On Brownian motions with alternating drifts

Consider the model of random evolution on the real line consisting in a Brownian motion with alternating drift, where the random times separating consecutive reversals of the drift perform an alternating renewal process. This model arises as a suitable extension of the standard Brownian motion and of a motion at constant speed on the real line, whose direction is reversed at every event of a Poisson process. We obtain the probability law of the resulting stochastic process, with explicit expressions of the transition densities in the special case of exponentially distributed inter-renewal times.