# TCC Course on Large Deviation Theory

**Content:** The theory of large deviations is concerned with estimating the probability of rare events. In many large stochastic systems a "law of large number" holds, i.e. asymptotically the system behaves in a

deterministic way. The theory of large deviations is concerned with the question of how quickly the probability to see behaviour different from the one predicted by the "law of large numbers" decays to zero. It turns out that in many cases this decay takes place extremely quickly, namely exponentially, and we will be concerned with finding the rate of exponential decay.

This theory was initiated in the 30s by Cramer and Sanov, but the modern theory is based on the abstract formalism developed by Donsker/Varadhan and Freidlin/Wentzell in the seventies. It has proven to be applicable for example in statistical physics or statistics.

The first half will contain the classical theorems of Large deviation theory, i.e. Cramer Thm (i.e LD for iid sequence), Sanov Theorem (LD for empirical measure), Schilder Theorem (LD for Brownian motion), Gärtner-Ellis Theorem (LD for correlated sequences). I will explain in some detail the Varadhan-Bryc Theorem, that an LDP is equivalent to the behaviour of certain exponential integrals and exponential tightness. This first part will not require a lot of background. It should be enough if know the content of an introductory probability course and the definition of Brownian motion (perhaps Girsanov (Cameron-Martin) Thm, but I will explain that).

In the second half of the course I want plan to discuss LD for time continuous processes. In particular, I want to discuss LD for Markov processes (Donsker Varadhan Theory) and LD for small noise perturbation of dynamical systems (Freidlin-Wentsell Theory). I plan to introduce a modern technology (discussed for example in the recent book by Feng and Kurtz) to proof these estimates. I also plan to discuss some applications to PDE.

**Validation:**

The validation will be through a short essay. Throughout the course I will mention some papers on topics that are related to the course, and I will ask the students who are interested in validation to choose one of these papers, to read it and explain the content of the paper and the relation to the course in a short ~10 page essay.

**Literature:**

Wolfgang König * Große Abweichungen, Techniken und Anwendungen,* * Lecture notes (in German), available online at *

*http://www.wias-berlin.de/people/koenig/www/GA.pdf
*

Frank den Hollander *Large deviations*

Amir Dembo, Ofer Zeitouni* Large deviations techniques and applications*

Jean-Dominique Deuschel, Daniel Stroock* Large Deviations *

Jin Feng, Thomas Kurtz *Large deviations for stochastic processes*

**Lectures:**