(H_1, \cdots, H_d). We obtain conditions satisfied by α\alpha and HH so that the square integrable solution uu exists uniquely

    Generalized Wiener Process and Kolmogorov's Equation for Diffusion induced by Non-Gaussian Noise Source

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    We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker-Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov-Feller equation for discontinuous Markovian processes, and the fractional Fokker-Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.Comment: 8 pages. in press, Fluctuation and Noise Letters, 200

    Linear SPDEs with harmonizable noise

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    Using tools from the theory of random fields with stationary increments, we introduce a new class of processes which can be used as a model for the noise perturbing an SPDE. This type of noise (called harmonizable) is not necessarily Gaussian, but it includes the spatially homogeneous Gaussian noise introduced in Dalang (1999), and the fractional noise considered in Balan and Tudor (2010). We derive some general conditions for the existence of a random field solution of a linear SPDE with harmonizable noise, under some mild conditions imposed on the Green function of the differential operator which appears in this equation. This methodology is applied to the study of the heat and wave equations (possibly replacing the Laplacian by one of its fractional powers), extending in this manner the results of Balan and Tudor (2010) to the case H<1/2H<1/2.Comment: 31 page