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

Abstract

Consider $n$ independent Goldstein-Kac telegraph processes $X_1(t), \dots ,X_n(t), \; n\ge 2, \; t\ge 0,$ on the real line $\Bbb R$. Each the process $X_k(t), \; k=1,\dots,n,$ 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 $x_k^0=X_k(0)\in\Bbb R$ and whose evolution is controlled by a homogeneous Poisson process $N_k(t)$ of rate $\lambda_k>0$. The governing Poisson processes $N_k(t), \; k=1,\dots,n,$ are supposed to be independent as well. Consider the linear form of the processes $X_1(t), \dots ,X_n(t), \; n\ge 2,$ defined by $$L(t) = \sum_{k=1}^n a_k X_k(t) ,$$ where $a_k, \; k=1,\dots,n,$ 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 $2^n$ 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. 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 $S^{\pm}(t)=X_1(t) \pm X_2(t)$ of two independent telegraph processes with arbitrary parameters, are also posed.