The explicit probability distribution of the sum of two telegraph processes

Abstract

We consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and $X_2(t)$ on the real line $\Bbb R$, both developing with finite constant speed $c>0$, that, at the initial time instant $t=0$, simultaneously start from the origin $0\in\Bbb R$ and whose evolutions are controlled by two independent homogeneous Poisson processes of the same rate $\lambda>0$. Closed-form expressions for the transition density $p(x,t)$ and the probability distribution function $\Phi(x,t)=\text{Pr} \{ S(t)<x \}, \; x\in\Bbb R, \; t>0,$ of the sum $S(t)=X_1(t)+X_2(t)$ of these processes at arbitrary time instant $t>0$, are obtained. It is also proved that the shifted time derivative $g(x,t)=(\partial/\partial t+2\lambda)p(x,t)$ satisfies the Goldstein-Kac telegraph equation with doubled parameters $2c$ and $2\lambda$. From this fact it follows that $p(x,t)$ solves a third-order hyperbolic partial differential equation, but is not its fundamental solution. The general case is also discussed.