Math 7770: Probability and Analysis in Infinite-Dimensional Spaces

Meets

Personnel

• Instructor: Nate Eldredge
  • Email: neldredge@math.cornell.edu
  • Office hours: TBA, or by appointment (send email)
  • Office: 593 Malott Hall

Course overview

In finite-dimensional spaces like Rⁿ, we have a lot of nice structure for doing analysis: most notably, the translation-invariant Lebesgue measure. We also know a lot about how analysis and probability are related, such as the relationship between the Laplace operator and Brownian motion. In infinite-dimensional spaces, such as Banach spaces, we can no longer have a translation-invariant measure. However, we can have Gaussian measures, and this is enough structure to construct operators and stochastic processes that are reasonable analogues of their classical brethren.

We will start with a quick look at Gaussian measures in finite dimensions to get a feel for the properties we want to extend. Then we'll look at Gaussian measures on infinite-dimensional spaces, including Banach and Hilbert spaces (which yield so-called abstract Wiener spaces), and possibly other topological vector spaces. We'll see a certain Hilbert space emerge (the Cameron-Martin space) which will show up repeatedly as we proceed. We'll also see how to reverse the process and construct a Gaussian measure from a "covariance matrix".

Then we'll see how to compute derivatives and Laplacians in abstract Wiener space, and how to construct a Brownian motion on such a space.

Possible topics for later in the course may include:

• Logarithmic Sobolev inequalities: How to relate smoothness and integrability.
• Quasi-regular Dirichlet forms in infinite dimensions, and the processes who love them. Analogues of the Ornstein-Uhlenbeck process make sense on abstract Wiener space, and can be studied with the machinery of Dirichlet forms.
• 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, such as the probabilistic proof of Hormander's hypoellipticity theorem
• Curved Wiener spaces; differential geometry in infinite dimensions

Prerequisite

Lecture notes

The lecture notes from this course are now posted on arXiv (id 1607.03591).

Other references

• Bruce Driver's probability notes: PDF (see Part VIII)
• H. H. Kuo, Gaussian Measures in Banach Spaces. Electronic version available free from Cornell IPs at Springerlink (link, includes downloadable PDF chapters). Also available fairly cheap in paperback, search your favorite book sites (mine is addall.com) for ISBNs 978-3540-07173-0 and 978-1419-64580-8 (there are two equivalent editions from different printers).
• D. Nualart, The Malliavin Calculus and Related Topics. Electronic version available free from Cornell IPs at Springerlink (link, includes downloadable PDF chapters). Springerlink will also sell you a nice paperback for $25; click the "Buy a print copy" link or else here.
• V. Bogachev, Gaussian Measures. I will put this on reserve at the library.
• Gross, Leonard. Abstract Wiener measure and infinite dimensional potential theory. 1970 Lectures in Modern Analysis and Applications, II pp. 84-116 Lecture Notes in Mathematics, Vol. 140. Springer, Berlin. PDF (Springerlink)
• Driver, Bruce K.; Gordina, Maria. Heat kernel analysis on infinite-dimensional Heisenberg groups. J. Funct. Anal. 255 (2008), no. 9, 2395-2461. Link to journal.
• Üstünel, A. S. Analysis on Wiener space and Applications. arXiv

Homework

Exams