Mail Address 251 Mercer St. New York, NY 10012, U.S.A. Phones 212.998.3334 (voice) 212.995.4121 (fax) Email [email protected]

S. Sethuraman and S.R.S. Varadhan, "Large deviations for the current and tagged particle in 1D nearest-neighbor symmetric simple exclusion", Annals of Probability 41, no. 3A, 1461-1512 (2013) New York University

2 S. R. S. VARADHAN 1. Introduction The theory of large deviations deals with rates at which probabilities of certain events decay as a natural parameter in the problem varies. It is best to think of a speci c example to clarify the idea. Let us suppose that we have nindependent and identically distributed random variables fX

Sections available now on .ps and .pdf format Information about the course .ps file. Information about the course .pdf file Chapter 1 of Notes .ps file

Probability Theory S.R.S.Varadhan Courant Institute of Mathematical Sciences New York University August 31, 2000

Chapter 6 Banach Alegebras, Wiener’s Theorem There is a theorem due to Wiener that asserts the following. Theorem 6.1. Suppose f(x) on the d-torus has an absoutely convergent

8BMO The space of functions of Bounded Mean Oscillation (BMO) plays an impor-tant role in Harmonic Analysis. A function f,inL1(loc)inRd is said to be a BMO function if sup x;r inf a 1 jB j Z y2Bx;r jf(y)− ajdy = kuk BMO < 1 (8.1) where B x;r is the ball of radius r centered at x,andjB x;rj is its volume. Remark.

Prerequisites: Probability 1, i.e. the content of the book Probability Theory by S.R.S. Varadhan. Textbooks: Our reference text will be Stochastic Processes, by S.R.S. Varadhan. Homework: Every Tuesday for the next Tuesday. Grading: problem sets (50%) and final (50%). A tentative schedule for this course is: Jan 21. 1.1 Continuous time processes.

Y. Kifer and S.R.S. Varadhan, "Nonconventional limit theorems in discrete and continuous time via martingales", Annals of Probability 42, no. 2, 649-688 (2014)

Chapter 5 Hardy Spaces. 5.1 Stationary Gaussian Processes. Consider a collection{X j: −∞ <j<∞} of random variables such that the joint distribution of any finite collection is a multivariate Gaussian distribu-