Exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity-jump processes

This paper examines a class of linear hyperbolic systems which generalizes the Goldstein–Kac model to an arbitrary finite number of speeds vivi with transition rates μijμij. Under the basic assumptions that the transition matrix is symmetric and irreducible, and the differences vi−vjvi−vj generate all the space, the system exhibits a large-time behavior described by a parabolic advection–diffusion equation. The main contribution is to determine explicit formulas for the asymptotic drift speed and diffusion matrix in term of the kinetic parameters vivi and μijμij, establishing a complete connection between microscopic and macroscopic coefficients. It is shown that the drift speed is the arithmetic mean of the velocities vivi. The diffusion matrix has a more complicate representation, based on the graph with vertices the velocities vivi and arcs weighted by the transition rates μijμij. The approach is based on an exhaustive analysis of the dispersion relation and on the application of a variant of the Kirchoff's matrix tree Theorem from graph theory.