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

Wednesday, 16.15-17.15 (T03 R03 D05)

**Summer Semester 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)

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

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.

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.

** **

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)

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**

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.

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.

**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

Monika Meise (Universität Duisburg-Essen):
**Shape
restricted smoothing**

**no talk because of the mini-workshop on dualities at the Hausdorff Center Bonn **

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.

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.

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

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.

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)

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

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