Section: New Results
Theory of point processes
In a joint work with MirOmid HajiMirsadeghi, Sharif University, Department of Mathematics, F. Baccelli studied a class of nonmeasure preserving dynamical systems on counting measures called pointmaps. This research introduced two objects associated with a point map $f$ acting on a stationary point process $\Phi $:

The $f$probabilities of $\Phi $, which can be interpreted as the stationary regimes of the action of $f$ on $\Phi $. These probabilities are defined from the compactification of the action of the semigroup of pointmap translations on the space of Palm probabilities. The $f$probabilities of $\Phi $ are not always Palm distributions.

The $f$foliation of $\Phi $, a partition of the support of $\Phi $ which is the discrete analogue of the stable manifold of $f$, i.e., the leaves of the foliation are the points of $\Phi $ with the same asymptotic fate for $f$. These leaves are not always stationary point processes. There always exists a pointmap allowing one to navigate the leaves in a measurepreserving way.
Two papers on the matter available. The first one is under revision for Annals of Probability.