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19242101 Stochastik IV "Stochastic Homogenisation"

Summer Term 2021

Lecture & exercise: Dr. Ana Djurdjevac


Time and place: Please note, that the lecture & exercise will 

 start in the second week
  • Lecture: Mondays, 14--16h, online, starting on April 19th.
  • Exercise Session: Wednesdays, 10--12h, online, starting on April 21st.
  • Final Exam:  to be announced in due course

Criteria:  weekly exercises, regular participation in the excercise session, exam


Requirements: Prerequisites are Analysis I-III and Stochastics I and II. Functional analysis is helpful but not required.

Exercises 

Problem sets will be put online every Wednesday and can be found under Assignements in the MyCampus/Whiteboard portal. Solutions (in pairs!) are due on Wednesday of the following week  –  please submit either by uploading the solutions via Whiteboard or by handing them in via email.

Course Overview/ Content:

"Stochastic Homogenisation"
First we will recall some basic results about Markov processes and ergodic theory that will be needed in the rest of the course. We will then study the equations on a microscale level with random (periodic) coefficients and characterize the solutions as the scale of the microstructure tends to zero. In the second part of the course we will consider averaging and homogenization of stochastic differential equations. We will first derive the limiting equations and then prove the corresponding convergence theorems. Along the way we will comment on some applications of the stochastic homogenization theory. Moreover, we will mention the numerical methods in stochastic homogenization. In particular, approximations of effective coefficients.

Topics:

  • Basics about Markov processes and ergodic theory;
  • Homogenization of partial differential equations with random coefficients;
  • Averaging for Markov chains;
  • Avergaing for SDEs: derivation of the limiting equation and convergence theorem;
  • Homogenization for SDEs: derivation of the limiting equation and convergence theorem;
  • Applications;
  • Numerical methods

References

  • Pavliotis, Grigoris, and Andrew Stuart: Multiscale methods: averaging and homogenization. Springer Science & Business Media, 2008.
  • Bensoussan, Alain, Jacques-Louis Lions, and George Papanicolaou: Asymptotic analysis for periodic structures. Vol. 374. American Mathematical Soc., 2011.