B03 – Optimal transport and random measures
The theory of optimal transportation, a concept originating from economics and probability, in recent years has undergone an impressive development. It became one of the hot topics of current research with deep results and important applications, among others to functional analysis, PDEs and Riemannian geometry. This project will be devoted to new probabilistic applications. The focus will be on optimal transports between random measures, mainly, between point processes and the Lebesgue measure. This is intended as a link between the two very active fields of current research: optimal transport and
The point processes to be considered will be ‘perturbations’ of the Poisson point process. Of particular interest will be Gibbs measures describing the invariant distribution of infinitely many interacting Brownian motions (iBM) with Ruelle type interaction potentials as well as the recently constructed quasi-Gibbs measures describing the invariant distribution of iBM with logarithmic interaction potentials. Particular cases of the latter are the Dyson and the Ginibre point processes both of which play a prominent role in random matrix theory. All these point processes may be regarded as perturbations of the Poisson point process via interaction. Other important classes of point processes to be considered will be compound Poisson point processes as well as Poisson point processes on homogeneous, non-Euclidean spaces.
Poisson point processes as well as Poisson point processes on homogeneous, non-Euclidean spaces. Another aim of the project will be to derive a more detailed understanding of the random geometry induced by an optimal coupling between the Lebesgue measure and a point process as well as sharp estimates for the transportation cost per unit mass. For the latter, not only asymptotic results but also sharp estimates for finite system sizes are aimed for. These questions will be of interest both for the (‘unperturbed’) Poisson point process as well as for any of the generalizations mentioned above.