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

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.

  1. The Finite-Dimensional Theory
  2. The Infinite-Dimensional Theory
  • The latest version of the notes (Dec 3, 2018).
  • Grading. Grades are based on attendence, and fairly regular assignments.

    Basic References.