Stochastic foundation of normal and anomalous Cattaneo-type transport

We investigate the connection of the Cattaneo equation and the stochastic continuous time random walk (CTRW) theory. We show that the velocity model in a CTRW scheme is suited to derive the standard Cattaneo equation, and allows, in principle, for a generalisation to anomalous transport. As a result for a broad waiting time distribution with diverging mean, we find a strong memory to the initial condition of the system: The ballistic behaviour subsists also for long times. Only if a characteristic waiting time exists, a non-ballistic, enhanced motion is found in the limit of long times. No transition to subdiffusion can be found.