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 Roland Meizis (Universität Duisburg-Essen) May 29 Daniel Pieper (Universität Duisburg-Essen) Jun 5 Richard Kraaij (Ruhr Universität Bochum) Jun 12 Anita Winter (Universität Duisburg-Essen) Jun 19 No talk because of SPP 1590 summer school Jul 3 Lionel Lenôtre (CMAP - Ecole Polytechnique) Jul 17 Robert Link (Universität Duisburg-Essen)

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). May 15 Roland Meizis (Universität Duisburg-Essen) Convergence of metric two-level measure spaces We extend the notion of metric measure spaces to so-called metric two-level measure spaces (m2m spaces): An m2m space $(X, r, \nu)$ is a Polish metric space $(X, r)$ equipped with a two-level measure $\nu \in \mathcal{M}_f(\mathcal{M}_f(X))$, i.e. a finite measure on the set of finite measures on $X$. We define the set $\mathbb{M}^{(2)}$ of (equivalence classes of) m2m spaces and provide a complete metric on this set. Furthermore, we introduce a convergence determining set of test functions on $\mathbb{M}^{(2)}$, which is suitable for defining generators of Markov processes on $\mathbb{M}^{(2)}$. The framework introduced in this talk is motivated by applications in biology. It is well suited for modeling the random evolution of the genealogy of a population in a hierarchical system with two levels, for example, host-parasite systems or populations which are divided into colonies. May 29 Daniel Pieper (Universität Duisburg-Essen) Altruistic defense traits in structured populations: Many-demes limit in the sparse regime We discuss spatially structured Wright-Fisher type diffusions modelling the frequency of an altruistic defense trait. These arise as the diffusion limit of spatial Lotka-Volterra type models with a host population and a parasite population, where one type of host individuals (the altruistic type) is more effective in defending against the parasite but has a weak reproductive disadvantage. For the many-demes limit (mean-field approximation) hereof, we prove a propagation of chaos result in the case where only a few diffusions start outside of an accessible trap. In this "sparse regime", the system converges in distribution to a forest of trees of excursions from the trap. Jun 5 Richard Kraaij (Ruhr Universität Bochum) Fluctuations for a dynamic Curie-Weiss model of self-organized criticality The Curie-Weiss model of self-organized criticality was introduced by Cerf and Gorny(2014,2016) as a modification of the Curie-Weiss model of ferromagnetism that drives itself into a criticalstate. We consider a dynamic variant of this model, i.e. a system of interacting SDE's, and study its dynamical fluctuations. Based on joint work with Francesca Collet(Delft) and Matthias Gorny(Paris-Sud). Jun 12 Anita Winter (Universität Duisburg-Essen) Aldous move on cladograms in the diffusion limit In this talk we are interested in limit objects of graph-theoretic trees as the number of vertices goes to infinity. Depending on which notion of convergence we choose different objects are obtained. One notion of convergence with several applications in different areas is based on encoding trees as metric measure spaces and then using the Gromov-weak topology. Apparently this notion is problematic in the construction of scaling limits of tree-valued Markov chains whenever the metric and the measure have a different scaling regime. We therefore introduce the notion of algebraic measure trees which capture only the tree structure but not the metric distances. Convergence of algebraic measure trees will then rely on weak convergence of the random shape of a subtree spanned a sample of nite size. We will be particularly interested in binary algebraic measure trees which can be encoded by triangulations of the circle. We will show that in the subspace of binary algebraic measure trees sample shape convergence is equivalent to Gromov-weak convergence when we equip the algebraic measure tree with an intrinsic metric coming from the branch point distribution. The main motivation for introducing algebraic measure trees is the study of a Markov chain arising in phylogeny whose mixing behavior was studied in detail by Aldous (2000) and Schweinsberg (2001). We give a rigorous construction of the diffusion limit as a solution of a martingale problem and show weak of the Markov chain to this diffusion as the number of leaves goes to infinity. Jul 3 Lionel Lenôtre (CMAP - Ecole Polytechnique) A simulation method called GEARED and its application to Skew Diffusions The GEARED or GEneralized Algorithm based on REsolvent for Diffusion is a simulation method of Feller's processes whatever they admit a representation as a stochastic differential equation or not. The only requirements is to be able to sample a particular random variable whose density is given by the resolvent kernel of the stochastic process that one wants to simulate. This mainly means that an analytical form of the resolvent kernel is required. In this talk, we present the GEARED method and provide and application of it on Skew Diffusions with piecewise constant coefficients. On this particular case of Feller's processes, we show through numerical experiments that the GEARED method has a quite fast convergence and that it conserves important properties in physics such a good repartition of mass. Jul 17 Robert Link (Universität Duisburg-Essen) Existence and uniqueness of solutions of infinite dimensional Kolmogorov equations It is well known from the Feynman-Kac formula that a classical solution of the Kolmogorov backward equation can be written as the expectation of the solution of the corresponding SDE. In 2015 M. Hairer, M. Hutzenthaler, and A. Jentzen gave a finite dimensional example of a Kolmogorov backward equation with globally bounded and smooth coefficients and a smooth initial function with compact support such that the unique viscosity solution is not locally Hölder continuous. Moreover, they proved in the finite dimensional case that under suitable assumption the Kolmogorov backward equation has a unique viscosity solution which can be represented as the expectation of the solution of the corresponding SDE. In the talk I will generalize this result to infinite dimensional Hilbert spaces and SPDEs. Therefore I will use a more general notation of viscosity solution introduced by H. Ishii and show that under suitable assumptions the expectation of the solution of an SPDE is the unique viscosity solution of the corresponding Kolmogorov backward equation.

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