Random walks on homogeneous spaces

Tel Aviv University, Spring 2016


Grading Here is the exercise sheet (final version).

If you are taking the course for credit, please submit the exercise sheet by August 1, 2016. The best way to submit the sheet is by sending it to me by email (if you prefer to submit a hand-written project, send me a scan).




Notes In the lecture of 7.3.16 I made some imprecise remarks about the Riesz representation theorem, here is the precise formulation of the results. Details in the book W. Rudin, Real and Complex Analysis , Chapter 2.


References
  • Y. Benoist and J.-F. Quint, Mesures stationnaires et fermes invariants des espaces homogenes I, Ann. Math. 2011.
    This is the first in a series of breakthrough papers, the aim of the course is to prove the main results of this paper. Here is an English translation of the paper.

  • Y. Benoist and J.-F. Quint, Introduction to random walks on homogeneous spaces, Takagi surveys, Japan J. Math. 2012.
    This is a survey in English describing the main ideas and steps of proof of the paper above. Warmly recommended!

  • Y. Benoist and J.-F. Quint, Random walks on reductive groups, to appear.
    A book containing much of the material on random matrix products which we will need -- and more.

  • Y. Benoist and J.-F. Quint, Recurrence on the space of lattices, ICM Proc. Seoul 2014
    A survey of how one deals with the issue of `escape of mass'.

    The above can all be accessed via Yves Benoist's webpage.

  • M. Einsiedler and T. Ward, Ergodic theory with a view toward number theory, Springer 2011.
    A carefully written textbook with useful chapters on ergodic theorems, conditional expectations, etc. Also contains a lot of interesting material we will not cover.

  • A. Furman, Random walks on groups and random transformations, Handbook of dynamical systems, Vol. 1A, North Holland 2002.
    A detailed and well-written survey of many of the results on random matrix products.