Velocity-jump processes with a finite number of speeds and their asymptotically parabolic nature

The paper examines a class of first order linear hyperbolic systems, proposed as a generalization of the Goldstein-Kac model for velocity-jump processes and determined by a finite number of speeds and corresponding transition rates. It is shown that the large-time behavior is described by a corresponding scalar diffusive equation of parabolic type, defined by a diffusion matrix for which an explicit formula is given. Such representation takes advantage of a variant of the Kirchoff's matrix tree Theorem applied to the graph associated to the system and given by considering the velocities as verteces and the transition rates as weights of the arcs.