PL2US Workshop

Third edition: February 21-22, 2013
Universität des Saarlandes

(Details on the first and second editions, held respectively on 2011 in Kaiserslautern and 2012 in Luxembourg, are here and here.)

#### What it is?

- This two-day workshop, held once a year, brings together the members of the probability groups of the following four universities: Technische Universität Kaiserslautern, University of Luxembourg, Université de Lorraine and Universität des Saarlandes. The main goal is to initiate new joint research projects, as well as to encourage young probabilists to take an active part in an international meeting.

- Every year, the workshop features a special lecture given by a guest speaker. This year's guest speaker is Philippe Biane from Marne-La-Vallée (France).

- The Scientific Committee of our workshop is composed of Christian Bender (Saarbrücken), Martin Grothaus (Kaiserslautern), Andreas Neuenkirch (Mannheim), Ivan Nourdin (Lorraine), Giovanni Peccati (Luxembourg), Roland Speicher (Saarbrücken), Anton Thalmaier (Luxembourg) and Samy Tindel (Lorraine).

- The workshop is generously supported by Université de la Grande Région

- The acronym PL2US is for Palatinat Lorraine LUxembourg and Saare.

#### Practical informations

- The registration is closed (since February 3rd).

- The current list of participants can be found here

- The conference will take place in the Günther Hotz Lecture Hall/Günther-Hotz-Hörsaal, building E2 2. Here is an interactive map of the campus and here is a pdf version of it. Here (in html) and here (in pdf) are some explanations on how to reach the university from the central station. (You may also find useful information on this webpage.)

- The conference dinner will be held at Restaurant Trattoria Toscana (Fröschengasse 18-22) on Thursday

- Here are three hotels suggested by the local organizers:
- Motel One (www.motel-one.com)
If you prefer an hotel a bit better equipped, you may try:
- Domicil Leidinger (www.domicil-leidinger.de)
If you prefer an hotel a bit lower priced, you may try:
- Hotel Schlosskrug (www.hotel-schlosskrug.de)

#### Program

Talks are 45 minutes long + 5 minutes for questions.

February 21st

15:00 Reception and registration in the foyer of the math building E2 4

16:00 Roland Speicher (Saarland University): Introduction to the workshop''

16:10 Carola Hodyas (Universität der Grossregion): A short presentation about UGR''

16:20 Aurélien Deya (Université de Lorraine): Rough paths and non-commutative stochastic calculus''

17:10 Coffee break

17:30 Torben Fattler (University of Kaiserslautern): A dynamical wetting model in (d+1)-dimension - Construction and analysis via Dirichlet forms''

18:20 Hatem Hajri (Luxembourg University): Tsirelson singularity and some SDEs''

19:10 End of the first day, followed by a dinner in Restaurant Trattoria Toscana (Fröschengasse 18-22) all together

February 22nd

9:00 Guest Speaker: Philippe Biane (Marne-La-Vallée): Free entropy''

9:50 Tobias Mai (Saarland University): Asymptotic eigenvalue distribution of polynomials in independent random matrices and operator valued free probability theory''

10:40 Coffee break

11:10 Angélo Koudou (Université de Lorraine): Independence properties of the Matsumoto-Yor type''

12:00 Patrik Stilgenbauer (University of Kaiserslautern ) : The Analysis of stochastic fiber lay-down models: New results about hypocoercivity and geometric Langevin equations on manifolds''

12:50 Lunch

14:00 Ehsan Azmoodeh(Luxembourg University): Parameter estimation in a fractional Ornstein-Uhlenbeck model''

14:50 Robert Knobloch (Saarland University): Limit theorems for random characteristics associated with fragmentation processes''

15:40 End of the workshop

#### Abstracts

Ehsan Azmoodeh (Luxembourg University)

Parameter estimation in a fractional Ornstein-Uhlenbeck model

We study estimation problem of drift parameter in a Langevin type equation with fractional Ornstein-Uhlenbeck process of the second kind as its solution. We show that the least squares estimator provides a consistent estimator. Moreover, using multiple Wiener integrals technique, we prove a central limit theorem. The talk is based on two closely related joint works with Igor Morlanes and Lauri Viitasaari.

Philippe Biane (Marne-La-Vallée)

Free entropy

I will give an overview of free entropy, and discuss some recent joint work with Yoann Dabrowski on the additivity problem.

Aurélien Deya (Université de Lorraine)

Rough paths and non-commutative stochastic calculus

Within the last 15 years, Lyons' rough paths theory (RPT) has shed new light on the classical (i.e., commutative) stochastic calculus. After a brief review of the main principles and results of this approach, we will show how to adapt ideas from RPT in the setting of non-commutative probability theory, with a special attention to the free Brownian case. The talk is based on a joint work with René Schott.

