Linear combinations of the telegraph random processes driven by partial differential equations

Abstract

Consider [Formula: see text] independent Goldstein–Kac telegraph processes [Formula: see text] on the real line [Formula: see text]. Each process [Formula: see text] describes a stochastic motion at constant finite speed [Formula: see text] of a particle that, at the initial time instant [Formula: see text], starts from some initial point [Formula: see text] and whose evolution is controlled by a homogeneous Poisson process [Formula: see text] of rate [Formula: see text]. The governing Poisson processes [Formula: see text] are supposed to be independent as well. Consider the linear combination of the processes [Formula: see text] defined by [Formula: see text] where [Formula: see text] are arbitrary real nonzero constant coefficients. We obtain a hyperbolic system of [Formula: see text] first-order partial differential equations for the joint probability densities of the process [Formula: see text] and of the directions of motions at arbitrary time [Formula: see text]. From this system we derive a partial differential equation of order [Formula: see text] for the transition density of [Formula: see text] in the form of a determinant of a block matrix whose elements are the differential operators with constant coefficients. Initial-value problems for the transition densities of the sum and difference [Formula: see text] of two independent telegraph processes with arbitrary parameters, are also posed.