Universität Duisburg-Essen
Fakultät Mathematik, Campus Essen
Anita Winter

10.th May 2010

Probability Seminar
Summer Semester 2011
Wednesday, 16.15-17.15 (T03 R03 D05)

Summer Semester 2011

• 06.04.2011,
• Anita Winter (Universität Duisburg-Essen): Coalescent processes arising in the study of diffusive clustering

Abstract: We study the spatial coalescent on $\Z^2$. In our setting, partition elements are located at the sites of $Z^2$ and undergo local delayed coalescence and migration. The system starts in either locally finite configurations or in configurations containing countably many partition elements per site. Our goal is to determine the longtime behavior with an initial population of countably many individuals per site restricted to a box $\Lambda^{\alpha,t}:=[-t^{\alpha/2}, t^{\alpha/2}]^2 \cap \Z^2$ and observed at time $t^\beta$ with $1 \geq \beta \geq \alpha\ge 0$. We study both asymptotics, as $t\to\infty$, for a fixed value of $\alpha$ as the parameter $\beta\in[\alpha,1]$ varies and for a fixed $\beta$, as the parameter $\alpha\in [0,\beta]$ varies. This exhibits the genealogical structure of the mono-type clusters arising in 2-dimensional Moran and Fisher-Wright systems. A new random object, the so-called {\em coalescent with rebirth}, is constructed via a look-down procedure and shown to arise in the space-time limit of the coalescent restricted to $\Lambda^{\alpha,t}$ with $\alpha\in[0,1]$ and observed at time time goes to infinity. (this is joint work with Andreas Greven and Vlada Limic)

• 04.05.2011,
• Andrej Fischer (Universität Köln): Stochastic tunneling in a two-locus system with recombination

• 18.05.2011,
• Wolfgang Löhr (Universität Duisburg-Essen): Measures and Continuous Functions on the Space of Metric Measure Spaces

Abstract: We introduce the space of metric measure spaces (mm-spaces) and its embedding into the space of distance matrix distributions. This classical embedding can be extended to the space of measures on the space of mm-spaces, which has been shown recently by Depperschmidt, Greven and Pfaffelhuber, and will be shown here in a different way. We also explain that the space of mm-spaces is not locally compact and the algebra of polynomials is not dense in the space of bounded continuous functions.

• 01.06.2011,
• Siva Athreya (Indian Statistical Institute, Bangalore): Blowup and Conditionings of $\psi$-super Brownian Exit Measures

Abstract: We extend earlier results on conditioning of super-Brownian motion to general branching rules. We obtain representations of the conditioned process, both as an $h$-transform, and as an unconditioned superprocess with immigration along a branching tree. Unlike the finite-variance branching setting, these trees are no longer binary, and strictly positive mass can be created at branch points. This construction is singular in the case of stable branching. We analyze this singularity first by approaching the stable branching function via analytic approximations. In this context the singularity of the stable case can be attributed to blowup of the mass created at the first branch of the tree. Other ways of approaching the stable case yield a branching tree that is different in law. To explain this anomaly we construct a family of martingales whose backbones have multiple limit laws.

• 08.06.2011,
• 15.06.2011,
• Anja Sturm (Universität Göttingen): Long-term behavior of subcritical contact processes

Abstract: We consider the long-time behavior of the law of a contact process started with a single infected site, distributed according to counting measure on the lattice. This distribution is related to the configuration as seen from a typical infected site and gives rise to the definition of so-called eigenmeasures, which are possibly infinite measures on the set of non empty configurations that are preserved under the dynamics up to a multiplicative constant. We show that contact processes on general countable groups have in the subcritical regime a unique spatially homogeneous eigenmeasure. We also discuss possible applications of this result, in particular regarding the behavior of the exponential growth rate of the process as a function of its death rate. This is joint work with Jan Swart (UTIA Prague)

