Resumen de Point-map-probabilities of a point process and Mecke’s invariant measure equation
Francois Baccelli, Mir-Omid Haji Mirsadeghi
A compatible point-shift FF maps, in a translation invariant way, each point of a stationary point process ΦΦ to some point of ΦΦ. It is fully determined by its associated point-map, ff, which gives the image of the origin by FF. It was proved by J. Mecke that if FF is bijective, then the Palm probability of ΦΦ is left invariant by the translation of −f−f. 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 −f−f? The point-map-probabilities of ΦΦ 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 FF is bijective, the point-map-probability of ΦΦ boils down to the Palm probability of ΦΦ. When it is not bijective, there exist cases where the point-map-probability of ΦΦ 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 FF-pre-images of all orders is also provided. The results are illustrated by a few examples.
