Stability results for a general class of interacting point processes dynamics, and applications

The focus in this article is on point processes on a product space R×L that satisfy stochastic differential equations with a Poisson process as one of the driving processes. The questions we address are that of existence and uniqueness of both stationary and non stationary solutions, and convergence (either weakly or in variation) of the law of non-stationary solutions to the stationary distribution. Theorems 1 and 3 (respectively, 2 and 4) provide sufficient conditions for these properties to hold and extend previous results of Kerstan (1964) (respectively, Brémaud and Massoulié (1996)) to a more general framework. Theorem 5 provides yet another set of sufficient conditions which, although they apply only to a very specific instance of the general model, enable to drop the Lipschitz continuity condition made in Theorems 1–4. These results are then used to derive sufficient ergodicity conditions for models of (i) loss networks, (ii) spontaneously excitable random media, and (iii) stochastic neuron networks.