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© 2012

Fluctuations in Markov Processes

Time Symmetry and Martingale Approximation

Benefits

  • Unique in mathematical literature covers the most advanced theories about martingale approach to central limit theorems

  • Develops techniques that allow to deal with applications in statistical mechanics and engineering

  • Is of interest to probabilists, mathematical physicists and analysts

Book

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 345)

Table of contents

  1. Front Matter
    Pages I-XVII
  2. General Theory

    1. Front Matter
      Pages 1-1
    2. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 3-32
    3. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 33-79
    4. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 81-135
    5. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 137-151
  3. Simple Exclusion Processes

    1. Front Matter
      Pages 153-153
    2. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 155-197
    3. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 199-240
    4. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 241-274
    5. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 275-289
  4. Diffusions in Random Environments

    1. Front Matter
      Pages 291-291
    2. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 293-329
    3. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 331-343
    4. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 345-373
    5. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 375-435
    6. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 437-454
    7. Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 455-473
  5. Back Matter
    Pages 475-491

About this book

Introduction

Diffusive phenomena in statistical mechanics and in other fields arise from markovian modeling and their study requires sophisticated mathematical tools. In infinite dimensional situations, time symmetry properties can be exploited in order to make martingale approximations, along the lines of the seminal work of Kipnis and Varadhan. The present volume contains the most advanced theories on the martingale approach to central limit theorems. Using the time symmetry properties of the Markov processes, the book develops the techniques that allow us to deal with infinite dimensional models that appear in statistical mechanics and engineering (interacting particle systems, homogenization in random environments, and diffusion in turbulent flows, to mention just a few applications). The first part contains a detailed exposition of the method, and can be used as a text for graduate courses. The second concerns application to exclusion processes, in which the duality methods are fully exploited. The third part is about the homogenization of diffusions in random fields, including passive tracers in turbulent flows (including the superdiffusive behavior).
 
There are no other books in the mathematical literature that deal with this kind of approach to the problem of the central limit theorem. Hence, this volume meets the demand for a monograph on this powerful approach, now widely used in many areas of probability and mathematical physics. The book also covers the connections with and application to hydrodynamic limits and homogenization theory, so besides probability researchers it will also be of interest to mathematical physicists and analysts.

Keywords

60F05, 60J25, 60K35, 60K37, 60H25, 60G60, 35B27, 76M50 Markov processes central limit theorems exclusion processes fluid mechanics homogenization martingales

Authors and affiliations

  1. 1.Institute of MathematicsMaria Curie-Sklodowska UniversityLublinPoland
  2. 2.Instituto de Matemática Pura e AplicadaJardim BotânicoBrazil
  3. 3.CEREMADE, CNRS UMR 7534Université Paris - DauphineParis CX 16France

Bibliographic information

  • Book Title Fluctuations in Markov Processes
  • Book Subtitle Time Symmetry and Martingale Approximation
  • Authors Tomasz Komorowski
    Claudio Landim
    Stefano Olla
  • Series Title Grundlehren der mathematischen Wissenschaften
  • Series Abbreviated Title Grundlehren math. Wissensch.
  • DOI https://doi.org/10.1007/978-3-642-29880-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 2012
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-642-29879-0
  • Softcover ISBN 978-3-642-42847-0
  • eBook ISBN 978-3-642-29880-6
  • Series ISSN 0072-7830
  • Edition Number 1
  • Number of Pages XVIII, 494
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Probability Theory and Stochastic Processes
    Mathematical Physics
  • Buy this book on publisher's site
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Reviews

From the reviews:

“The authors provide a monographic survey of the martingale approximation for additive functionals of Markov processes and central limit theorems (CLTs) and invariance principles following from this. … Each chapter closes with a section of historical and bibliographical comments. … these historical surveys make the book a long-lasting source of information about development of ideas, related fields and results, and historical data. The book is highly recommended to researchers and postgraduate students working in various fields of interacting particle systems, random environments and homogenization.” (Bálint Tóth, Mathematical Reviews, November, 2013)