• 04.08.2011, 17.00,
• Lior Bary-Soroker (Universität Duisburg-Essen): Golais and Probability

Abstract: Évariste Galois died in a dual at the age of 20 on May 31, 1832. The mathematics he managed to do until that led, a decade after his death, to Modern Algebra. In this talk I will try to explain what is Galois theory. In particular I hope to give some insights to the famous theorem of Galois saying that the general equation of degree 5 has no root formula. If time permits, I'll discuss two connections with probability (or one, or zero but then we can discuss on coffee). As tempting as it is, I'm not going to discuss history, if one is interested in the story of his life, there are many books on this subjects, or wikipedia.....

Winter Semester 2010/11

• 19.10.2010,
• Wolfgang Löhr (Universität Duisburg-Essen): Complexity Measures of Discrete-Time Stochastic Processes, Continuity and Ergodic Decomposition

Abstract: In complex system sciences, one tries to quantify different kinds of complexity'' of processes. The resulting complexity measures are then used for data analysis and modelling. In my work, I provide a rigorous mathematical framework for one of these complexity measures, namely statistical complexity, and some related quantities. As a main result, I obtain functional properties such as lower-semi continuity and behaviour under ergodic decomposition. An important tool is the prediction process introduced by Frank Knight in 1975.

• 26.10.2010,
• Anton Klimovsky (Hausdorff-Zentrum Bonn): Universal macroscopic behavior of evolving genealogies of spatial Lambda-Fleming-Viot processes

Abstract: We consider a class of stochastic processes -- the so-called spatial Lambda-Fleming-Viot processes -- that describe the evolution of the genealogies in the spatially extended populations with migration and occasionally large (i.e., comparable to the population size) reproduction events. What reproduction mechanisms can be observed in these processes on the macroscopic level? We argue that, in the regime when the migration mechanism mixes the spatially extended population well, the macroscopic reproduction behavior is rather universal and is described by the Kingman coalescent. Joint work in progress with A. Greven and A. Winter.

• 02.11.2010,
• no talk because of the SFB/TR 12 meeting

Wolfgang Löhr (Universität Duisburg-Essen): Complexity Measures of Discrete-Time Stochastic Processes, Continuity and Ergodic Decomposition II

Abstract: Continuation of the talk from 19.10.2010

• 16.11.2010,
• Monika Meise (Universität Duisburg-Essen): Shape restricted smoothing

• 23.11.2010,
• no talk because of the mini-workshop on dualities at the Hausdorff Center Bonn

• 30.11.2010,
• Lorenz Pfeiffroth (TU München): Frogs in a random environment on Z

Abstract: The frog model in a fixed environment can be described as follows. Let G be a graph and take one vertex as origin. Initially there is a number of sleeping frogs at each vertex except the origin. At the origin there is one active frog which jumps according to a random walk on $G$. If an active frog jumps to a vertex where sleeping frogs are, they get awake and move according to the same random walk, independently from everything else. The idea of this model is that every active frog has some information and it shares it with the sleeping frogs for the first time when they meet. Alves, Machado and Popov proved a recurrence criterion if the graph is Z^d or T_d and the underlying random walk is a symmetric simple random walk. The first time other underlying random walks were investigated was by Gantert and Schmidt in 2008. The random walk was a simple random walk in Z with drift to the right. In the first part of this talk we consider a more general setting of underlying random walks. I.e. the only assumption for our random walk is that he is transient to the right. The question, we are interested in, is if the origin is visited infinitely often by active frogs with probability 1 or not. This is not a trivial question in this setting because all random walks in this model won't eventually visit the negative integers. But intuitively spoken if there are enough frogs on the positive integers, which will be activated surely, the change of visiting the negative integers is increasing and thus also the origin. So we expect if there are enough frogs on the right of the origin the model will be recurrent. We give a necessary and sufficient condition that this will happened. Also we show that our result is a generalization of the model, which Gantert and Schmidt investigate, and present a 0-1 law for this model. Now the question naturally arise is if we take the jumping probability random, can we derive analogue conditions for the recurrence of such a model. The second part of this talk deals with that kind of problem. We give recurrence criteria for such a model. If we take the starting configuration of sleeping frogs also as random, we derive a 0-1 law too and show that the recurrence of such a model only depends on the distribution of the starting configuration and it does not depend on the distribution of the jumping probability of the underlying random walk. In the last part I sketch the proof of the recurrence criteria for a frog model in a fixed and random environment, respectively.

