Markov Infinitely-Divisible Stationary Time-Reversible Integer-Valued Processes

We characterize all stationary time reversible Markov processes whose finite dimensional marginal distributions (of all orders) are infinitely divisible– MISTI processes, for short. Aside from two degenerate cases (iid and constant), in both discrete and continuous time every such process with full support is a branching process with Poisson or Negative Binomial marginal univariate distributions and a specific bivariate distribution at pairs of times. As a corollary, we prove that all nondegenerate stationary integer valued processes constructed by the Markov thinning process fail to have infinitely divisible multivariate marginal distributions, except for the Poisson.