Course 18.445, Spring 2014

This course is an introduction to Stochastic processes and Markov processes, random walks, poisson processes, birth and death processes, continuous time processes, with a particular emphasis on their long time behaviour.

This course will require only basic notions in probability theory, 18.440 should be more than enough.

Grading will be based on 8 homeworks (tentative dates for posting the HW on this webpage: Feb 11, 20, March 4, 13,20, April 1, 10, 22; due dates one week later.) Homeworks will be given at the end of the class on the due date. The 7 first homeworks will be 10% of the final grade each and the last will be 30%.

Office hours: Alice Guionnet (E17-310) Tuesday 2:00PM-4:00PM, Xin Sun (E18-401H) Wednesday 3:30PM-5:30 PM

Course: Tuesday-Thursday 11-12:30 PM, E25-111


An introduction to Markov Processes, second edition, Springer, by Daniel Stroock

Markov Chains and Mixing times, AMS, by David Levin, Yuval Peres and E. Wilmer.

Problem set 1, due Thursday, February 20

Problem set 2, due Thursday, February 27

Problem set 3, due Thursday, March 13

Problem set 4, due Thursday, March 20

Problem set 5, due Thursday, April 3

Problem set 6, due Thursday, April 10

Problem set 7, due Thursday, April 24

Problem set 8 (and last), due Thursday, May 8

We will follow Stroock book up to chapter 5, and end the course with some elements of chapter 6 and/or of Levin-Peres-Wilmer book. Program:

--February: -02/4--02/6: Section 1.1 of Stroock book (recurrence/transience of random walk on Z).

-02/11--02/13: Section 1.2 (random walk on /transience of random walk on Z^d). Random walk on finite sets. Beginning of Chapter 2: definition of Markov chains in a discrete state space/discrete time.

-02/18--02/20: Doeblin theory.

-02/25--02/27: Return times and the stationnary measure (end of chapter 2), Classification of states, criteria, stationary probabilities (beginning chapter 3)

-03/04--03/06: Classification of states, criteria, aperiodicity

-03/11--03/13: aperiodicity, stationary probabilities, Wilson algorithm (end of chapter 3)

-03/18--03/20: End of Wilson algorithm (end of chapter 3 of Stroock book), Coupling techniques: Chapter 4 and beginning of chapter 5 in Levin-Peres-Wilmer's book.

-03/31--04/03: Coupling and strong stationary times, end of chapter 5 and chapter 6 in Levin-Peres-Wilmer's book.

-04/08--04/10: end of chapter 6 and chapter 7 in Levin-Peres-Wilmer's book. 04/15--04/17:Mixing in Expander graphs (chapter 12-13 in Levin-Peres-Wilmer), beginning of chapter 4 in Stroock book.

. 04/24:end of chapter 4 in Stroock book.

-04/28--05/01: Chapter 5 in Stroock book