Time-fractional telegraph equations and telegraph processes with Brownian time

We study the fundamental solutions to time-fractional telegraph equations of order 2alpha. We are able to obtain the Fourier transform of the solutions for any alpha 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.