Compactification of the Action of a Point-Map on the Palm Probability of a Point Process

A compatible point-shift f maps, in a translation invariant way, each point of a stationary point process Φ to some point of Φ. It is fully determined by its associated point-map, gf , which gives the image of the origin by f . The initial question of this paper is whether there exist probability measures which are left invariant by the translation of −gf . The point-map probabilities of Φ 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 is uniquely defined, and if it satisfies certain continuity properties, it then provides a solution to the initial question. Point-map probabilities are shown to be a strict generalization of Palm probabilities: when the considered point-shift f is bijective, the point-map probability of Φ boils down to the Palm probability of Φ. When it is not bijective, there exist cases where the point-map probability of Φ is absolutely continuous with respect to its Palm probability, but there also exist cases where it is singular with respect to the latter. A criterium of existence of the point-map probabilities of a stationary point process is also provided. The results are illustrated by a few examples.