Approach to stationarity for birth and death on flows

This work considers the approach to stationarity of a Markov process of birth and death on stochastic flows. The process takes values in the space of counting measures, so its stationary states are point processes representing equilibrium distributions of points of unit mass. Stationary states of Poisson equilibria are explored in the context of a very natural balance among rates of birth, motion and killing. The rate of convergence to a Poisson equilibrium is shown to decay at the rate at which a certain transient diffusion escapes finite sets; similar rates apply in the general case for which equilibrium is shown necessarily to be a Cox process with directing measure hinged on the integral of a discounted one-point motion on the flow.