## Gemeinsames Kolloquium der Arbeitsgruppen Stochastik

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

**Termine im Sommersemester 2018**

**Freitag, 18.05.2018**, TU Darmstadt, S1|05 Raum 24 (Maschinenhaus)

**15:15 Friedrich Götze (Universität Bielefeld): Asymptotic Expansions in Entropic Limit Theorems**

We discuss the convergence in classical (central) limit theorems measured in relative entropy based divergences. This includes Entropy, Fisher-Information as well as Renyi-Divergences. Analogues of these expansions of divergences relative to the Wigner law in non-commutative probability are discussed as well.

**16:45 Michael Scheutzow (TU Berlin): Generalized couplings and ergodicity**

The coupling method is a classical tool to establish uniqueness and stability of an invariant measure of a Markov process. In this talk we recall the concept of a generalized coupling and show how it can be used to prove ergodicity (including rates of convergence). The approach is particularly useful in the analysis of stochastic delay equations and some class of SPDEs. This is joint work with Alex Kulik (Kiev) and Oleg Butkovsky (Haifa/Berlin).

**Freitag, 22.06.2018**, Universität Frankfurt, Institut für Mathematik, Robert-Mayer-Str. 10, Raum 711 groß

**15:15 Louigi Addario-Berry (McGill University, Montreal)**: **The front location for branching Brownian motion with decay of mass**

Consider a standard branching Brownian motion whose particles have varying mass. At time t, if a total mass m of particles have distance less than one from a fixed particle x, then the mass of particle x decays at rate m. The total mass increases via branching events: on branching, a particle of mass m creates two identical mass-m particles.

One may define the front of this system as the point beyond which there is a total mass less than one (or beyond which the expected mass is less than one). This model possesses much less independence than standard BBM, and martingales are hard to come by. However, using careful tracking of particle trajectories and a PDE approximation to the particle system, we are able to prove an almost sure law of large numbers for the front speed. We also show that, almost surely, there are arbitrarily large times at which the front lags distance ~ c t^{1/3} behind the typical BBM front. At a high level, our argument for the latter may be described as a proof by contradiction combined with fine estimates on the probability Brownian motion stays in a narrow tube of varying width.

This is joint work with Sarah Penington and Julien Berestycki.

**16: 45 Julien Berestycki (University of Oxford): The hydrodynamic limit of two variants of Branching Brownian motion**

In this talk, I'll consider two variants of branching Brownian motion (BBM): with decay of mass (as in Louigi's talk) and with selection. In the BBM with selection, the number of particles is fixed at some number N and is kept constant by killing the leftmost particle at each branching event. Both models are motivated by considerations from ecology and evolutionary biology.

A particle system has a hydrodynamic limit when, as the number of particles tends to infinity, the behaviour of the system becomes well approximated by the solution of a partial differential equation. In this case I will show that the behaviour of the BBM with decay of mass is governed by the non-local version of the celebrated Fisker-KPP equation while the BBM with selection tends to the solution of a new free boundary problem also in the Fisher-KPP class that we study.

This is based on joint work with Louigi Addario-Berry and Sarah Penington on the one hand and Eric Brunet and Sarah Penington on the other.

**Freitag 06.07.2018**, Universität Mainz, Institut für Mathematik, Staudingerweg 9, Hilbertraum (Raum 05-432)

**15:15 Daniel Valesin (Rijksuniversiteit Groningen): The asymmetric multitype contact process**

We study a class of interacting particle systems known as the multitype contact process on Z^d. In this model, sites of Z^d can be either empty or occupied by an individual of one of two species. Individuals die with rate one and send descendants to neighboring sites with a rate that depends on their (the parent's) type. Births are not allowed at sites that are already occupied. We assume that one of the types has a birth rate that is larger than that of the other type, and larger than the critical value of the standard contact process. We prove that, if initially present, the stronger type has a positive probability of never going extinct. Conditionally on this event, it takes over a ball of radius growing linearly in time. We also completely characterize the set of stationary distributions of the process and prove a complete convergence theorem.

Joint work with Pedro L.B. Pantoja and Thomas Mountford.

**16:45 Jan Swart (UTIA, Prag): The mean-field dual of systems with cooperative branching**

In this talk we consider interacting particle systems where pairs of particles can give birth to new particles, and in addition particles die with a certain rate. We are interested in the random map that describes how the state at a given time depends on the initial state, for well-mixing populations in the limit that the size of the population is large. We will reveal an interesting link to Random Tree Processes

(RTPs) as studied by Aldous and Bandyopadhyay, which are a sort of Markov chains with a tree-like time parameter. In particular, we will discuss endogeny of RTPs related to systems with cooperative branching.

This is joint work with Tibor Mach (Prague) and Anja Sturm (Göttingen).

** **

------------------------------------------------------------------------------------------------------------------------------------------------------------

### Wegbeschreibungen

zur Anreise an die Uni Mainz finden Sie unter https://www.mathematik.uni-mainz.de/anfahrt/ , zur Anreise an die Uni Frankfurt unter https://www.uni-frankfurt.de/38074653/campus_bockenheim und https://www.uni-frankfurt.de/38093742/Campus_Bockenheim-pdf.pdf, zur Anreise an die TU Darmstadt unter http://www3.mathematik.tu-darmstadt.de/fb/mathe/wir-ueber-uns/adresse-und-lageplan/anreise.html

### Termine in früheren Semestern finden Sie hier.

Last update: 15.5.2018, stochastik@uni-mainz.de