# Probability Seminar Essen

## Probability Seminar Essen

Covers a wide range of topics in Probability and its applications.

 Apr 17 Anton Klimovsky (Universität Duisburg-Essen) Apr 24 Anselm Hudde (Universität Duisburg-Essen) May 8 Fernando Cordero (Universität Bielefeld) May 15 May 29 Roland Meizis (Universität Duisburg-Essen) Jun 5 Daniel Pieper (Universität Duisburg-Essen) Jun 12 Jun 19 Jun 26 Jul 3 Jul 10 Jul 17

## Abstracts:

 Apr 17 Anton Klimovsky (Universität Duisburg-Essen) Stochastic population models on evolving networks Many models of complex systems can be seen as a system of many interacting (micro)variables evolving in time. We focus on the situation, where the network of interactions between the variables is complex and possibly itself evolves in time. We discuss a modeling framework for interacting particle systems on evolving networks based on such familiar ingredients as exchangeability and Markovianity. In some simple cases, we discuss the genealogies of such population models. (Based on joint work in progress with Jiří Černý.) Apr 24 Anselm Hudde (Universität Duisburg-Essen) A perturbation Theory and applications to numerical approximation of SDEs In this talk we will discuss a perturbation theory which can be applied to find strong $L^2$-convergence rate for approximations schemes of SDEs and SPDEs. May 8 Fernando Cordero (Universität Bielefeld) On the stationary distribution of the block-counting process in populations with mutation and selection The $\Lambda$-Wright-Fisher model is a population model subject to selection, mutation and neutral reproduction (described by a finite measure $\Lambda$ on $[0,1]$). The block-counting process traces back the number of potential ancestors of a sample of the population at present. In absence of selection and mutation the latter coincides with the $\Lambda$-coalescent. Selection and mutation translate into additional branching and pruning. Under some conditions the block-counting process is positive-recurrent and its stationary distribution is described via a linear system of equations. In this talk, we first characterise the measures $\Lambda$ leading to a geometric stationary distribution, the Bolthausen-Sznitman model being the most prominent example having this feature. For a general measure $\Lambda$, we show that the probability generating function of the stationary distribution of the block-counting process satisfies an integro-differential equation. We solve the latter for the Kingman model and the star-shaped model. (Based on joint work with M. Möhle).

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