Strong law of large numbers for continuous random dynamical systems

We study a dynamical system generalizing continuous iterated function systems and stochastic differential equations disturbed by Poisson noise. The aim of this paper is to study stochastic processes whose paths follow deterministic dynamics between random times, jump times, at which they change their position randomly. Continuous random dynamical systems can be used as a description of many physical and biological phenomena. We prove the existence of an exponentially attractive invariant measure and the strong law of large numbers for continuous random dynamical systems. We illustrate the usefulness of our criteria for asymptotic stability by considering a general dd-dimensional model for the intracellular biochemistry of a generic cell with a probabilistic division hypothesis (see Lasota and Mackey, 1999).