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2012 Probabilistic representation of fundamental solutions to $\frac{\partial u}{\partial t} = κ_m \frac{\partial^m u}{\partial x^m}$
Enzo Orsingher, Mirko D'Ovidio
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Electron. Commun. Probab. 17: 1-12 (2012). DOI: 10.1214/ECP.v17-1885

Abstract

For the fundamental solutions of heat-type equations of order $n$ we give a general stochastic representation in terms of damped oscillations with generalized gamma distributed parameters. By composing the pseudo-process $X_m$ related to the higher-order heat-type equation with positively skewed stable r.v.'s $T^j_{1/3}$, $j=1,2, ..., n$ we obtain genuine r.v.'s whose explicit distribution is given for $n=3$ in terms of Cauchy asymmetric laws. We also prove that $X_3(T^1_{1/3}(...(T^n_{(1/3)}(t))...))$ has a stable asymmetric law.

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Enzo Orsingher. Mirko D'Ovidio. "Probabilistic representation of fundamental solutions to $\frac{\partial u}{\partial t} = κ_m \frac{\partial^m u}{\partial x^m}$." Electron. Commun. Probab. 17 1 - 12, 2012. https://doi.org/10.1214/ECP.v17-1885

Information

Accepted: 30 July 2012; Published: 2012
First available in Project Euclid: 7 June 2016

zbMATH: 1272.35005
MathSciNet: MR2965747
Digital Object Identifier: 10.1214/ECP.v17-1885

Subjects:
Primary: 60G52
Secondary: 35C05

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