Infinite dimensional Gaussian measures: Gaussian processes and Gaussian fields
Student No.:50
Time:Tue/Wed 13:00-14:50, 2015-08-05 ~ 2015-08-26(except for public holidays)
Instructor:Linan Chen  [McGill University]
Place:Conference Room 4, Floor 2, Jin Chun Yuan West Building
Starting Date:2015-8-5
Ending Date:2015-8-26



This course aims to be a general introduction to the theory and applications of infinite dimensional Gaussian measures. We will start with a brief review of the structural properties of the classical Gaussian processes, e.g., the Brownian motion and the Ornstein-Uhlenbeck process. Then we will extend from these classical examples to general infinite dimensional Gaussian measures and give a thorough exposition of the theory of the abstract Wiener spaces. 

The second part of the course focuses on the applications. In particular, we will give a mathematically rigorous treatment of Gaussian free fields (in 2D as well as in higher dimensions). We hope to explore some recent developments in the study of Gaussian free fields, such as the geometry of the random fields, the construction of the 2D quantum gravity measure, and a mathematical interpretation of the KPZ ()Knizhnik- Polyakov- Zamolodchikov) formula.

Depending on the backgrounds and the interests of the audience, the topics of the course are subject to change.




Real analysis: measure and integration, distribution theory. 

Functional analysis.

Probability theory. 

Although not required, knowledge of stochastic processes will be an advantage.




No textbook is required. The course material is based on the lecture notes. 

The references include (but not limited to):

-          "Real Analysis and Probability” (2nd ed.) by R. Dudley, Cambridge University Press, 2002.

-          "Probability Theory- An Analytic View” (2nd ed.) by D. Stroock, Cambridge University Press, 2011.

-        "Abstract Wiener Space” by L. Gross (Proc. 5th Berkeley Symp. on Math.