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  • © 2012

Fluctuations in Markov Processes

Time Symmetry and Martingale Approximation

  • 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
  • Includes supplementary material: sn.pub/extras

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

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Table of contents (14 chapters)

  1. Front Matter

    Pages I-XVII
  2. General Theory

    1. Front Matter

      Pages 1-1
    2. A Warming-Up Example

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 3-32
    3. Central Limit Theorems

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 33-79
    4. Random Walks in Random Environment

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 81-135
    5. Bounds and Variational Principles for the Asymptotic Variance

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 137-151
  3. Simple Exclusion Processes

    1. Front Matter

      Pages 153-153
    2. The Simple Exclusion Process

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 155-197
    3. Self-diffusion

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 199-240
    4. Equilibrium Fluctuations of the Density Field

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 241-274
    5. Regularity of the Asymptotic Variance

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 275-289
  4. Diffusions in Random Environments

    1. Front Matter

      Pages 291-291
    2. Diffusions in Random Environments

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 293-329
    3. Variational Principles for the Limiting Variance

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 331-343
    4. Diffusions with Divergence Free Drifts

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 345-373
    5. Diffusions with Gaussian Drifts

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 375-435
    6. Ornstein–Uhlenbeck Process with a Random Potential

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 437-454
    7. Analytic Methods in Homogenization Theory

      • Tomasz Komorowski, Claudio Landim, Stefano Olla
      Pages 455-473
  5. Back Matter

    Pages 475-491

About this book

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 also to mathematical physicists and analysts.

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)

Authors and Affiliations

  • Institute of Mathematics, Maria Curie-Sklodowska University, Lublin, Poland

    Tomasz Komorowski

  • Instituto de Matemática Pura e Aplicada, Jardim Botânico, Brazil

    Claudio Landim

  • CEREMADE, CNRS UMR 7534, Université Paris - Dauphine, Paris CX 16, France

    Stefano Olla

Bibliographic Information

Buy it now

Buying options

eBook USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access