The explicit probability distribution of the sum of two telegraph processes

الأصل · EN

We consider two independent Goldstein-Kac telegraph processes X₁(t) and X₂(t) on the real line R, both developing with finite constant speed c>0, that, at the initial time instant t=0, simultaneously start from the origin 0∈ R and whose evolutions are controlled by two independent homogeneous Poisson processes of the same rate λ>0. Closed-form expressions for the transition density p(x,t) and the probability distribution function Φ(x,t)=Pr { S(t)<x }, x∈ R, t>0, of the sum S(t)=X₁(t)+X₂(t) of these processes at arbitrary time instant t>0, are obtained. It is also proved that the shifted time derivative g(x,t)=(∂/∂ t+2λ)p(x,t) satisfies the Goldstein-Kac telegraph equation with doubled parameters 2c and 2λ. 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.