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 $f$ 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 point-map 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 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.