Alternative probabilistic representations of Barenblatt-type solutions

Abstract

A general class of probability density functions \[u(x,t)=Ct^{-αd}\left (1-\left (\frac{\|x\|}{ct^α}\right )^β\right )_+^γ,\quad x\in \mathbb{R}^d,t>0,\] 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.

Published at https://doi.org/10.15559/20-VMSTA151 in the Modern Stochastics: Theory and Applications (https://vmsta.org/) by VTeX (http://www.vtex.lt/)