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19208111 Masterseminar Stochastics "Averaging and Homogenization in Multiscale Systems"

Winter Term 2024/2025

Lecturer: Lucio Galeati,  Immanuel Zachhuber

  • Time and place: Wednesdays, 12--14h, SR 009, Arnimallee 6

If you want to participate, please write an email to lgaleati@zedat.fu-berlin.de

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.

Contents:  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.
We will follow selected chapters from the book "Multiscale Methods - Averaging and Homogenization" by Grigorios A. Pavliotis & Andrew M. Stuart. 

Talks

Date Subject Speaker
16.10. preliminary discussion Lucio Galeati
23.10. no seminar  
30.10. Background on SDEs Immanuel Zachhuber
06.11. Averaging and invariant manifolds for ODEs Erwin Heiser
13.11. Averaging for SDEs: derivation of limit equation Carlos Villanueva
20.11. Averaging for SDEs: applications Irmak Metin
27.11. Averaging for SDEs: proof of the convergence the- orem Mihail Birsan
04.12. Homogenisation: introduction, formal derivation and first properties N.N.
11.12. Homogenisation: rigorous proof Ana Damnjanovic
18.12. Homogenisation: applications N.N.
2025    
08.01. N.N. TBA
15.01. N.N. TBA
22.01. N.N. TBA
29.01. N.N. TBA
05.02. N.N. TBA
12.02. N.N. TBA

Literature As mentioned, our main reference will be the selected chapters from the book by Pavliotis and 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.