## Topics in Probability: Gaussian Analysis |

Time & Place. | TH 10:50 a.m.-12:20 a.m. | JWB 108 |
---|---|---|

Instructor. | Davar Khoshnevisan | JWB 102/JWB 237 |

**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.

- 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

- The
Canonical Gaussian Measure on \(\mathbb{R}^n\);
Assignment from Chapter 1 (due Sept 14);
- The Infinite-Dimensional Theory

**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.