Francesco Caravenna

Pavia-Bicocca-Indam Ph.D. program in Mathematics

PhD Course, Spring 2020

Topics from the Gaussian World

Francesco Caravenna and Maurizia Rossi

Overview

Gaussian distributions play a central role in probability and analysis. We aim to illustrate this role through a selection of topics, ranging from basic properties to more advanced and recent results. The emphasis will be both on the general theory and on concrete models.

The course is aimed at students interested in probability and/or analysis. Prerequisites are a course in probability theory and some functional analysis (Hilbert spaces, linear operators). No special knowledge of the "Gaussian world" will be assumed.

Interested participants are invited to contact us via email.

Abstract

After a general introduction on the basic properties of Gaussian random variables, the core of the course will be divided in two parts.

The first part is an introduction to the theory of Gaussian approximation, described as the art of estimating the distance between a given probability distribution and a Gaussian. This classical topic has developed enormously in recent years, based on the fruitful combination of two techniques: Malliavin calculus (an infinite-dimensional differential calculus) and Stein's method. These techniques will be introduced in a smooth and self-contained way, alongside the important notion of Wiener chaos. As an application of the general theory, we will prove Central Limit Theorems for the zeros of random trigonometric polynomials.

The second part of the course is devoted to Gaussian stochastic processes, with a focus on processes defined in the Euclidean space R^d. We will start with Brownian motion, investigating some of its fascinating properties, both classical and more advanced. We will then present the Gaussian Free Field, a generalization of Brownian motion to higher dimensional domains, which plays an important role in statistical physics. We will finally introduce White Noise and discuss some basic applications to stochastic PDEs. All these processes will be shown to fit the same general framework, known as Abstract Wiener Space, which allows to express them as generalized random Fourier series.

Lectures

Due to the Coronavirus emergency, the course is given on-line using the Zoom Meeting program. Interested participants are invited to contact us via email.

The course is given in Italian . Recorded lectures are available upon request.

• Lecture 1 (Thu 19 March). Part 1 (Caravenna). Introduction and motivation. Presentation of the second part of the course (Gaussian processes in R^d). Reminders on Gaussian random variables. Notes 1.1
  Part 2 (Rossi). Moments and cumulants. The problem of moments. Presentation of the first part of the course (Gaussian approximation). Notes 1.2

First part of the course

• Lecture 2 (Wed 25 March, Rossi). Probability metrics (Kolmogorov, Total Variation, Wasserstein, Fortet-Mourier). Stein's method for Normal approximation in dimension one. Applications: proof of the Berry-Esseen Theorem. Notes 2
• Lecture 3 (Thu 2 Apr, Rossi). Malliavin calculus in dimension one (derivative, divergence, Ornstein-Uhlenbeck operators). Poincaré inequalities (I and II order). Notes 3
• Lecture 4 (Wed 8 Apr, Rossi). Hermite polynomials. Isonormal Gaussian Processes (Wiener-Ito integrals). Wiener chaos. Notes 4
• Lecture 5 (Thu 16 Apr, Rossi). Malliavin calculus and isonormal Gaussian processes (derivative and divergence operators). Multiple integrals as Hermite polynomials. Notes 5
• Lecture 6 (Thu 23 Apr, Rossi). Stroock-Varadhan formula. Ornstein-Uhlenbeck operator and Mehler formula. Gaussian approximation in Wiener chaos (Fourth Moment Theorem, Poincaré inequalities) and applications. Notes 6

Second part of the course

• Lecture 7 (Wed 29 Apr, Caravenna). Definition of Brownian motion (BM). Cameron-Martin space for Gaussian vectors in R^d. BM as a random series in H¹₀. Markov processes. Reflected brownian motion. Lévy M-B theorem. Notes 7
• Lecture 8 (Wed 6 May, Caravenna). Hausdorff measure and dimensions. Lower bounds: mass distribution and energy principles. Hausdorff dimension of the Range of BM in R^d and of the Zeros of BM in R. Notes 8
• Lecture 9 (Wed 13 May, Caravenna). Bridge of Gaussian random walk. Discrete Gaussian Free Field (GFF) on a bounded domain of Z^d. Properties of the GFF: covariance, Green function, Markov property. Discrete Dirichlet problem. Notes 9
• Lecture 10 (Wed 20 May, Caravenna). Reminders on the discrete Gaussian Free Field (GFF). Scaling limit of the discrete GFF and heuristics. The continuum Green function. The continuum GFF on a bounded domain of R^d. Notes 10
• Lecture 11 (Wed 27 May, Caravenna). The continuum GFF as a random distribution: construction of the GFF as a random element of H^-s for s > d/2 - 1. Key features of the 2d GFF (statements). Abstract Wiener space (hints). Notes 11

When

A dozen lectures from March to May 2020.

1. Thu 19 March 14:30
2. Wed 25 March 14:30
3. Thu 2 April 14:30
4. Wed 8 April 14:30
5. Thu 16 April 14:30
6. Thu 23 April 14:30
7. Wed 29 Apr 14:30
8. Wed 6 May 14:30
9. Wed 13 May 14:30
10. Wed 20 May 14:30
11. Wed 27 May 14:30

References

• I. Nourdin, G. Peccati
  Normal approximations with Malliavin calculus: from Stein's method to universality
  Chapters 1-3 & 5, Cambridge University Press (2012)
• A. Granville, I. Wigman
  The distribution of the zeros of random trigonometric polynomials
  American Journal of Mathematics 133 (2011), 295-357
  (also on arXiv.org: 0809.1848)
• J. Angst, G. Poly
  Variations on Salem-Zygmund results for random trigonometric polynomials. Application to almost sure nodal asymptotics
  Preprint (2019), arXiv.org: 1912.09928
• P. Morters, Y. Peres
  Brownian Motion
  Cambridge University Press (2010)
  (also here)
• N. Berestycki
  Introduction to the Gaussian Free Field and Liouville Quantum Gravity
  Lecture Notes (2016)
• W. Werner, E. Powell
  Lecture notes on the Gaussian Free Field
  Lecture Notes (2020), arXiv.org: 2004.04720
• S. Sheffield
  Gaussian free fields for mathematicians
  Probab. Theory Relat. Fields 139 (2007), 521-541
  (also on arXiv.org: math/0312099)
• M. Biskup
  Extrema of the two-dimensional Discrete Gaussian Free Field
  In: Random Graphs, Phase Transitions, and the Gaussian Free Field (M. Barlow and G. Slade eds.), Springer Proceedings in Mathematics & Statistics 304 (2020) 163-407
  (also on arXiv.org: 1712.09972)
• M. Hairer
  An Introduction to Stochastic PDEs
  Lecture Notes (2009)

What else?

For more information, do not hesitate to contact us via email.

Course details (PDF)