Symmetric Telegraph Process on the Line

The symmetric Goldstein-Kac telegraph process describes the motion of a particle on the real line moving at some finite constant speed and alternating between the two possible directions of travel (positive or negative) at random homogeneous Poisson-paced time instants. We define the process and obtain the Kolmogorov equations for the joint probability densities of the particle’s position and its direction at arbitrary time instant. By combining these equations we derive the telegraph equation for the transition density of the motion. The characteristic function of the telegraph process is obtained as the solution of a respective Cauchy problem. The explicit form of the transition density of the process is given as a generalised function containing a singular and an absolutely continuous part. The convergence in distribution of the telegraph process to the homogeneous Brownian motion under Kac’s scaling condition is established and explicit formulae for the Laplace transforms of the transition density and of the characteristic function of the telegraph process are obtained.