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 x L that satisfy stochastic differential equations with a Poisson proce ss as one of the driving processes. The questions we address are that of existence and uniqueness of both stationary and non stationary solu tions, and convergence (either weakly or in variation) of the law of n on-stationary solutions to the stationary distribution. Theorems 1 and 3 (respectively, 2 and 4) provide sufficient conditions for these pro perties to hold and extend previous results of Kerstan (1964) (respect ively, Bremaud and Massoulie (1996)) to a more general framework. Theo rem 5 provides yet another set of sufficient conditions which, althoug h they apply only to a very specific instance of the general model, en able 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. (C) 1998 Elsevier Science B.V. All rights reserved.