Stochastics Seminar

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.