Section: New Results
Theory of point processes
In a joint work with Mir-Omid Haji-Mirsadeghi, Sharif University, Department of Mathematics, F. Baccelli studied a class of non-measure preserving dynamical systems on counting measures called point-maps. This research introduced two objects associated with a point map acting on a stationary point process :
The -probabilities of , which can be interpreted as the stationary regimes of the action of on . These probabilities are defined from the compactification of the action of the semigroup of point-map translations on the space of Palm probabilities. The -probabilities of are not always Palm distributions.
The -foliation of , a partition of the support of which is the discrete analogue of the stable manifold of , i.e., the leaves of the foliation are the points of with the same asymptotic fate for . These leaves are not always stationary point processes. There always exists a point-map allowing one to navigate the leaves in a measure-preserving way.
Two papers on the matter available. The first one is under revision for Annals of Probability.