• 07.12.2010,
• Andre Depperschmidt (Hausdorff Zentrum Bonn): Tree-valued Fleming-Viot process with mutation and selection

Abstract: In population genetics Moran models are used to describe the evolution of types in a population of a fixed size N. The type of individuals may change due to mutation. Furthermore, due to selection the offspring distribution of an individual depends on its current type. As N tends to infinity the empirical distribution of types converges to the Fleming-Viot process. At fixed times the genealogy of such populations can be constructed using the ancestral selection graph (ASG) of Krone and Neuhauser, which generalizes the Kingman coalescent. As the population evolves its genealogy evolves as well. We construct a tree-valued version of the Fleming-Viot process with mutation and selection (TFVMS) using a well-posed martingale problem. This extends the construction of the neutral tree-valued process given in (Greven, Pfaffelhuber and Winter, 2010). For existence we use approximating tree-valued Moran models and for uniqueness a Girsanov-type theorem on marked measure spaces, the state spaces of TFVMS. Furthermore we study the long-time behavior of TFVMS using duality. Finally, in a concrete example, we compare the Laplace transforms of pairwise genealogical distances in equilibrium of TFVMS and the neutral tree-valued process. This is joint work with Andreas Greven and Peter Pfaffelhuber.

• 14.12.2010,
• Vladimir Osipov (Universität Duisburg-Essen): Ultra-metric models of protein conformational dynamics

• 11.01.2011,
• Guillaume Voisin (Universität Duisburg-Essen): Local time of a diffusion in a Levy environment

Abstract: Diffusions in random environment can be viewed as a limit in time and space of random walks in discrete random environment. In the recurrent case, discrete and continuous diffusions have localization properties. The local time process on some well chosen points of the medium gives a better idea of this localization. We get the asymptotic law of the local time process at the favorite point of the diffusion.

• 18.01.2011,
• Anita Winter (Universität Duisburg-Essen): Brownian motion on real trees

Abstract: The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. We use Dirichlet form methods to construct Brownian motion on any given locally compact real tree equipped with a Radon measure. We specify a criterion under which the Brownian motion is recurrent or transient. (this is joint work with Siva Athreya and Michael Eckhoff)

• 25.01.2011,
• Alexa Manger (Universität Duisburg-Essen): Association of Ito processes

Abstract: Association is a special kind of positive dependence. In the special case of It\^o processes we find conditions for the association and we can conclude the association of their hitting times which gives applications in risk management

• 10.02.2011, 14.15
• Anita Winter (Universität Duisburg-Essen): A multitype branching model with local self-regulation

Abstract: We consider a spatial multi-type branching model in which individuals migrate in $Z^d$ according to random walks and reproduce according to a branching mechanism which can be sub-, super- or critically depending on carrying capacities and the local intensity of individuals of the different types. In this talk we will focus on the diffusion limit of small mass, locally many individuals and rapid reproduction in the exchangeable set-up where non of the parameters involved in the model are type dependent. In Etheridge (2006) it has been shown that there are parameter regimes allowing for survival in all dimensions. In this talk we present duality relations which allow for monotonicity statements in the parameters with regards whether or not the different surviving types can coexist. (joint work with Andreas Greven, Peter Pfaffelhuber, Anja Sturm and Iljana Z\"ahle)

Interested students and colleges are very welcome!

anita (dot) winter `at' due (dot) de