## 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.