PUMA
Istituto di Scienza e Tecnologie dell'Informazione     
Pagnini G., Paradisi P. A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation. In: Fractional Calculus and Applied Analysis, vol. 19 (2) pp. 408 - 440. De Gruyter, 2016.
 
 
Abstract
(English)
The stochastic solution with Gaussian stationary increments is established for the symmetric space-time fractional diffusion equation when 0 < β < α ≤ 2, where 0 < β ≤ 1 and 0 < α ≤ 2 are the fractional derivation orders in time and space, respectively. This solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with respect to previous approaches such as the scaling limit of the continuous time random walk, the parametric subordination and the subordinated Langevin equation. A new integral representation formula for the fundamental solution of the space-time fractional diffusion equation is firstly derived. It is then shown that, in the symmetric case, a stochastic solution can be obtained by a Gaussian process with stationary increments and with a random wideness scale variable distributed according to an arrangement of two extremal Levy stable densities. This stochastic solution is self-similar with stationary increments and uniquely defined in a statistical sense by the mean and the covariance structure.
URL: http://https://www.degruyter.com/view/j/fca.2016.19.issue-2/fca-2016-0022/fca-2016-0022.xml
DOI: 10.1515/fca-2016-0022
Subject Anomalous diffusion
Fractional diffusion equation
Gaussian processes
Self-similar stochastic process
Signal processing
G.3 PROBABILITY AND STATISTICS
I.5.4 PATTERN RECOGNITION. Applications
60G20 Probability theory and stochastic processes. Generalized stochastic processes
60G22 Probability theory and stochastic processes. Fractional processes, including fractional Brownian motion
82C31 Statistical mechanics, structure of matter. Stochastic methods (Fokker-Planck, Langevin, etc.)


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