Section: New Results

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

In collaboration with Mir-Omid Haji-Mirsadeghi (Sharif University, Iran) [50] , we analyzed the compactification of Palm probabilities by the action of a point–shift. A point-shift maps, in a translation invariant way, each point of a stationary point process $\Phi$ to some point of $\Phi$. The initial motivation of this paper is the construction of probability measures, defined on the space of counting measures with an atom at the origin, which are left invariant by a given point-shift $f$. The point-shift probabilities of $\Phi$ are defined from the action of the semigroup of point-shift translations on the space of Palm probabilities, and more precisely from the compactification of the orbits of this semigroup action. If the point-shift probability is uniquely defined, and if $f$ is continuous with respect to the vague topology, then the point-shift probability of $\Phi$ provides a solution to the initial question. Point-shift probabilities are shown to be a strict generalization of Palm probabilities: when the considered point-shift $f$ is bijective, the point-shift probability of $\Phi$ boils down to the Palm probability of $\Phi$. When it is not bijective, there exist cases where the point-shift probability of $\Phi$ is the law of $\Phi$ under the Palm probability of some stationary thinning $\Psi$ of $\Phi$. But there also exist cases where the point-shift probability of $\Phi$ is singular w.r.t. the Palm probability of $\Phi$ and where, in addition, it cannot be the law of $\Phi$ under the Palm probability of any stationary point process $\Psi$ jointly stationary with $\Phi$. The paper also gives a criterium of existence of the point-shift probabilities of a stationary point process and discusses uniqueness. The results are illustrated through several examples.