On Spatial Point Processes with Uniform Births and Deaths by Random Connection

This paper is focused on a class of spatial birth and death process of the Euclidean space where the birth rate is constant and the death rate of a given point is the shot noise created at its location by the other points of the current configuration for some response function $f$. An equivalent view point is that each pair of points of the configuration establishes a random connection at an exponential time determined by $f$, which results in the death of one of the two points. We concentrate on space-motion invariant processes of this type. Under some natural conditions on $f$, we construct the unique time-stationary regime of this class of point processes by a coupling argument. We then use the birth and death structure to establish a hierarchy of balance integral relations between the factorial moment measures. Finally, we show that the time-stationary point process exhibits a certain kind of repulsion between its points that we call $f$-repulsion.