Linear forms of the telegraph random processes driven by partial differential equations
Abstract
Consider n independent Goldstein-Kac telegraph processes X1(t), … ,Xn(t), \; n 2, \; t 0, on the real line R. Each the process Xk(t), \; k=1,…,n, describes a stochastic motion at constant finite speed ck>0 of a particle that, at the initial time instant t=0, starts from some initial point xk0=Xk(0)∈ R and whose evolution is controlled by a homogeneous Poisson process Nk(t) of rate λk>0. The governing Poisson processes Nk(t), \; k=1,…,n, are supposed to be independent as well. Consider the linear form of the processes X1(t), … ,Xn(t), \; n 2, defined by L(t) = Σk=1n ak Xk(t) , where ak, \; k=1,…,n, are arbitrary real non-zero constant coefficients. We obtain a hyperbolic system of 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 2n 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. The weak convergence of L(t) to a homogeneous Wiener process, under Kac's scaling conditions, is proved. Initial-value problems for the transition densities of the sum and difference S(t)=X1(t) X2(t) of two independent telegraph processes with arbitrary parameters, are also posed.