Nonlocal in-time telegraph equation and telegraph processes with random time

In this paper we study the properties of a non-markovian version of the telegraph process, whose non-markovian character comes from a nonlocal in-time evolution equation that is satisfied by its probability density function. In the first part of the paper, using the theory of Volterra integral equations, we obtain an explicit formula for its moments, and we prove that the Carleman condition is satisfied. This shows that the distribution of the process is uniquely determined by its moments. We also obtain an explicit formula for the moment generating function. In the second part of the paper, we prove that the distribution of this process coincides with the distribution of a process of the form T(|W(t)|) where T(t) is the classical telegraph process, and |W(t)| is a random time whose distribution is related to a nonlocal in-time version of the wave equation. To this end, we construct the probability density function via subordination from the distribution of the classic telegraph process. Our results exhibit a strong interplay between this type of processes and subdiffusion theory.