Mutual Service Processes in R Existence and Ergodicity

Consider a set of objects, abstracted to points of a spatially stationary point process in R, that deliver mutually 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 balance integral relations between the factorial moment measures and show that the time-stationary process exhibits a repulsivity property. AMS Classification: 60D05, 60G55, 60D05, 60G10, 60G17, 60J25, 05C80, 70F45.