## 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.
• 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.