We study the fundamental solutions to time-fractional telegraph equations of order $2\alpha$.We are able to obtain the Fourier transform of the solutions for any and to give a representation of their inverse, in terms of stable densities. For the special case $\alpha=1/2$, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order $1/2$ with respect to time.
Time-fractional telegraph equations and telegraph process with Brownian time / Orsingher, Enzo; Beghin, Luisa. - In: PROBABILITY THEORY AND RELATED FIELDS. - ISSN 0178-8051. - STAMPA. - 128:(2004), pp. 141-160. [10.1007/s00440-003-0309-8]