Topics in Probability: Gaussian Analysis
Math 7880-1, Spring 2015
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, \[ \mathbb{P}_n(A) := \int_A \frac{\exp\left(-\frac12\|x\|^2\right)}{(2\pi)^{n/2}}\,{\rm d}x, \] 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)

• Chapter 1 (The canonical Gaussian measure on \(\mathbb{R}^n\))
• Chapter 2 (Calculus in Gauss space)
• Chapter 3 (Harmonic Analysis)
• Chapter 4 (Heat Flow)
• Chapter 5 (Integration by Parts and Its Applications)
• Chapter 6 (Gaussian Processes)
• Chapter 7 (Regularity Theory)

• 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, The Generic Chaining, Springer, New York, 2005.