Rhein-Main Kolloquium Stochastik





Shared colloquium of the probability and statistics workgroups

TU Darmstadt / Goethe-Universität Frankfurt / Gutenberg-Universität Mainz


Winter term 2021/2022, hybrid events or online via Zoom only


January 14, 2022, 15 c.t. CET, Rhein-Main-Kolloquium Stochastik, Gutenberg-Universität Mainz, online via Zoom

15:15-16:15: René Schilling (TU Dresden)

16:15-16:45: Virtual coffee break

16:45-17:45:  Andreas Kyprianou (University of Bath)


René Schilling: Some martingales for Levy processes

We show the structure of all martingales of the form $f(X_t)- Ef(X_t)$ or $g(X_t)/E g(X_t)$ where $X_t$ is a Levy process. This is connected with Cauchy's functional equation and the Liouville theorem for Levy processes. (Joint work with Franziska Kühn)

Andreas Kyprianou: Asymptotic moments of spatial branching processes

We introduce a very general class of non-local branching particle processes and non-local superproesses for which the asymptotic moments can be computed explicitly as a function of time (our results are agnostic to either category of process). The method we use is extremely robust and we are able to provide similar results for the asymptotic moments of occupation functionals as a function of time.

Online via Zoom - No registration required for this event

Access to links to the talks and coffee break is available at: stochastik@uni-mainz.de


Friday, December 3, 2021, 3 pm CET ct,  Rhein-Main-Kolloquium Stochastik, Goethe Universität Frankfurt/Main, Online event via Zoom

15:15-16:15: Nina Gantert (TU München)

16:15-16:45: (virtual) coffee break

16:45-17:45: Paolo Dai Pra (Università degli Studi di Verona)

Nina Gantert (TU München)

Sharp concentration for the largest and smallest fragment in a k-regular self-similar fragmentation

Abstract: We study the asymptotics of the k-regular self-similar fragmentation process. For α>0 and an integer k≥2, this is the Markov process (I_t)_{t≥0} in which each It is a union of open subsets of [0,1), and independently each subinterval of It of size u breaks into k equally sized pieces at rate u^α. Let k^{−mt} and k^{−Mt} be the respective sizes of the largest and smallest fragments in I_t. By relating (I_t)_{t≥0} to a branching random walk, we find that there exist explicit deterministic functions g(t) and h(t) such that |mt−g(t)|≤1 and |Mt−h(t)|≤1 for all sufficiently large t. Furthermore, for each n, we study the final time at which fragments of size k^{-n} exist. In particular, by relating our branching random walk to a certain point process, we show that, after suitable rescaling, the laws of these times converge to a Gumbel distribution as n→∞.

Based on joint work with Piotr Dyszewski, Samuel G. G. Johnston, Joscha Prochno and Dominik Schmid

Paolo Dai Pra (Università degli Studi di Verona)

Self-sustained oscillations in interacting systems: an overview and some recent advances

Abstract: Self organized collective periodic behavior is seen to emerge in several different contexts: from neuroscience to tectonic plates movements, from population dynamics to epidemiology. A large variety of stochastic models have been proposed to capture this phenomenon at a mathematical level, showing that it may be induced by a combination of factors including noise, dissipation, loss of Markovianity and/or of time reversibility. Most of the present literature concerns mean-field models, where the thermodynamic limit is well understood at a dynamical level, and the emergence of oscillations can be seen from the macroscopic evolution equations. In models with short range interaction it is much harder to understand how self organization at microscopic level may produce large scale rhythms. Some partial results have been obtained for a non-reversible modification of the nearest neighbour Ising model.




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