Regular speakers and their current areas of interest:

Olav Kallenberg: stochastic analysis and dynamical systems, with applications to statistical mechanics and biological evolution

Ming Liao: stochastic processes in Lie groups

Erkan Nane: fractional diffusions and iterated processes

Jurek Szulga

**2015 - 2016**

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September 2

Speaker: **Dr. Ming Liao**

Title: Convolution of probability measures on Lie groups and homogeneous spaces

Abstract: This is the first of several talks on the subject in the title. The convolution of measures on Euclidean spaces is well known. A convolution semigroup of probability measures is the distribution of a Levy process, that is, a process with independent and stationary increments. This notion naturally extends from a Euclidean space to a Lie group G, and may also be formulated on a more general homogeneous space X = G/K, which is a manifold X under the transitive action of a Lie group G.

I will talk about some basic properties of convolutions and convolution semigroups, some relations between convolution semigroups on G and on X = G/K, the problem of embedding an infinitely divisible distribution on a convolution semigroup, and a Levy-Khinchin type formula on symmetric spaces, which are a special type of homogeneous spaces. A closely related stochastic process will be mentioned from time to time. Some of these results are from my own work, old and more recent.

August 26

Speaker: **Olav Kallenberg**

Title: Local time, excursions, and regeneration

Abstract: A random process $X$ is said to be regenerative at a state $a$, if it enjoys the strong Markov property at visits to $a$. The set of times $t\geq 0$ with $X_t=a$ is typically perfect and nowhere dense and supports a singular and diffuse random measure $\xi$, called the local time of $X$ at $a$. Furthermore, the set of excursions of $X$ from $a$ is given by a Poisson process on the time scale of $\xi$. Our aim is to study the local hitting and conditioning properties of $X$, as described in terms of the density of $E\xi$ and the Palm kernel of $X$ with respect to $\xi$. The talk is based on some work done in stages throughout my career.