Eigenvalues, Inequalities, and Ergodic Theory / Edition 1

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A problem of broad interest – the estimation of the spectral gap for matrices or differential operators (Markov chains or diffusions) – is covered in this book. The area has a wide range of applications, and provides a tool to describe the phase transitions and the effectiveness of random algorithms. In particular, the book studies a subset of the general problem, taking some approaches that have, up till now, only appeared largely in the Chinese literature.

Eigenvalues, Inequalities and Ergodic Theory serves as an introduction to this developing field, and provides an overview of the methods used, in an accessible and concise manner. The author starts with an overview chapter, from which any of the following self-contained chapters can be read.

Each chapter starts with a summary and, in order to appeal to non-specialists, ideas are introduced through simple examples rather than technical proofs. In the latter chapters readers are introduced to problems and application areas, including shastic models of economy.

Intended for researchers, graduates and postgraduates in probability theory, Markov processes, mathematical physics and spectrum theory, this book will be a welcome introduction to a growing area of research.

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Editorial Reviews

From the Publisher
From the reviews of the first edition:

"The estimation of the spectral gap for matrices or differential operators (Markov chain or diffusions) – is covered in this book, and the book surveys, in popular way, the main progress made in the field. … Each chapter starts with a summary, and ideas are introduced through simple examples rather than technical proofs, which is one of the advantages of the book … . The book may be useful for researchers, graduates and postgraduates in probability theory, Markov processes, mathematical physics and spectrum theory." (Anatoliy Swishchuk, Zentralblatt MATH, Vol. 1079, 2006)

"The main topics of this book, as indicated in the title, are eigenvalues, inequalities, and ergodic theory. … Chen is an experienced researcher who has won prizes in China and spoken at the International Congress of Math, and he is an extensive writer of lecture notes. … The math is clearly explained and the book is pleasant to read. … it deserves a place on your bookshelf." (Rick Durrett, SIAM Review, Vol. 48 (1), 2006)

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Product Details

  • ISBN-13: 9781849969383
  • Publisher: Springer London
  • Publication date: 12/13/2010
  • Series: Probability and Its Applications Series, #464
  • Edition description: Softcover reprint of hardcover 1st ed. 2005
  • Edition number: 1
  • Pages: 228
  • Product dimensions: 0.51 (w) x 6.14 (h) x 9.21 (d)

Meet the Author

Mu-Fa Chen is Professor of Mathematics at Beijing Normal University, in the People’s Republic of China, and Member of the Chinese Academy of Sciences. In 1999 he received a Prize for Progress on Sciences and Technology from the Ministry of Education and a National Prize on Natural Sciences from the Ministry of Science and Technology in the People’s Republic of China.

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Table of Contents

Preface.- An Overview of the Book.- Optimal Markovian Couplings.- New Variational Formulas for the First Eigenvalue.- Generalized Cheeger's Method.- Ten Explicit Criteria in Dimension One.- Poincaré-type Inequalities in Dimension One.- Functional Inequalities.- A Diagram of Nine Types of Ergodicity.- Reaction-Diffusion Processes.- Shastic Models of Economic Optimization.- Appendix A: Some Elementary Lemmas.- Appendix B: Examples of Ising Model on Two-Four Sites.- References.- Author Index.- Subject Index.

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