Topics in Probability: Gaussian Analysis Math 7880-2, Fall 2018 University of Utah

Course Synopsis. Let \(\mathbb{P}_n\) denote the canonical Gaussian measure - or the standard multivariate normal - on \(\mathbb{R}^n\); that is, \[ \mathbb{P}_n(E) := \int_E \frac{\exp\left(-\frac12\|x\|^2\right)}{(2\pi)^{n/2}}\,{\rm d}x \hskip1in \text{for all Borel sets \(E\) in \(\mathbb{R}^n\)}. \] This is an object that you have seen, say in the context of the classical central limit theorem. And some of you have studied many of the elementary properties of \(\mathbb{P}_n\) in your graduate, and sometimes also undergraduate, courses. In this course we study some of the deeper structure of the "Gauss space" \((\mathbb{R}^n\,,\mathcal{B}(\mathbb{R}^n)\,,\mathbb{P}_n)\). En route we take a tour through a good deal of beautiful mathematics, and also encounter a few applications worthy of note.

Prerequisites. Basic measure-theoretic probability at the level of Math. 6040.

Text. The material of the course is based in part on a set of notes that Tom Alberts and I are in the process of writing. There are chapter links below; they will become live in due time, and will continuously be updated. Also, here is a link to the table of contents.

Topics. Here are some of the topics that we will likely cover.

1. The Finite-Dimensional Theory
   • The Canonical Gaussian Measure on \(\mathbb{R}^n\); Assignment from Chapter 1 (due Sept 14); Announcement: No lectures on Sept 11 and 13th.
   • Calculus in Gauss Space; Assignment from Chapters 2-3 (due October 9);
   • Harmonic Analysis
   • Heat Flow
   • Integration by Parts and Its Applications
2. The Infinite-Dimensional Theory
   • Gaussian Processes
   • Regularity Theory
   • Level Sets

Grading. Grades are based on attendence, and fairly regular assignments.

Basic References.

• Dudley, Richard, M., A Course in Empirical Processes, École d'été de probabilités de Saint-Flour, XII-1982, pp. 1-142, Lecture Notes in Math. 1097, Springer, Berlin, 1984.
• Ledoux, Michel, The Concentration of Measure Phenomenon, American Math. Society, Providence, RI, 2001.
• Ledoux, Michel, and Michel Talagrand, Probability in Banach Spaces, Springer, Berlin, 1991. Reprinted in 2014 in the Classics in Math. Series.
• Marcus, Michael B., and Jay Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press, Cambridge, UK, 2006.
• Nourdin, Ivan, and Giovanni Peccati, Normal Approximations with Malliavin Calculus, Cambridge University Press, Cambridge, UK, 2012.
• Nualart, David, Malliavin Calculus and Related Topics, Springer, New York, 2006 [second edition].
• Sanz-Solé, Marta, Malliavin Calculus, EPFL Press, Lausanne, 2005.
• Talagrand, Michel, Upper and Lower Bounds for Stochastic Processes, Springer, New York, 2014.