Abstract
By using stochastic calculus for pure jump martingales, we study a class of infinite-dimensional birth-and-death processes. A technique, based on a relative entropy condition, which is adopted from diffusion process theory, enables us to also handle the corresponding processes obtained by reversing the direction of time. The duality between the processes of forward and backward time, respectively, is considered for Markov processes, defined by a prescribed family of upward and downward jump rates. A new characterization is obtained of the probability measures, which are invariant for the stochastic evolution associated with a specific set of jump rates. It leads to the conclusion that, if phase transition occurs, then all measures, with a given set of local conditional distributions in common, are invariant provided one of them is.
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