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.

• 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.- Stochastic Models of Economic Optimization.

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)

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.

• 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.- Stochastic Models of Economic Optimization.

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)

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.

• 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.- Stochastic Models of Economic Optimization.

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)