Math 777 — Fall 2001 Stochastic Processes

You might have come across one of these problems:

Say you toss a coin one million times. You certainly expect about half of these tosses to be heads and half to be tails. How likely is it that this does not happen and you get, say, three times as many heads as you get tails?

You add certain potential V to the heat equation, creating point spectrum. What is the long time behavior of the solutions?

Brownian motion has continuous paths starting at zero. How likely is it to make a big excursion in just a short time?

If independent particles lie in a box, their distribution is just a product one. But, say you add the condition that the total energy has to be a given constant. What is the new distribution?

All these problems involve computing the exponential decay of the probability of certain rare events (events that are not supposed to happen in the limit, according to a law of large numbers). These are the large deviations. In this course we will review the basic tools used in these (and other) simple problems, explore the "general theory" and study its application to more interesting problems.

We will mostly follow the material from Dembo and Zeitouni's Large Deviations Techniques and Applications (2nd edition, Springer, 1998).

The course requires some knowledge of probability (up to Brownian motion and SDEs) and basic PDEs.