Modern Stochastics: Theory and Applications

Keywords: Anomalous diffusion

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Alternative probabilistic representations of Barenblatt-type solutions

Alessandro De Gregorio

Roberto Garra

https://doi.org/10.15559/20-VMSTA151

Pub. online: 23 Mar 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 1 (2020), pp. 97–112

Abstract

A general class of probability density functions

$u(x,t)=C{t^{-\alpha d}}{\left(1-{\left(\frac{\| x\| }{c{t^{\alpha }}}\right)^{\beta }}\right)_{+}^{\gamma }},\hspace{1em}x\in {\mathbb{R}^{d}},t>0,$

is considered, containing as particular case the Barenblatt solutions arising, for instance, in the study of nonlinear heat equations. Alternative probabilistic representations of the Barenblatt-type solutions

$u(x,t)$ are proposed. In the one-dimensional case, by means of this approach,

$u(x,t)$ can be connected with the wave propagation.

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