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