Abstract
Consider a set of objects, abstracted to points of a spatially stationary point process in $$\mathbb {R}^d$$Rd, that deliver to each other a service at a rate depending on their distance. Assume that the points arrive as a Poisson process and leave when their service requirements have been fulfilled. We show how such a process can be constructed and establish its ergodicity under fairly general conditions. We also establish a hierarchy of integral balance relations between the factorial moment measures and show that the time-stationary process exhibits a repulsivity property.
Highlights
This paper was initially motivated by our study of peer to peer file-sharing in spatial scenarios, where the transmission speed between two peers depends on their distance [1]
The peers1 join as a Poisson rain in R2, serve each other at rates given by some decreasing function f of their distances, and leave when their individual service requirements have been fulfilled
We show that the intensity measure of this point process on R is locally finite under the condition given above
Summary
This paper was initially motivated by our study of peer to peer file-sharing in spatial scenarios, where the transmission speed between two peers depends on their distance [1]. The authors study the evolution of the correlation functions of the process as elements of a Banach space and show that this uniquely determines the evolution of the distribution of the spatial birth and death process on certain compact sets of time They develop conservation equations for higher order moment measures [6, 4]. Combining these with differential equations describing the time evolution, we prove in Section 6 that on any compact set of space, the time of last influence of Z0 is integrable This allows to develop a coupling from the past argument, proving the fundamental existence and ergodicity result of the paper.
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