DYNAMICAL-SYSTEMS DEFINED ON POINT-PROCESSES

This paper introduces a new stochastic process in which the iterates o f a dynamical system evolving in discrete time coincide with the event s of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dyn amical map is linear. The flow of probability density functions genera ted by the stochastic process is analysed in detail, and the relations hip between the flow and the solutions of the linear Boltzmann equatio n is investigated. It is shown that the flow is a semigroup if and onl y if the point process defining the stochastic process is Poisson, the reby providing a new characterization of the Poisson process.