### Inhalt des Dokuments

# Berliner Kolloquium Wahrscheinlichkeitstheorie und Seminar RTG 1845

**Sommerstemester 2015**

Datum | Sprecher GK-Seminar 17:15-18:00 | Sprecher Kolloquium (18:00-19:00) | Ort |
---|---|---|---|

22. 4. 2015 | (keiner, Ausstellung) | Herbert Spohn (München) 17:15-18:15 | U Potsdam Haus 8, Raum 0.58 |

6.5.2015 | Maite Wilke Berenguer (TUB) | Patrick Cattiaux (Toulouse) | TU Berlin MA 041 |

20. 5. 2015 | (kein Kolloquium, Euler-Lecture am 22. 5.) | U Potsdam | |

3.6.2015 | Alberto Chiarini (TUB) | Sabine Jansen (Bochum) | TU Berlin MA 041 |

17.6.2015 | Matti Leimbach (TUB) | Jan Swart (Prag) | TU Berlin MA 041 |

1.7.2015 | Alexandros Saplaouras (TUB) | Yuri Kifer (Jerusalem) | TU Berlin MA 041 |

15.7.2015 | Jennifer Krüger/Eva Lang (TUB) | Dirk Blömker (Augsburg) | U Potsdam |

Das Kolloquium im Sommersemester findet an der TU Berlin (MA 041) und der U Potsdam statt. Organisation: S. Roelly (UP) und N. Kurt (TUB).

# Abstracts

22. April

Kolloquium: Herbert Spohn (TU München)

The KPZ (Kardar-Parisi-Zhang) equation and its universality class

Abstract: The one-dimensional KPZ equation is a stochastic PDE which describes the dynamics of surface growth. It is one representative of a much larger universality class. I will discuss a few models in this class and explain how they are connected. They all share the common feature to be stochastic integrable.

6. Mai:

RTG-Seminar: Maite Wilke Berenguer (TUB)

Lipschitz PercolationAbstract:

Kolloquium: Patrick Cattiaux (Toulouse)

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

Abstract: 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 $f$.

If $f \in \mathbb L^{2}(\mu)$ the expected appropriate normalization is $\sqrt{\Var(S_t)}$ (expected to be of order $\sqrt t$), and the expected limit is then a standard gaussian. If $f \in mathbb L^p(\mu)$ ($1<p<2$) one expects in some cases some stable limit after appropriate normalization. It turns out that the mixing rate of the process (equivalently the rate of convergence to equilibrium) is of particular importance for these results to hold true. We shall recall some of the main recent (and less recent) results in this direction and explain how the mixing rate enters into the game.

We shall also discuss a particular class of examples for which depending on whether the convergence to equilibrium is quick enough or not, anomalous limit (with some variance breaking) or anomalous normalization appear. At the level of the invariance principle instead of the simple CLT theorem, the expected limiting process becomes a fractional Brownian motion instead of the usual one. These examples correspond to a special class of kinetic P.D.E.'s with heavy tails equilibria.

3. Juni:

RTG-Seminar: Alberto Chiarini (TUB)Extremes of the supercritical Gaussian Free FieldAbstract: We show that the rescaled maximum of the discrete Gaussian Free Field (DGFF) in dimension larger or equal to 3 is in the maximal domain of attraction of the Gumbel distribution. A finer description of the maximum can also be obtained, that is, the associated extremal process converges to a Poisson point process. These results holds both for the infinite-volume field as well as the field with zero boundary conditions. The proofs follow from an interesting application of the Stein-Chen method from Arratia et al. (1989).Joint work with Alessandra Cipriani (WIAS) and Rajat Subhra Hazra (Indian Statistical Institute)

Kolloquium: Sabine Jansen (Bochum)

Non-colliding Ornstein-Uhlenbeck bridges and symmetry breaking in a quantum 1D Coulomb systemAbstract:

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). The proof uses the Feynman-Kac formula to map the quantum model to asystem of non-colliding Ornstein-Uhlenbeck bridges, and then applies the Krein-Rutman theorem (an infinite-dimensional version of Perron-Frobenius). The talk is based on joint work with Paul Jung (University of Alabama at Birmingham).

17. Juni

RTG-Seminar: Matti Leimbach (TUB)

Porous medium equation with proliferation

Abstract:

Motivated by mathematical oncology, we present two PDE's, the viscous poruous medium equation and the Fisher-Kolmogorov-Petrovskii-Piskunov model (FKPP). Formally, we derive the viscous porous medium equation as the limit of the empirical measure of a system of interacting particles with intermediate interaction-range and large amplitude. We illustrate that also the FKPP model is a limit of a particle system with certain branching mechanism.

At the end, we briefly discuss a combination of these two PDE's, the porous medium equation with proliferation. This is based on ongoing research with Franco Flandoli.

Kolloquium: Jan Swart (Prag)

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

Abstract:

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. Examples of such systems are

the modified Bak-Sneppen model, introduced (as a variation of the original

1993 model) by Meester and Sarkar (2012), Barabási's (2005) queueing system

and a variation on the latter due to Gabrielli and Caldarelli (2009), a model

for the evolution of the state of an order book on a stock market, introduced

by Stigler (1964) and independently by Luckock (2003), and a two models for

canyon formation introduced by me (2014). All these systems employ a version

of the rule "kill the lowest particle" and seem to exhibit self-organized

criticality at a critical point that marks the boundary between an interval

where all particles are eventually removed and an interval where particle stay

in the system forever.

1. Juli

RTG-Seminar: Alexandros Saplaouras (TU Berlin)

Towards a robustness result for BSDEs with jumpsMotivated by the robustness of BSDEs with respect to the Brownian motion, see \cite{BDM}, we want to prove that the same holds when the BSDE is taken with respect to a square integrable, quasi-left-continuous martingale $M$. The robustness of a BSDE stands for the following property: having a suitable martingale approximation $M^n$ of $M$, then the solutions of the BSDEs driven by $M^n$, converge to the solution of the BSDE driven by $M$. In order to obtain the result, we need to overcome two intermediate problems. The first is to guarantee the existence and uniqueness of solutions of BSDEs driven by $M^n$. In this case, the predictable quadratic covariation of $M^n$ may have jumps, hence the Lebesgue-Stieltjes integral is not necessarily a continuous process. In this work we improve a general result of existence and uniqueness for BSDEs, see \cite{EKH}, where the Lebesgue-Stieltjes integral is with respect to a continuous, predictable and increasing process. Our improvement consists in allowing the integrator of the Lebesgue-Stieltjes integral having (suitably small) jumps, i.e. being a c\`adl\`ag, predictable and increasing process. The second problem consists in proving that the corresponding stochastic and Lebesgue-Stieltjes integrals with respect to $M^n$ and the predictable quadratic covariations $\pqc{M^n}$ respectively converge to the stochastic and Lebesgue-Stieltjes integral with respect to $M$ and the predictable quadratic covariation $\pqc{M}$ respectively. Once this second obstacle is overcome, we could proceed to proving the desired result. As a byproduct of this result, the convergence of the Euler scheme for BSDEs is obtained, where $M^n$ is the time discretization of $M$.

Kolloquium: Yuri Kifer, (Hebrew University, Jerusalem)

Further advances in nonconventional limit theorems

15. Juli (Beginn 16:30/17:30, Ort: U Potsdam)

RTG-Seminar: Eva Lang (TUB)

A multiscale analysis of traveling waves in stochastic neural fields

Kolloquium: Dirk Blömker (Augsburg)Stochastic dynamics near a change of stability (Amplitude- and Modulation-Equations)