Probability distributions for the run-and-tumble models with variable speed and tumbling rate

In this paper we consider a telegraph equation with time-dependent coefficients, governing the persistent random walk of a particle moving on the line with a time-varying velocity c(t) and changing direction at instants distributed according to a non-stationary Poisson distribution with rate λ(t). We show that, under suitable assumptions, we are able to find the exact form of the probability distribution. We also consider the space-fractional counterpart of this model, finding the characteristic function of the related process. A conclusive discussion is devoted to the potential applications to run-and-tumble models.