Mathematics 275C - Spring 2011
Stochastic Processes
- Time and place: MWF at 2 in MS5137
- Instructor: Thomas M. Liggett
- Office hours: MWF 1-2 in MS 7919
- Text: "Continuous Time Markov Processes: An Introduction"
by T. M. Liggett. We will cover Chapters 2 (Continuous time Markov chains), 3 (Continuous time
Markov processes), and
4 (Interacting particle systems).
- Prerequisite: Mathematics 275B (including some background on
Brownian motion) or equivalent.
- Note: Math 275C in Spring, 2010, covered Chapters 1,5 and 6 of the text.
There will be minimal overlap with last year's course, so students from last year
are welcome to take it again for credit.
- Grading: Grades will be based on homework. There will be
no exams.
Topics
- Continuous time Markov chains. While the emphasis will be on continuous time,
I will start with some background material on discrete time Markov chains. After
discussing some examples, we will move on to the contruction of a chain from its
infinitesimal (in time) description. Finally, we will discuss the important issues
of stationary measures, recurrence, and transience.
- Feller processes. These are continuous time Markov processes on a general
state space. (Markov chains have a countable state space.) Using Brownian motion and
continuous time Markov chains as motivating examples, we will cover the construction
of Feller processes from their infinitesimal description. This topic is known as semigroups
(which describe the evolution of processes over time) and generators (which provide the
infinitesimal description). The main result is known as the Hille-Yosida Theorem. There
will be applications to variants of Brownian motion.
- Interacting particle systems. These are Feller processes that arise in such diverse
areas as statistical physics, biology, economics, traffic flow, and Monte Carlo studies. They
are constructed from their generators via the Hille-Yosida Theorem, and therefore
provide new applications of the material in Chapter 3. The main issues in
their study are quite different from those relevant to more classical processes such as
Brownian motion. We will discuss the long time behavior of the processes, and in particular,
the extent to which small changes in the local evolution rules have large
effects on this asymptotic behavior.
Homework
No late homework will be accepted.
- Due Wed. April 13: Homework 1
- Due Wed. May 4: Problems 2.32,2.36,2.38,2.42,2.46,2.55.
- Due Wed. May 18: Problems 2.65,2.69,3.5,3.7,3.20,3.21.
- Due Wed. June 1: Problems 3.36,3.61,3.65,4.1,4.5(a,d),4.9.