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 $\mathbb{R}$. 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 $x_k^0=X_k(0)\in \mathbb{R}$ 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.

23 pages

References in corpus (1)

• Statistical analysis of the inhomogeneous telegrapher's process