Run-and-Tumble Motion: The Role of Reversibility

We study a model of active particles that perform simple random walk and on top have preferred direction determined by an internal state which is modelled stationary Markov process. First we calculate the limiting diffusion coefficient. Then show `active part' coefficient in some sense maximal for reversible processes. Further, obtain large deviations principle particle terms rate function empirical process corresponding to Again free energy are (pointwise) optimal Finally, case with two states, Fourier-Laplace transform distribution, moment generating can be computed explicitly. Along way provide several examples.