Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation

ISBN-13: 9783642428470 / Angielski / Miękka / 2014 / 494 str.

Tomasz Komorowski; Claudio Landim; Stefano Olla

Opis

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.

Recenzje

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)

Spis treści

Preface.- Part I: General Theory.- 1.A Warming-up Example.- 2.Central Limit Theorems.- 3.RandomWalks in Random Environment.- 4.Bounds and Variational Principles for the Asymptotic Variance.- Part II: Simple Exclusion Processes.- 5.The Simple Exclusion Process.- 6.Self Diffusion.- 7.Equilibrium Fluctuations of the Density Field.- 8.Regularity of the Asymptotic Variance.- Part III: Diffusions in Random Environments.- 10.Variational Principles for the Limiting Variance.- 11.Diffusions with Divergence Free Drifts.- 12.Diffusions with Gaussian Drifts.- 13.Ornstein-Uhlenbeck Process with a Random Potential.- 14.Analytic Methods in Homogenization Theory.- References.- Notation.- Subject Index.

Blurb

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.