Topics in Probability: Gaussian Analysis
Math 7880-1, Spring 2015
University of Utah

Time & Place: MWF 9:40-10:30 a.m. LCB 222
Instructor: Davar Khoshnevisan JWB 102

Course Synopsis. Let \(\mathbb{P}_n\) denote the canonical Gaussian measure - or the standard multivariate normal - on \(\mathbb{R}^n\); that is, \[ \mathbb{P}_n(A) := \int_A \frac{\exp\left(-\frac12\|x\|^2\right)}{(2\pi)^{n/2}}\,{\rm d}x, \] for all Borel sets \(A\) 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 courses such as 6010 and 6020 [linear models]. In this course we study some of the deeper structure of the "Gauss space" \((\mathbb{R}^n\,,\mathcal{B}(\mathbb{R}^n)\,,\mathbb{P}_n)\). We will also see that our analysis of \(\mathbb{P}_n\) yields a much better understanding of the theory of Gaussian processes [which we will introduce as well].

Lecture notes. (Read them at your own risk)

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

Basic References.