Graduate Course "Weak convergence methods for large deviations"

Large deviation theory deals with the decay of the probability of increasingly unlikely events. It is one of the key techniques of modern probability, a role which is emphasised by the award of the 2007 Abel prize to S.R.S. Varadhan, one of the pioneers of the subject. A relatively recent approach to this theory is based on the analogy to weak convergence of probability measures and was developed by Puhalskii, O'Brien and Vervaat, de Acosta and others. We base our study of the method on the recent book by Feng and Kurtz `Large deviations for stochastic processes'.

In the course I will use the first three lectures to present background and basic principles of large deviation theory. Then the participants take over and present the core material of the book (Chapters 5 to 8) and a selection of examples (Chapter 9 to 13). Here is a provisional schedule:

Lecture 1 (Oct 6): Introduction to large deviations I
Lecture 2 (Oct 13): Introduction to large deviations II
Lecture 3 (Oct 20): Introduction to large deviations III
Lecture 4 (Nov 3rd): Large deviations for Markov processes and nonlinear semigroup convergence (Chapter 5)
Lecture 5 (Nov 10th): Large deviations and nonlinear semigroup convergence using viscosity solutions (Chapter 6)
Lecture 6 (Nov 17th): Extension of viscosity solution methods (Chapter 7)
Lecture 7 (Nov 24th): The Nisio semigroup and a control representation of the rate function (Chapter 8)
Lecture 8 (Dec 1st): Example I: Small perturbation problems (Chapter 10)
Lecture 9 (Dec 15th): Example II: Random evolutions (Chapter 11)

Course material:

We will provide copies of the relvant sections of the book to those presenting lectures. Lecture notes for the first three lectures are available here to download.

Further material for self-study:

Here is are lecture notes of a graduate course I gave in 2008 presenting a short introduction to large deviations . This is different from the introduction I give in the first three lectures, mostly as it contains additional material and motivation.

An introduction to viscosity solutions by Federica Dragoni can be found here found here.

Prerequisites:

An undergraduate course in measure-theoretic probability and good personal motivation.

The course is scheduled for Wednesdays 10:15 to 12:05 in 4W1.7.

Last changed on 8.10.2010.

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