Section: New Results
Point-map-probabilities of a point process and Mecke’s invariant measure equation
[4] A compatible point-shift maps, in a translation invariant way, each point of a stationary point process to some point of . It is fully determined by its associated point-map, , which gives the image of the origin by . It was proved by J. Mecke that if is bijective, then the Palm probability of is left invariant by the translation of . The initial question motivating this paper is the following generalization of this invariance result: in the nonbijective case, what probability measures on the set of counting measures are left invariant by the translation of ? The point-map-probabilities of are defined from the action of the semigroup of point-map translations on the space of Palm probabilities, and more precisely from the compactification of the orbits of this semigroup action. If the point-map-probability exists, is uniquely defined and if it satisfies certain continuity properties, it then provides a solution to this invariant measure problem. Point-map-probabilities are objects of independent interest. They are shown to be a strict generalization of Palm probabilities: when is bijective, the point-map-probability of boils down to the Palm probability of . When it is not bijective, there exist cases where the point-map-probability of is singular with respect to its Palm probability. A tightness based criterion for the existence of the point-map-probabilities of a stationary point process is given. An interpretation of the point-map-probability as the conditional law of the point process given that the origin has -pre-images of all orders is also provided. The results are illustrated by a few examples.