Time-fractional telegraph equations and telegraph processes with brownian time
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- Orsingher, E. & Beghin, L. Probab. Theory Relat. Fields (2004) 128: 141. doi:10.1007/s00440-003-0309-8
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We study the fundamental solutions to time-fractional telegraph equations of order 2α. We are able to obtain the Fourier transform of the solutions for any α and to give a representation of their inverse, in terms of stable densities. For the special case α=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.