Torben Fattler (University of Kaiserslautern)

A dynamical wetting model in (d+1)-dimension - Construction and analysis via Dirichlet forms

We give a Dirichlet form approach for the construction of a distorted Brownian motion in E:=[0,∞)n, nN, where the behavior on the boundary is determined by the competing effects of reflection from and pinning at the boundary. In providing a Skorokhod decomposition of the constructed process we are able to justify that the stochastic process is solving the underlying stochastic differential equation weakly in the sense of N. Ikeda and Sh. Watanabe for $\mathcal{E}$-quasi every starting point. In particular, our considerations enable us to construct a dynamical wetting model (also known as Ginzburg--Landau dynamics) on a bounded set DNZd.

Hatem Hajri (Luxembourg University)

Tsirelson singularity and some SDEs

We review some aspects of stochastic calculus on star graphs: Walsh Brownian motion, spidermartingales, Itô's formula, Walsh filtration... We then study some SDEs on these graphs and their associated stochastic flows. This is joint work with Olivier Raimond.

Robert Knobloch (Saarland University)

Limit theorems for random characteristics associated with fragmentation processes

This talk is concerned with a strong law of large numbers for (Zηφ )η in (0,1), the process counted with a random characteristic φ. For any η this stochastic process is a sum involving independent copies of φ, each associated with a block of a given fragmentation process. For a large class of random characteristics φ we prove almost sure convergence and L1-convergence of Zηφ as η decreases to 0. In particular, we consider two examples of random characteristics for which we obtain almost sure limit theorems that extend previously known L1-convergence results.

Angelo Koudou (Université de Lorraine)

Independence properties of the Matsumoto-Yor type

We prove that, under smoothness assumptions, there are essentially four decreasing functions f defined on the positive real line with the following property: there exist independent, positive random variables X and Y such that the variables f(X+Y) and f(X)-f(X+Y) are independent. The first one is f(x)=1/x. In this case, referred to in the literature as the Matsumoto-Yor property, the law of X is the generalized inverse Gaussian distribution while Y is gamma distributed. We provide the associated densities in the three other cases, among which the Kummer distribution. We also write such an independence property in the case where X and Y are random matrices, where X follows a matrix Kummer distribution and Y a Wishart distribution.

Tobias Mai (Saarland University)

Asymptotic eigenvalue distribution of polynomials in independent random matrices and operator valued free probability theory

Free probability theory was invented around 1986 by D. Voiculescu originally as a tool for the theory of operator algebras. But later on, deep connections to random matrix theory were found, particularly providing an effective way to understand the asymptotic eigenvalue distribution of sums and products of independent random matrices of many types. Nevertheless, it turned out to be a rather intricate problem to deal with more general polynomials in independent random matrices. In my talk, I will explain how the "linearization trick" in a new version due to G. W. Anderson gives in combination with the operator valued generalization of free probability theory an uniform approach for problems of this kind. Some examples will show that the algorithm is easily accessible for numerical computations. This is joint work with S. Belinschi and R. Speicher.

Patrik Stilgenbauer (University of Kaiserslautern)

The Analysis of stochastic fiber lay-down models: New results about hypocoercivity and geometric Langevin equations on manifolds

The so called fiber lay-down models arise in the production process of nonwovens. These new mathematical models have been developed in recent years and provide an interesting interplay between functional analysis, stochastics and differential geometry. In particular, the models are formulated as stochastic differential equations on manifolds. The quality of the nonwoven material is influenced and determined by the convergence to equilibrium of the fiber lay-down process. Especially, one is interested in the speed of convergence. To study the long-time behaviour, mathematical difficulties arising since the equations and their associated generators are degenerate.
In this talk we present an extension of the modern, powerful and simple abstract Hilbert space strategy for proving hypocoercivity that has been developed originally by Dolbeault, Mouhot and Schmeiser. Such hypocoercivity methods imply an exponential decay to equilibrium with explicit computable rate of convergence. Our extension is now made for studying the long-time behavior of some strongly continuous semigroup generated by a degenerate Kolmogorov (backward) operator L. Additionally, we introduce several domain issues into the framework. Necessary conditions for proving hypocoercivity need then only to be verified on some fixed operator core of L. Furthermore, the setting is also suitable for covering existence and construction problems as required in many applications. Afterwards, we apply the extended framework to prove hypocoercivity of our fiber lay-down process.
Moreover, we present a class of new geometric Langevin type equations on regular submanifolds and explain their relation to the fiber lay-down model. In this way, we will get to know a fascinating interaction between applied and pure mathematics.