In this talk, we speak about a framework for studying point-map invariant measures. A compatible point-map on the set of counting measures is a mapping on discrete multi-subsets of Rd: to each point of the discrete set, it associates another point of the same set, carrying the multiplicity; compatibility means that the image of a point depends only on the support of the counting measure (the counting measure after removing the multiplicities) “seen from” (i.e. shifted to) this point.
It is a well known fact in the literature that whenever a compatible point-map is bijective on all simple counting measures, the Palm version of any stationary simple point process is left invariant by the action of such a compatible and bijective point-map.
We focus on the case of a not necessarily bijective point-map f. Its goal is to associate, to any stationary point process , a companion point process with an almost sure mass at zero and with a law which is invariant under the action of f. We introduce the notion of Point-map Palm (or here f-Palm) version of the point process , which satises the desired invariance property when it exists and we give sucient conditions for it to exist.
Joint work with F. Baccelli.