**Time and place:**MWF 11 in MS 5233 in the Winter and MS 6221 in the Spring**Instructor:**Thomas M. Liggett**Office hours:**MWF 1-2 in MS 7919- During the
**Spring quarter**, there are two advanced courses in probability -- this one and 285K/1 taught by Marek Biskup at 10am in MS 6221. The courses will be coordinated in the following sense: Both hours will be used by T. Liggett on May 3, 5, 8, 15, 17, 19; both hours will be used by M. Biskup on April 21, May 1, 12, 26, and June 2, 9. On other lecture days, Biskup and Liggett will use one hour each.**NOTE**This arrangement is cancelled, effective May 15.The subject of this course is a part of probability theory that is used to model situations that arise in many areas, including physics, biology, economics, and even sociology. Typically, in the resulting models, there are an infinite number of agents (atoms, bacteria, people...) that change their states in continuous time, according to stochastic rules that involve dependence among the agents. Without this dependence, the state of each agent would evolve according to a finite or countable state Markov chain. With the dependence, one has to keep track of the states of all the agents, and this results in a Markov process on an uncountable space of configurations. Typical questions are: What are all the stationary distributions for the process? Does the limiting distribution (as time tends to infinity) exist, and if so, how does this limit depend on the initial distribution?

In this course, I will discuss several of the most important classes of models, including voter models, contact processes, and exclusion processes. I will begin by discussing the construction problem (including a treatment of semigroups and generators) and some of the main tools used in the field (including coupling and duality).

Much of the material will come from my two books:

(1) "Interacting Particle Systems", Springer "Grundlehren" series #276 (1985) (reprinted in 2004 in their "Classics in Mathematics" series),

and

(2) "Stochastic Interacting Systems", Springer "Grundlehren" series #324 (1999).

(I have asked that both books be placed on reserve in both the Mathematics Reading Room, and the EMS Library. I also have several extra copies of the 1985 book, which I am willing to lend out).

(3) Four years ago, I wrote some introductory lecture notes for a school in Trieste, Italy. I will distribute copies of these notes in class.

In order to follow the lectures, one should have a good grounding in graduate level real analysis and probability theory. It is particularly important to be comfortable with Markov chains and martingales.

I expect that there will be both graduate students and some faculty in the course. I will try to teach it in such a way that both groups can benefit from it. Before each lecture, I will list below the topic of the lecture, and a more detailed reference for it. Students are expected to attend all lectures. Faculty are welcome to come or not according to the topic to be covered.

There will be occasional problems assigned. HW 1 is due January 20. HW 2 is due February 6. HW 3 is due February 27. HW 4 is due March 15. HW 5 is due April 24. HW 6 is due May 22.

**January 9, 11, 13.**Introduction. Models (voter, contact, and exclusion), questions, and some answers. References: (1) pages 1-5 and (3) pages 1-10.**January 18, 20, 23.**Continuous time Markov chains, Chapman-Kolmogorov equations, differentiability of transition functions, Q-matrix, minimal solution to the backward equations. Reference: pages 142-147 and 237-246 of "Markov Chains" by David Freedman.**January 25, 27, 30, February 1.**Semigroups and generators for Markov processes on compact or locally compact spaces; Hille-Yosida theorem. References: (1) pages 7-20. The main theorems I will not prove are: Hille-Yosida Theorem (see pages 10-20 of "Markov Processes, Characterization and Convergence" by S. Ethier and T. Kurtz) and the construction of a Markov process with good properties from the semigroup (see pages 46-50 of "Markov Processes and Potential Theory" by R. Blumenthal and R. Getoor or pages 164-170 of Ethier and Kurtz).**February 3, 6, 8.**The construction problem for spin systems; examples. Reference: (1) pages 20-34.**February 10, 13.**Duality for diffusions, Markov chains, voter model, contact process, exclusion process. References: (1) pages 84-88 and 157-163 and (3) section 1.3.**February 15.**Duality continued. Coupling proof of the Choquet-Deny Theorem. Reference: (1) pages 69-70.**February 17, 22, 24, 27, March 1.**Analysis of the voter model via duality. I will discuss the case in which p(x,y) are the transition probabilities for a random walk on Z^d. References: (1) pages 226-239 (general case) and (3) sections 2.1 and 2.2 (random walk case).**March 3, 6, 8.**Monotonicity, correlation inequalities, and coupling for spin systems. References: (1) pages 70-82 and 124-144, and (3) pages 12-15.**March 10, 13, 15, 17.**Reversibility; nearest particle systems. References: (1) pages 90-94 and 330-343.**April 3,5,7,10.**The contact process -- bounds on the critical value. References: (1) pages 265-275, 307-308, and 462.**April 12.**Application of the smoothing process to the contact process to get an improved upper bound for the critical value in high dimensions. Reference: (1) pages 442-449.**April 14,17,19.**The one dimensional contact process. Reference: (1) pages 281-287.**April 24.**Review of the Ising model, Gibbs states, stochastic Ising models, phase transition. Reference: (1) pages 179-204.**April 26,28.**The invariant measures for stochastic Ising models. Reference: (1) pages 88-89 and 213-222.**May 1.**No class, since Marek will use both hours.**May 3.**Class cancelled due to illness.**May 5.**Two hours. 10am: Complete proof of invariant implies reversible. 11am: Start symmetric exclusion. Reference: (1) pages 361-370.**May 8.**Two hours. Symmetric exclusion, continued. Reference: (1) pages 398-402.**May 10.**11am: Tagged particle problem; product stationary distributions in the general (not necessarily symmetric) case. Reference: Annals of Probability 33 (2005) pages 2258-2260.**May 12.**No class, since Marek will use both hours.**May 15.**Product stationary distributions for general exclusion process, continued.**May 17.**Coupling for general exclusion processes. Reference: (1) pages 382-390.**May 19.**Invariant measures for exclusion processes on Z^1 without drift. Reference: (1) pages 391-392.**May 22,24.**Extremality of product measures; invariant measures for exclusion processes on Z^1 with drift. References: (2) pages 216-218, AoP 4(1976), pages 353-355, and AoP 30(2002), pages 1539-1575.**May 26.**Convergence theorems for exclusion processes on Z^1 with drift. Reference: (2) pages 220-261 and 302-303.**May 31, June 2, 5.**Construction of blocking stationary distributions for one dimensional exclusion processes with drift. Reference: AoP (2002), pages 1556-1575.**June 7,9.**Convergence theorems for one dimensional exclusion processes with drift. Reference: AoP (1977), pages 795-801.