  We study the stochastic dynamics of a particle with two distinct motility
states. Each one is characterized by two parameters: one represents the average
speed and the other represents the persistence quantifying the tendency to
maintain the current direction of motion. We consider a run-and-tumble process,
which is a combination of an active fast motility mode (persistent motion) and
a passive slow mode (diffusion). Assuming stochastic transitions between the
two motility states, we derive an analytical expression for the time evolution
of the mean square displacement. The interplay of the key parameters and the
initial conditions as for instance the probability of initially starting in the
run or tumble state leads to a variety of transient regimes of anomalous
transport on different time scales before approaching the asymptotic diffusive
dynamics. We estimate the crossover time to the long-term diffusive regime and
prove that the asymptotic diffusion constant is independent of initially
starting in the run or tumble state.
