2016
Journal article
Open Access
A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
Pagnini G., Paradisi P.
Mathematical Physics (math-ph) 60G22 60G20 Mathematical Physics Anomalous diffusion 26A33 FOS: Mathematics Fractional diffusion equation 82C31 Statistical Mechanics (cond-mat.stat-mech) FOS: Physical sciences Probability (math.PR) Analysis Gaussian processes Mathematics - Probability Self-similar stochastic process Signal processing Condensed Matter - Statistical Mechanics Applied Mathematics
The stochastic solution with Gaussian stationary increments is established for the symmetric space-time fractional diffusion equation when 0 < ? < ? <= 2, where 0 < ? <= 1 and 0 < ? <= 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 Lévy 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. Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental solution of the symmetric space-time fractional diffusion equation.
Source: Fractional Calculus & Applied Analysis (Online) 19 (2016): 408–440. doi:10.1515/fca-2016-0022
Publisher: Institute of Mathematics & Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
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BibTeX entry
@article{oai:it.cnr:prodotti:368816,
title = {A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation},
author = {Pagnini G. and Paradisi P.},
publisher = {Institute of Mathematics \& Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria},
doi = {10.1515/fca-2016-0022 and 10.48550/arxiv.1603.05300},
journal = {Fractional Calculus \& Applied Analysis (Online)},
volume = {19},
pages = {408–440},
year = {2016}
}CNR authorsLaboratories
0000-0002-1036-4583CNR ExploRA
ISTI Repository
DOI
ISTI Repository
DOI
10.1515/fca-2016-0022
10.48550/arxiv.1603.05300
arXiv.org e-Print Archive
Fractional Calculus and Applied Analysis
www.degruyter.com