STOCHASTIC ANALYSIS

Meets Wednesdays, 2 PM in Parker 236

 

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



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. 

Last updated: 08/31/2015