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Time-fractional telegraph equations and telegraph processes with Brownian time. (English) Zbl 1049.60062
The Fourier transforms of the fundamental solutions of the time-fractional telegraph equation 2α u t 2α +2λ α u t α =c 2 2 u x 2 , 0<α1, with initial conditions u(x,0)=δ(x) for 0<α1/2 and u(x,0)=δ(x), u t (x,0)=0 for 1/2<α1, are determined in terms of Mittag-Leffler functions. In the case α=1/2, the fundamental solution is shown to be the distribution of a telegraph process with Brownian time. In the special case c,λ such that c 2 /λ1, this distribution turns out to be a law of the iterated Brownian motion.
MSC:
60H30Applications of stochastic analysis
33E12Mittag-Leffler functions and generalizations
42A61Probabilistic methods in Fourier analysis
26A33Fractional derivatives and integrals (real functions)
60G52Stable processes