Günter Last (KIT)

Stochastic analysis for the Dirichlet-Ferguson process

The Dirichlet--Ferguson process $\zeta$ is a random, purely discrete, probability measure, whose finite-dimensional distributions are Dirichlet distributions.
It can be defined on a general state space and has numerous applications, such as in population genetics.
We shall present the fundamental chaos expansion by Peccati (2008), providing an explicit formula for the kernel functions.
Then we proceed with developing a Malliavin calculus for $\zeta$. To this end we introduce a gradient, divergence and a generator which act as linear operators on $\zeta$-measurable random variables or random fields and which are linked by some basic formulas such as integration by parts.
While this calculus is strongly motivated by Malliavin calculus for isonormal Gaussian processes and the general Poisson process, the strong dependence properties of $\zeta$ require considerably more combinatorial efforts. We will identify our generator as the generator of the Fleming--Viot process and to describe the associated Dirichlet form explicitly in terms of the chaos expansion. If time permits, we shall also present a short direct proof of the Poincaré inequality.

The talk is based on joint work with Babette Picker (Karlsruhe).

Jan 26

2:30-
6:00 pm

at RUB

Vitali Wachtel (Bielefeld)
Berry-Esseen inequality for random walks conditioned to stay positive

Patricia Alonso Ruiz (Jena)
An isoperimetric inequality for (fractal) spaces with different micro- and macrostructure

David Criens (Freiburg)
Stochastic control problems with irregular coefficients and $L_d$-drift

Dec 15

Panagiotis Spanos (University of Bochum)

Percolation on transitive graphs of polynomial growth.

In this talk, we will discuss a special model of percolation, called spread-out percolation. Specifically, let $G$ be a vertex-transitive graph equipped with the graph metric, and let $r>0$ be a range parameter. We define $G_r$ to be the graph with the same vertex set as $G$, where two vertices are connected by an edge whenever their distance in $G$ is at most $r$. The spread-out percolation on $G$ with degree parameter $\lambda$ and range $r$ is the Bernoulli percolation on $G_r$ in which each edge is retained with probability $\lambda / \deg(G_r)$. We prove that for graphs of superlinear polynomial growth,
$$
p_c = \frac{1+o(1)}{\deg(G_r)}
\quad\text{as } r\to\infty,
$$
that is, the critical degree parameter for the existence of an infinite cluster tends to $1$ as $r$ grows. This theorem generalises a result of Penrose for $\mathbb{Z}^d$. The model was introduced by Hara and Slade in the study of mean-field behaviour. We will present open problems as well as new results in the area. Based on joint work with M. Tointon.

Oct 27

2:30-
6:00 pm

at RUB

Helene Götz (Dortmund)
Large deviations for spectral measures of random matrices under nonstandard scaling

Philipp Tuchel (Bochum)
Asymptotic behavior of random projections of high-dimensional $\ell_p^N$ balls

Roman Gambelin (Essen)
An extension of the algebraic Aldous diffusion