Apr 17 
Anton Klimovsky (Universität DuisburgEssen)
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 DuisburgEssen)
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 blockcounting process in populations with mutation and selection
The $\Lambda$WrightFisher model is a population model subject to selection, mutation and neutral reproduction (described by a finite measure $\Lambda$ on $[0,1]$). The blockcounting 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 blockcounting process is positiverecurrent 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 BolthausenSznitman 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 blockcounting process satisfies an integrodifferential equation. We solve the latter for the Kingman model and the starshaped model.
(Based on joint work with M. Möhle).

May 15 
Roland Meizis (Universität DuisburgEssen)
Convergence of metric twolevel measure spaces
We extend the notion of metric measure spaces to socalled metric twolevel measure spaces (m2m spaces): An m2m space $(X, r, \nu)$ is a Polish metric space $(X, r)$ equipped with a twolevel 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, hostparasite systems or populations which are divided into colonies.

May 29 
Daniel Pieper (Universität DuisburgEssen)
Altruistic defense traits in structured populations: Manydemes limit in the sparse regime
We discuss spatially structured WrightFisher type diffusions modelling the frequency of an altruistic defense trait. These arise as the diffusion limit of spatial LotkaVolterra 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 manydemes limit (meanfield 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 CurieWeiss model of selforganized criticality
The CurieWeiss model of selforganized criticality was introduced by Cerf and Gorny(2014,2016) as a modification of the CurieWeiss 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(ParisSud).

Jun 12 
Anita Winter (Universität DuisburgEssen)
Aldous move on cladograms in the diffusion limit
In this talk we are interested in limit objects of graphtheoretic 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 Gromovweak topology. Apparently this notion is problematic in the construction of scaling limits of treevalued 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 Gromovweak 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 DuisburgEssen)
Existence and uniqueness of solutions of infinite dimensional Kolmogorov equations
It is well known from the FeynmanKac 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.
