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Zbl 1049.60062
Orsingher, Enzo; Beghin, Luisa
Time-fractional telegraph equations and telegraph processes with Brownian time. (English)
[J] Probab. Theory Relat. Fields 128, No. 1, 141-160 (2004). ISSN 0178-8051; ISSN 1432-2064/e

The Fourier transforms of the fundamental solutions of the time-fractional telegraph equation $\frac{\partial^{2\alpha}u}{\partial t^{2\alpha} }+ 2\lambda \frac{\partial^\alpha u}{\partial t^\alpha}= c^2\frac{\partial^2u}{\partial x^2 }$, $0<\alpha\leq 1$, with initial conditions $u(x,0)=\delta(x)$ for $0<\alpha\leq 1/2$ and $u(x,0)=\delta(x)$, $u_t(x,0)=0$ for $1/2<\alpha\leq 1$, are determined in terms of Mittag-Leffler functions. In the case $\alpha= 1/2$, the fundamental solution is shown to be the distribution of a telegraph process with Brownian time. In the special case $c,\lambda\to\infty$ such that $c^2/\lambda\to 1$, this distribution turns out to be a law of the iterated Brownian motion.
[Ilya Pavlyukevitch (Berlin)]

MSC 2000:: *60H30 Appl. of stochastic analysis
33E12 Mittag-Leffler functions and generalizations
42A61 Probabilistic methods in Fourier analysis
26A33 Fractional derivatives and integrals (real functions)
60G52 Stable processes

Keywords: telegraph equation; fractional derivatives; stable laws; fractional heat; wave equation; iterated Brownian motion; Mittag-Leffler function

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