Universität Duisburg-Essen
Fachbereich Mathematik, Campus Essen
Anita Winter

08th April 2017

Special Topic Course with integrated block seminar (2SWS+2SWS)
Sommer Semester 2015
Tuesday, 10.15-11.45; WSC-S-U-3.03

Gaussian Processes and Local Times

A realization of discrete GFF on a square grid $60 \times 60$

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 the case of the plane, we can visualize the random field as a random surface in 3 dimensional space, where the height (or depth, if negative) of the surface at each point above (below) the reference level is shown. (See the above picture.)

Gaussian free field (GFF) is a generalization of Brownian motion to such spatially indexed settings. Similarly to Brownian motion, GFF is a centered Gaussian process which satisfies a spatial generalization of the Markov property -- the Gibbs property (sometimes also called Markov random field property). In this course we develop relationships between local times of strongly symmetric Markov processes and the Gaussian Free Field. This was done for Brownian motion over 50 years ago in the famous Ray-Knight theorems. In the first part of the class we discuss Brownian motion with an emphasis on those definitions and properties which we will use in the second part to generalize to a much larger class of Markov processes.

### Literature

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

When does the class start?
The class starts on Tuesday, 18th 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 6CTS as Vertiefungsmodul Stochastik. Alternatively, you can present a paper in the block seminar. In that case you get 9CTS as either Vertiefungsmodul Stochastik or Master Seminar.

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