Alternative probabilistic representations of Barenblatt-type solutions

Abstract

Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: Alternative probabilistic representations of Barenblatt-type solutions, Authors: Alessandro De Gregorio, Roberto Garra , 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.