Poisson process with different Brownian clocks

In this paper, different types of Poisson processes N subordinated to random time processes X, depending on Brownian motion, are analysed. In particular, the processes X considered here are the elastic Brownian motion B-el, the Brownian sojourn time on the positive half-line Gamma(+)(t), the first-passage time T-t (through the level t) of a Brownian motion, with or without drift, and the gamma-Bessel process R-gamma, for gamma > 0. In all these cases, we obtain the explicit state probability distributions p(k)(t) = Pr{N(X(t)) = k}, k >= 0, t > 0, their governing difference-differential equations and some moments. The connections among different models and, in particular, of N(R-gamma(t)) with birth and death processes are obtained and discussed. The models presented have probability distributions governed by higher order or fractional difference-differential equations. In the case of the Poisson process with Bessel times, equations with time-dependent coefficients appear.