# Probability Seminar Essen

The seminar will start on November 3rd in form of video conference.

If you like to participate we ask you to enroll at the Moodle-Course or write an e-mail to Dagmar Goetz to get the Zoom-link.

Nov 10 |
Matthias Erbar (University of Bielefeld) - organized jointly with the SPP 2265-Research Seminar Ruhr. |

Nov 17 |
Sandra Palau (UNAM, Mexico) |

Nov 24 |
Zakhar Kabluchko (University of Münster) - organized jointly with the SPP 2265-Research Seminar Ruhr. |

Dec 1 |
Lukas Gonon (LMU, Munich) |

Dec 8 |
Lisa Hartung (University of Mainz) - organized jointly with the SPP 2265-Research Seminar Ruhr. |

Dec 15 |
Jiequn Han (Princeton University) |

Dec 22 |
Arno Siri Jégousse (UNAM, Mexico) |

Jan 12 |
Eva Kopfer (University of Bonn) - organized jointly with the SPP 2265-Research Seminar Ruhr. |

Jan19 |
Gilles Bonnet (University of Bochum) - organized jointly with the SPP 2265-Research Seminar Ruhr. |

Jan 26 |
Christian Beck (WW University Münster) |

Feb 2 |
Alexander Drewitz and Bingxiao Liu (University of Cologne) - organized jointly with the SPP 2265-Research Seminar Ruhr. |

Feb 9 |
Felix Lindner (University of Kassel) |

#### Abstracts:

Nov 17 |
Sandra Palau (UNAM, Mexico)Branching processes in varying environment
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. |

Nov 24 |
Zakhar Kabluchko (University of Münster) 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. |

Dec 1 |
Lukas Gonon (LMU, Munich) 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. |

Dec 8 |
Lisa Hartung (University of Mainz) In this informal talk, after introducing the class of models I will give an overview of the existent results in the literature on the behaviour of branching Brownian motion among obstacles. These results mainly concern the survival probability in the case of hard obstacles and the population size in the case of mild obstacles. |

Dec 15 |
Jiequn Han (Princeton University) Developing algorithms for solving high-dimensional partial differential equations, controls, and games has been an exceedingly difficult task for a long time, due to the notorious "curse of dimensionality". In the first part of the talk, I will introduce the Deep BSDE method for solving high-dimensional parabolic PDEs. The algorithm builds on the reformulation of backward stochastic differential equations and utilizes deep neural networks as efficient approximators to unknown high-dimensional components. Numerical results of various examples, including multi-agent games, demonstrate the efficiency and accuracy of the proposed algorithms in high-dimensions. In the second part of the talk, I will introduce some convergence analysis of the Deep BSDE method in terms of a posteriori error estimation and an upper bound for the minimized objective function. |

Dec 22 |
Arno Siri Jégousse (UNAM, Mexico) 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. |

Jan 12 |
Eva Kopfer (University of Bonn) We study random perturbations of Riemannian manifolds by means of so-called Fractional Gaussian Field. The fields act on the manifolds via conformal transformation. Our focus will be on the regular case with Hurst parameter $H>0$, the celebrated Liouville geometry in two dimensions being borderline. We want to understand how basic geometric and functional analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap will change under the influence of the noise. |

Jan 19 |
Gilles Bonnet (University of Bochum) In this talk I will present aspects of the project "Concentration and Cumulants for Stabilizing Functionals on Point Processes". In particular, the notion of stabilizing functional will be introduced and several classical methods to obtain concentration bounds presented. |

Jan 26 |
Christian Beck (WW University Münster) Multilevel Picard approximation methods have been successfully used to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic PDEs (see, e.g., Hutzenthaler et al. https://royalsocietypublishing.org/doi/full/10.1098/rspa.2019.0630). In this talk we extend multilevel Picard approximations for semilinear elliptic PDEs, state that they can overcome the curse of dimensionality in the numerical approximation of semilinear elliptic PDEs, point out differences between the parabolic and the elliptic situation, and comment on possible directions of future research. |

Feb 2 |
Alexander Drewitz and Bingxiao Liu (University of Cologne) We are presenting our SPP project which focuses on the interplay between complex geometry and probability theory. More precisely, we aim to combine methods from complex geometry and geometric analysis with probabilistic techniques in order to study several problems concerning local and global statistical properties of zeros of holomorphic sections of holomorphic line bundles over Kähler manifolds. A particularly important instance of this setting is given by the case of random polynomials. We are interested in the asymptotics of the covariance kernels of the polynomial / sections ensembles, universality of their distributions, central limit theorems and large deviation principles in this context. |

Feb 9 |
Felix Lindner (University of Kassel) The dynamics of inextensible fibers in turbulent airflows is of interest, e.g, in the context of mathematical models for spunbond production processes of non-woven textiles. In this talk, a model based on higher-index stochastic differential-algebraic equations is presented. It involves an implicitly given Lagrange multiplier process, the explicit representation of which leads to an underlying stochastic ordinary differential equation with non-globally monotone coefficients. Strong convergence is established for a half-explicit drift-truncated Euler scheme which fulfills the algebraic constraint exactly. |

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