# Probability Colloquium

## Robustness of spatialpreferentialattachment networks

Speaker(s):
Peter Mörters (Bath)
Date:
Wednesday, November 18, 2015 - 6:15pm
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrarily small positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the unit circle and are connected to existing vertices with a probability favouring short spatial distances and high degrees.

## A weak dynamic principle for combined optimal stopping/ stochastic control with nonlinear expectations

Speaker(s):
Roxana Dumitrescu (HU Berlin)
Date:
Wednesday, November 18, 2015 - 5:15pm
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We study a combined optimal control/stopping problem under a nonlinear expectation E^f induced by a BSDE with jumps, in a Markovian framework. This study is the first one in this context which considers the case of a noncontinuous reward function.We establish a weak dynamic programming principle (DPP), which extends that obtained by Bouchard-Touzi in the case of linear expectations to the case of E^f -expectation and Borelian terminal reward function.

## Probability and Other Branches of Mathematics

Speaker(s):
Jordan Stoyanov (Newcastle University)
Date:
Wednesday, November 11, 2015 - 6:00pm
Location:
TU Berlin, Raum MA042, Straße des 17. Juni 136, 10623 Berlin

This lecture will be addressed to a wide audience: from undergraduate and graduate students in Mathematics, Statistics, Physics and Computer Science, to MSc and PhD students in these areas, and ... even to professionals. It will be shown that diverse and intriguing problems from branches of Mathematics such as Analysis, Combinatorics, Theory of Functions, Number Theory can be solved elegantly by using ideas and techniques from Probability. It is remarkable that sometime these are the only available solutions.

## Paracontrolled KPZ equation

Speaker(s):
Nicolas Perkowski (HU Berlin)
Date:
Wednesday, October 14, 2015 - 6:00pm
Location:
TU Berlin, Raum MA041, Straße des 17. Juni 136, 10623 Berlin

I will present several results that can be obtained by analyzing the KPZ equation using paracontrolled distributions. For example, we will see that the solution to KPZ is given by the value function of an optimal control problem, where a Brownian motion is steered through a white noise potential. We will also study discretizations la Sasamoto-Spohn and prove their convergence to the KPZ equation or a modified version of it.

This is joint work with Massimiliano Gubinelli.

## Stochastic mean curvature flow

Speaker(s):
Martina Hofmanová (TU Berlin)
Date:
Wednesday, October 14, 2015 - 5:00pm
Location:
TU Berlin, Raum MA041, Straße des 17. Juni 136, 10623 Berlin

Motion by mean curvature of embedded hypersurfaces in R^{N+1} is an important prototype of a geometric evolution law and has been intensively studied in the past decades. Mean curvature flow is characterized as a steepest descent evolution for the surface area energy and constitutes a fundamental relaxation dynamics for many problems where the interface size contributes to the systems energy. One of the main difficulties of the mean curvature flow is the appearance of topological changes and singularities in finite time.

## Stochastic dynamics near a change of stability (Amplitude- and Modulation-Equations)

Speaker(s):
Dirk Blömker (Universität Augsburg)
Date:
Wednesday, July 15, 2015 - 5:30pm
Location:
Universität Potsdam, House 8, Room 0.58

Modulation- or Amplitude-Equations are a universal tool to approximate solutions of complicated systems like partial or stochastic partial differential equations (SPDEs) near a change of stability, when there is no center manifold theory available. One can rely on the natural separation of time-scales at the bifurcation to show that the solution of the original equation is well described by the bifurcating pattern with an amplitude that is slowly modulated in time and also in space, if the underlying domain is sufficiently large.

## Further Advances in Nonconventional Limit Theorems

Speaker(s):
Yuri Kifer (Hebrew University Jerusalem)
Date:
Wednesday, July 1, 2015 - 6:00pm
Location:
TU Berlin, Raum MA041, Straße des 17. Juni 136, 10623 Berlin

Nonconventional limit theorems deal with the asymptotic behavior of sums of the form \sum_{n=1}^N F(\xi(q_1(n)), \xi(q_2(n)), ... \xi(q_l(n))) where F is a function, \xi(n); n \geq 0 is a stochastic process with some stationarity properties, in particular, it can be generated by a measure preserving transformation T in the form \xi(n) = f \circ T^n where f is a function. The functions q_j (n); j = 1,...,l take on nonnegative integer values on nonnegative integers and they satisfy some properties, for instance, they may have the form q_j(n) = jn.

## Rank-based Markov chains, self-organized criticality, and order book dynamics

Speaker(s):
Jan Swart (Prag)
Date:
Wednesday, June 17, 2015 - 6:00pm
Location:
TU Berlin, Raum MA041, Straße des 17. Juni 136, 10623 Berlin

In this talk, we will take a look at some systems of interacting particles on the real line, where the only spatial structure that is relevant for the dynamics is the relative order of the particles.

## Non-colliding Ornstein-Uhlenbeck bridges and symmetry breaking in a quantum 1D Coulomb system

Speaker(s):
Sabine Jansen (Bochum)
Date:
Wednesday, June 3, 2015 - 6:00pm
Location:
TU Berlin, Raum MA041, Straße des 17. Juni 136, 10623 Berlin

Jellium is a model where negatively charged electrons move in a uniform neutralizing background of positive charge. Eugene Wigner conjectured that at low density, the electrons should crystallize, i.e., form a periodic lattice. We prove that in dimension 1, in a quantum mechanics setup, this actually happens for all temperatures and densities, thereby extending low-density results by Brascamp and Lieb (1975) and classical results by Aizenman and Martin (1980).

## Central Limit Theorem for additive functionals of some Markov processes: anomalous results

Speaker(s):
Patrick Cattiaux (Toulouse)
Date:
Wednesday, May 6, 2015 - 6:00pm
Location:
TU Berlin, Raum MA041, Straße des 17. Juni 136, 10623 Berlin

In this talk we will consider an ergodic Markov process $X_t$ ($t \in \mathbb N$ or $t \in \mathbb R^+$)  with unique invariant probability $\mu$, and some additive functional $S_t=\sum_{k=1}^t \, f(X_k)$ or $S_t=\int_0^t \, f(X_s)ds$ for some $\mu$ centered $f$.