Section: New Results

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

With I. Norros (VTT Finland) and F. Mathieu (Bell Labs France), F. Baccelli has continued the line of thought on the geometry of Peer-to-Peer systems that was initiated in their Infocom 13 paper. This type of dynamics leads to 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. The research concentrated on space-motion invariant processes of this type. Under some natural conditions on $f$, one can construct the unique time-stationary regime of this class of point processes by a coupling argument. The birth and death structure can then be used to establish a hierarchy of balance integral relations between the factorial moment measures. One can also show that the time-stationary point process exhibits a certain kind of repulsion between its points that is called $f$-repulsion.

These results were published in [29] .