Universität Duisburg-Essen
Fachbereich Mathematik, Campus Essen
Anita Winter

22th March 2018

Special Topic Course with integrated block seminar (2SWS+2SWS)
Sommer Semester 2018
Tuesday, 14.15-15.45; WSC-S-U-4.03

Regularity of Gaussian processes (Vertiefungsmodul Stochastik)

Brownian motion is a centered Gaussian Markov process indexed by the half-line and having stationary independent increments. We think of the index set as of the time line along which Brownian motion evolves. Alternatively, to define Brownian motion, we could have just postulated the covariance structure and require that the path are a.s. continuous. Often, complementary to processes evolving in time, we want to model spatial processes, where to each point in space we attach a real random variable. (As an example, one can think, e.g., of the temperature distribution on a planar surface). Such spatially indexed collections of random variables are called random fields.

In this course we will consider Gaussian processes indexed by the positive real line, extend them to Gaussian fields. We will address the problem of finding necessary and sufficient conditions for the continuity of sample paths of a Gaussian process in terms of its covariance functions. This problem is well-posed since a centred Gaussian process is determined by its covariance function. The starting point will be Kolmogorov's criterion on a Hölder continuous version. This classical result gives necessary and sufficient conditions for continuity when the covariance function is sufficiently regular. We will then abandon Kolmogorov's approach to come up with a criterion on continuity developed by Dudley, by Fernique and by Talagrand.

### Literature

• Markus and Rosen "Markov processes, Gaussian processes and Local times" (2006)

When does the class start?
The class starts on Tuesday, 17th of April.

What should I know if I want to participate?
This special topic course (Vertiefungsmodul Stochastik) is offered as part of the Master Program in Mathematics. Nevertheless, if you are interested in probability theory you are very welcome to participate. Preknowledge in probability theory, stochastic processes and Brownian motion (as covered by Wahrscheinlichkeitstheorie II) are required. Don't hesitate to send me an e-mail for more information:

How many credits do I get?
This class has a block seminar integrated. The topics will usually be papers which rely on the material presented in the class. It will take place in the end of the summer semester break. If you only take the class and pass an oral exam you get 3CTS as Vertiefungsmodul Stochastik. Alternatively, you can present a paper in the block seminar. In that case you get 6CTS as either Vertiefungsmodul Stochastik or Master Seminar.

anita(dot)winter `at' due(dot)de