Continuous Time Markov Processes: An Introduction

by Thomas M. Liggett

Markov processes are among the most important stochastic processes for both theory and applications. This book develops the general theory of these processes and applies this theory to various special examples. The initial chapter is devoted to the most important classical example--one-dimensional Brownian motion. This, together with a chapter on continuous time Markov chains, provides the motivation for the general setup based on semigroups and generators. Chapters on stochastic calculus and probabilistic potential theory give an introduction to some of the key areas of application of Brownian motion and its relatives. A chapter on interacting particle systems treats a more recently developed class of Markov processes that have as their origin problems in physics and biology. This is a textbook for a graduate course that can follow one that covers basic probabilistic limit theorems and discrete time processes.

Genres: Mathematics

271 pages, Hardcover

First published June 22, 2010

Reviews

[Jaime · Rating 5 out of 5]

This grade is subject to change after my exam

[Kevin K. Gillette · Rating 4 out of 5]

This volume makes an excellent follow-on, at the early graduate level, to Karlin & Taylor's "Introduction to Stochastic Processes." The present monograph is probably too advanced for an undergraduate course in this subject, as it relies on extensive experience with measure theory, topology, and modern algebra (groups and semi-groups). Prof. Liggett leads the reader through a lengthy exposition of Brownian motion, Levy processes, Feller processes, and eventually arrives at the Ito calculus, with the Dirichlet problem rounding out the text. Liggett's exposition is clean and clear; my only complaint (and it's an exceedingly minor one) is that the reader is constantly being referred to the Appendix for many important foundational results, which might have been better presented as the first chapter of the book. Notwithstanding this comment, this monograph is really very well rendered. I highly recommend it to anyone wishing to explore the interplay between discrete and continuous-time Markov processes, myriad examples of how to construct Brownian motion processes, and those interested in how these processes are embedded in the framework of partial differential equations.