Matthias Erbar (University of Bielefeld)
Optimal transport, stationary point processes, and log gases

Abstract: In this talk I would like give an overview of the topics and aims of the project "Optimal transport for stationary point processes" within the SPP "Random geometric systems". I will give a brief introduction of the classical theory of optimal transport for finite measures and will outline how we envision to develop a counterpart to this theory in the setting of stationary point processes. I will also present first results in this direction from joint work with M. Huesmann and T. Lebl\'e which apply ideas from optimal transport in the study of log gases. The one-dimensional log gas in finite volume is a system of particles interacting via a repulsive logarithmic potential and confined by some external field. When the number of particles goes to infinity, their macroscopic empirical distribution approaches a deterministic limit shape. When zooming in one sees microscopic fluctuations around this limit which are described in the limit by a stationary point process, the Sine_\beta process constructed by Valko and Virag. We show that this process can be characterized as the unique minimizer of a renormalized free energy by leveraging strict convexity properties of this functional along interpolations built via optimal transport.

Abstract: In this talk we are going to study branching processes and its extension when the offspring distribution is varying over time. We are going to analyze its extinction probability. By using a two-spine decomposition technique we are going to give the law of the process conditioned on non extinction.
 

Zakhar Kabluchko (University of Münster)
Random beta tessellations

We shall define and study several families of random tessellations of the Euclidean space, the sphere and the hyperbolic space that generalize the classical Poisson-Voronoi and Poisson-Delaunay tessellations. The talk is based on a joint work with Anna Gusakova and Christoph Thäle.

Lukas Gonon (LMU, Munich)
Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models

We study the expression rates of deep neural networks for option prices written on high-dimensional baskets of risky assets, whose log-​returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process that ensure that the size of the DNN required to achieve a given approximation accuracy grows only polynomially with respect to the dimension of the Lévy process and the reciprocal of the approximation accuracy, thereby overcoming the curse of dimensionality and justifying the use of DNNs in financial modelling of large baskets in markets with jumps.
In addition, we exploit parabolic smoothing of Kolmogorov partial integrodifferential equations for certain multivariate Lévy processes to present alternative architectures of ReLU DNNs that provide higher approximation rates, however, with constants potentially growing exponentially with respect to the dimension. Under stronger, dimension-​uniform non-​degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the present results.
The talk is based on joint work with Christoph Schwab.

Arno Siri Jégousse (UNAM, Mexico)
Site frequency spectrum of the Bolthausen-Sznitman coalescent

In this talk, I will study the concept of site frequency spectrum (SFS), which happens to be one of the most relevant statistics in population genetics. It is of particular use for genealogical model selection and estimation in evolution models. The SFS is closely related to the shape of the genealogical tree of the observed sample of a population.
In the particular case of the Bolthausen-Sznitman coalescent, which is now accepted to be the null model for rapidly evolving populations or populations under strong selection, the limit behaviour of the SFS can be studied thanks to approximations with random walks. More interestingly, there exists a construction of the coalescent by means of random recursive trees which yields very precise approximations of the moments of the SFS. This technique also gives new asymptotic results, leading to a complete picture of the statistic.
This is a joint work with Götz Kersting (Frankfurt) and Alejandro Wences (Mexico City).