74,541 research outputs found

### Fractional diffusion in Gaussian noisy environment

We study the fractional diffusion in a Gaussian noisy environment as
described by the fractional order stochastic partial equations of the following
form: $D_t^\alpha u(t, x)=\textit{B}u+u\cdot W^H$, where $D_t^\alpha$ is the
fractional derivative of order $\alpha$ 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 $W^H$ a 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

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

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

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
$H<1/2$.Comment: 31 page

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