Math 595, Spring 2005

(Section LD, Course Reference Number 38183)

The study of *rare events* occupies
an important part in many probabilistic analyses. While there are many
types of rare events, the theory of *exponentially small* events
appears frequently enough that it deserves special consideration.
We will explore the circle of ideas known as *large deviations*,
which studies the structure and transformations of such exponentially
small events. We will start with a study of how quickly the
law of large numbers holds, develop some general theories, such as
the Gartner-Ellis theorem and the contraction principle, and then apply
them in a number of ways. Along the way, we will understand connections
with other areas of mathematics and engineering; for example, the connection
between the exit problem and singularly perturbed PDE's,
and the connection between empirical measures of Markov chains and
entropy.

- Lecture 1: Some ``Large Deviations'' problems
- Lecture 2: Laplace asymptotics
- Lecture 3: Abstract Calculations of Large Deviations
- Lecture 4: Two basic LDP's
- Lecture 6: Gartner-Ellis
- Lecture 5: Sanov's Theorem
- Lecture 7: Freidlin-Wentzell
- Lecture 8: Donsker-Varadhan results
- Level I
- Level II
- Level III
- Lecture 9: Gibbs Conditioning Principle
- Lecture 10: Queueing Theory (Tentative)
- Lecture 11: Hydrodynamic Limits (Tentative)

- Dembo and Zeitouni,
*Large Deviations Techniques and Applications*, 2nd ed., 1996, Springer-Verlag - Freidlin and Wentzell,
*Random Perturbations of Dynamical Systems*, 2nd ed., 1989, Springer-Verlag

**Instructor:** Richard Sowers

**Office:** 347 Illini Hall

**Phone:** (217) 333-6246

**email:** r-sowers@math.uiuc.edu

**Home page:** https://math.uiuc.edu/~r-sowers

**Class meets:** Mondays, Wednesdays, and Fridays 11-11:50 A.M. in 443 Altgeld Hall

**Office Hours:** MWF, 10-10:50