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Probabilistic representation of fundamental solutions to $\frac{\partial u}{\partial t} = κ_m \frac{\partial^m u}{\partial x^m}$


 
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1. Title Title of document Probabilistic representation of fundamental solutions to $\frac{\partial u}{\partial t} = κ_m \frac{\partial^m u}{\partial x^m}$
 
2. Creator Author's name, affiliation, country Enzo Orsingher; Sapienza University of Rome; Italy
 
2. Creator Author's name, affiliation, country Mirko D'Ovidio; Sapienza University of Rome; Italy
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Pseudo-process; higher-order heat equation; Airy functions; Cauchy distribution; stable laws; fractional diffusion equations
 
3. Subject Subject classification 60G52; 35C05
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2012-07-30
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1885
 
10. Identifier Digital Object Identifier 10.1214/ECP.v17-1885
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 17
 
12. Language English=en en
 
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