Occupation times and Bessel densities

Consider a Markov process with countably many states. In order to find a one-state occupation time distribution, we use a combination of Fourier and Laplace transforms in the way that allows for the inversion of the Fourier transform. We derive a closed-form expression for the occupation time distribution in the case of a simple continuous-time random walk on ZZ and represent the one-state occupation density of a reversible process as a mixture of Bessel densities.