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Probability law and flow function of Brownian motion driven by a generalized telegraph process. (English) Zbl 1322.60167

Summary: We consider a standard Brownian motion whose drift alternates randomly between a positive and a negative value, according to a generalized telegraph process. We first investigate the distribution of the occupation time, i.e. the fraction of time when the motion moves with positive drift. This allows to obtain explicitly the probability law and the flow function of the random motion. We discuss three special cases when the times separating consecutive drift changes have (i) exponential distribution with constant rates, (ii) Erlang distribution, and (iii) exponential distribution with linear rates. In conclusion, in view of an application in environmental sciences, we evaluate the density of a Wiener process with infinitesimal moments alternating at inverse Gaussian distributed random times.

MSC:

60J65 Brownian motion
60K15 Markov renewal processes, semi-Markov processes
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
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