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[We define the classic Goldstein-Kac telegraph process performed by a particle that moves on the real line with some finite constant speed and alternates between two possible directions of motion (positive or negative) at random homogeneous Poisson-paced time instants. We 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 absolutely continuous parts. The convergence in distribution of the telegraph process to the homogeneous Brownian motion under Kac’s scaling condition, is established. The explicit formulae for the Laplace transforms of the transition density and of the characteristic function of the telegraph process, are also obtained.]
Published: Oct 18, 2013
Keywords: Telegraph process; Kolmogorov equations; Transition density; Characteristic function; Rescaling; Laplace transform
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