Skip to content

Online ordering will be unavailable on Saturday 10 December 2022, 0800-1800 GMT.

To place an order, please contact Customer Services.

UK/ROW +44 (0) 1223 326050 | US 1 800 872 7423 or 1 212 337 5000 | Australia/New Zealand 61 3 86711400 or 1800 005 210, New Zealand 0800 023 520

Register Sign in Wishlist

Non-homogeneous Random Walks
Lyapunov Function Methods for Near-Critical Stochastic Systems


Part of Cambridge Tracts in Mathematics

  • Date Published: December 2016
  • availability: Available
  • format: Hardback
  • isbn: 9781107026698


Add to wishlist

Other available formats:

Looking for an inspection copy?

This title is not currently available on inspection

Product filter button
About the Authors
  • Stochastic systems provide powerful abstract models for a variety of important real-life applications: for example, power supply, traffic flow, data transmission. They (and the real systems they model) are often subject to phase transitions, behaving in one way when a parameter is below a certain critical value, then switching behaviour as soon as that critical value is reached. In a real system, we do not necessarily have control over all the parameter values, so it is important to know how to find critical points and to understand system behaviour near these points. This book is a modern presentation of the 'semimartingale' or 'Lyapunov function' method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks. Applications treat near-critical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. Spatially non-homogeneous random walks are explored in depth, as they provide prototypical near-critical systems.

    • Provides a deep understanding of the Lyapunov function method applied to a prototypical near-critical system
    • Foundational theory and applications are illustrated by examples and with remarks on intuition
    • Serves as a reference for experts as well as an instructional course for younger researchers
    Read more

    Reviews & endorsements

    'This is another impressive volume in the prestigious `Cambridge Tracts in Mathematics' series … The authors of this book are well-known for their long standing and well-recognized contributions to this area of research. Besides their own results published over the last two decades, the authors cover all significant achievements up to date … It is remarkable to see detailed `Bibliographical notes' at the end of each chapter. The authors have done a great job by providing valuable information about the historical development of any topic treated in this book. We find extremely interesting facts, stories and references. All this makes the book more than interesting to read and use.' Jordan M. Stoyanov, Zentralblatt MATH

    'This book gives a comprehensive account of the study of random walks with spatially non-homogeneous transition kernels. The main theme is to study recurrence versus transience and moments of passage times, as well as path asymptotics, by constructing suitable Lyapunov functions, which define semi-martingales when composed with the random walk. Of special interest are the Lamperti processes, which are stochastic processes on [0, ∞) with drift vanishing asymptotically on the order of 1/x as the space variable x tends to infinity. … Each chapter ends with detailed bibliographical notes.' Rongfeng Sun, Mathematical Reviews

    See more reviews

    Customer reviews

    Not yet reviewed

    Be the first to review

    Review was not posted due to profanity


    , create a review

    (If you're not , sign out)

    Please enter the right captcha value
    Please enter a star rating.
    Your review must be a minimum of 12 words.

    How do you rate this item?


    Product details

    • Date Published: December 2016
    • format: Hardback
    • isbn: 9781107026698
    • length: 382 pages
    • dimensions: 237 x 160 x 30 mm
    • weight: 0.73kg
    • contains: 20 b/w illus.
    • availability: Available
  • Table of Contents

    1. Introduction
    2. Semimartingale approach and Markov chains
    3. Lamperti's problem
    4. Many-dimensional random walks
    5. Heavy tails
    6. Further applications
    7. Markov chains in continuous time
    Glossary of named assumptions

  • Authors

    Mikhail Menshikov, University of Durham
    Mikhail Menshikov is Professor in the Department of Mathematical Sciences at the University of Durham. His research interests include percolation theory, where Menshikov's theorem is a cornerstone of the subject. He has published extensively on the Lyapunov function method and its application, for example to queueing theory.

    Serguei Popov, Universidade Estadual de Campinas, Brazil
    Serguei Popov is Professor in the Department of Statistics, Institute of Mathematics, Statistics and Scientific Computation, Universidad Estadual de Campinas, Brazil. His research interests include several areas of probability theory, besides Markov chains, including percolation, stochastic billiards, random interlacements, branching processes, and queueing models.

    Andrew Wade, University of Durham
    Andrew Wade is Senior Lecturer in the Department of Mathematical Sciences at the University of Durham. His research interests include, in addition to random walks, interacting particle systems, geometrical probability, and random spatial structures.

Sorry, this resource is locked

Please register or sign in to request access. If you are having problems accessing these resources please email

Register Sign in
Please note that this file is password protected. You will be asked to input your password on the next screen.

» Proceed

You are now leaving the Cambridge University Press website. Your eBook purchase and download will be completed by our partner Please see the permission section of the catalogue page for details of the print & copy limits on our eBooks.

Continue ×

Continue ×

Continue ×
warning icon

Turn stock notifications on?

You must be signed in to your Cambridge account to turn product stock notifications on or off.

Sign in Create a Cambridge account arrow icon

Find content that relates to you

Join us online

This site uses cookies to improve your experience. Read more Close

Are you sure you want to delete your account?

This cannot be undone.


Thank you for your feedback which will help us improve our service.

If you requested a response, we will make sure to get back to you shortly.

Please fill in the required fields in your feedback submission.