MATH 7770: Probability and Analysis in Infinite-Dimensional Spaces (Fall 2011)

Course Web Site: www.math.cornell.edu/~neldredge/7770/

This course will be an introduction to probability and analysis in infinite-dimensional state spaces.

In finite dimensions we know a lot about the relationship between Brownian motion (a Gaussian process with independent increments) and the Laplacian and its friends. We will begin the course by looking at abstract Wiener space (as introduced by our own Len Gross), which is essentially a Banach space with a Gaussian measure on it, and seeing how these relationships carry over (or not!). For this part of the course I plan to follow H. H. Kuo's lecture notes entitled Gaussian measures in Banach space, which are available online (from Cornell IP addresses) at http://www.springerlink.com/content/978-3-540-07173-0/

After this we can proceed based on the interests of students and instructor. Possible topics include:

Quasi-regular Dirichlet forms in infinite dimensions, and the processes who love them
Infinite-dimensional Lie groups modeled on abstract Wiener space, a la Driver-Gordina, with elliptic and hypoelliptic diffusions
Infinite-dimensional “Malliavin” calculus and its applications
Curved Wiener spaces; differential geometry in infinite dimensions

Prerequisites correspond roughly to MATH 6110 and 6710-6720. You should be familiar with measures, Banach/Hilbert spaces, and Brownian motion.