Stochastic Dynamical Systems

Math 553 - Fall 2014

Section information: Classes meet on Tuesdays and Thursdays from 1:00PM to 2:30PM at Cupples I 215.

Subject: This is a course on the theory of dynamical systems with an emphasis on probabilistic topics. More specifically, we are going to study the ergodic theory of both deterministic and random processes with a particular focus on entropy theory. If there is time and interest, we may also cover some rudiments of classical statistical physics.

Text: There is no official text. I plan to distribute my lecture notes in class as I write them, and will indicate on them the sources I use when preparing the lectures. I anticipate that the following texts will be useful: An Introduction to Ergodic Theory by Peter Walters (Springer, Graduate Texts in Mathematics volume 79, 1982); Introduction to the Modern Theory of Dynamical Systems by Anatole Katok and Boris Hasselblatt (Cambridge University Press, 1995); Entropy in Dynamical Systems by Tomasz Downarowicz (Cambridge University Press, 2011); Mathematical Theory of Nonequilibrium Steady States by Da-Quan Jian, Min Qian, Min-Ping Qian (Spriger Lecture Notes in Mathematics 1833, 2004).

Tentative list of topics:

• Some topics from probability theory:
  • Review of Lebesgue spaces, conditional expectations, stochastic processes.
  • General properties of measure preserving transformations and flows;
  • Martingale convergence and the Ergodic Theorem.


• Ergodic Theory
  • Classes of examples, isomorphism, spectral invariants;
  • Markov shifts;
  • Measure theoretic entropy, topological entropy;
  • Some smooth ergodic theory of hyperbolic systems; Pesin formula;
  • Rudiments of information theory, Ornstein's theorem.


• Statistical Mechanics
• Review of (geometric) concepts in mechanics;
• Classical statistical mechanics, equilibrium states;
• Classical lattice models;
• Ideas from the theory of nonequilibrium steady states.

Coursework: A variety of exercises will be proposed throughout the course, although I will not collect them. At the end of the course students (only those who are enrolled) should submit a short paper on a topic related to the course (suggested length between 5 and 10 pages) and will give a presentation on the topic of the paper (of approximately 30 minutes). I will be glad to propose themes if you like.