This is a Stochastics Student Seminar.

Below is a short description; further information can be found at the Homepage of the seminar.

Important practical info

The seminar takes place every Wednesday between 12.30-14.00 in room SR 009, Arnimallee 6.

Around one week before your presentation, please send us the current draft of the handout you will provide at the emails:

lgaleati@zedat.fu-berlin.de

immanuel.zachhuber@fu-berlin.de

You can use the same contacts to arrange a meeting with us to discuss anything about your part which is not clear or on which you would like some advice for the presentation.

Content
In many applications one must face multiscale systems, characterized by the (nonlinear) interactions of different (space-time) scales; one is often interested in reducing the complexity of the system, by deriving effective equations involving only one (or few) scales. This seminar focuses on such deriation by means of the mathematical techniques of averaging and homogenization; they can be interpreted as perturbative expansions around linear equations. Averaging is as a first order perturbation theory and a law of large numbers result, while homogenization is a second order theory which corresponds to a central limit theorem.

Literature

Our main reference will be the selected chapters from the book "Multiscale Methods - Averaging and Homogenization" by Grigorios A. Pavliotis & Andrew M. Stuart. Although its exposition is very nice, the book is intended for an applied audience and is not always fully rigorous. More rigorous but technical sources, which the students may use to complement the book, are the following:
- Chapter 12 of the book "Markov processes" by Ethier, Kurtz;
- Chapter II.3 of the book "Asymptotic Methods in the Theory of Stochastic Differential Equations" by Skorokhod;
- The lecture notes "Martingale approach to some limit theorems" by Papanicolau, Stroock, Varadhan.

Further information

Prerequisites:  Stochastics I und II; basic knowledge of linear PDEs (elliptic, parabolic) is needed. Either some background in Stochastic Analysis, or simultaneous attendance of Stochastics III, is advised.

Target group:  BMS students, Master students and advanced Bachelor students.