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

Consider n independent Goldstein-Kac telegraph processes X-1(t), ... , X n(t), n >= 2, t >= 0, on the real line R. Each process X-k(t), k = 1, ... , n, describes a stochastic motion at constant finite speed c(k) > 0 of a particle that, at the initial time instant t = 0, starts from some initial point x(k)(0) = X-k(0) is an element of R and whose evolution is controlled by a homogeneous Poisson process N-k(t) of rate lambda(k) > 0. The governing Poisson processes N-k(t), k = 1, ... , n, are supposed to be independent as well. Consider the linear combination of the processes X-1(t), ... , X-n(t), n >= 2, defined by L(t) = Sigma(n)(k=1) a(k) X-k(t), where ak, k = 1, ... , n, are arbitrary real nonzero constant coefficients. We obtain a hyperbolic system of 2(n) first-order partial differential equations for the joint probability densities of the process L(t) and of the directions of motions at arbitrary time t > 0. From this system we derive a partial differential equation of order 2(n) for the transition density of L(t) 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 S +/-(t) = X-1(t) +/- X-2(t) of two independent telegraph processes with arbitrary parameters, are also posed.