Math.Koll. am 10.01.07/Prof. Schreiber

[Institut für Mathematik]

Mathematisches Kolloquium

EINLADUNG

zu einem

VORTRAG

von

Prof. Bernd Schreiber (Wayne State University)

mit dem Thema:

''From Stationary to Nonstationary Stochastic Processes on Groups: How can
we extend the Spectral Analysis?''

Abstract:
Let $G$ be a locally compact group and $X = \{X_t,\ t\in G\}$ be a
stochastic process taking values in $L^2_H = L^2(\Omega,\mathcal{A},P;H)$,
the space of $H$-valued Bochner square-integrable functions on the
probability space $(\Omega,\mathcal{A},P)$. If $G$ is abelian with character
group $\widehat{G}$ and $X$ is stationary (in the wide sense), then the
classical theorem of Bochner implies that the covariance function of $X$ is
given in terms of the Fourier-Stieltjes transform of a positive measure
$\mu$ on $\widehat{G}$. This leads to the existence of an $L^2_H$-valued
``stochastic measure'' whose Fourier transform is $X$ and to many
applications in the sciences.

Similar results for nonabelian groups were obtained by Yaglom in the 1960's,
which we shall describe. In this case the theory of unitary representations
plays a role.

Numerous efforts have been made to study classes of nonstationary processes
on $G$ which preserve some analogues of the connection to the Fourier
transform enjoyed by stationary processes on abelian groups. In this talk
some of these classes and the relationships between them will be presented.
In particular, one can describe those processes for which a stochastic
measure whose Fourier transform is $X$ exists.

Zeit: Mittwoch, 10. Januar 2007, 15.45 Uhr (Kaffeejause), anschlieszend
16.15 Uhr Vortrag

Ort: Fakultaet fuer Mathematik der Universitaet Wien, Nordbergstr. 15,
Seminarraum C 2.09

Harald Rindler
Hans Georg Feichtinger