Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems

ISBN-13: 9783642331305 / Angielski / Twarda / 2012 / 280 str.

Pierre Collet; Servet Mart Nez; Jaime San Mart N.

Opis

Main concepts of quasi-stationary distributions (QSDs) for killed processes are the focus of the present volume. For diffusions, the killing is at the boundary and for dynamical systems there is a trap. The authors present the QSDs as the ones that allow describing the long-term behavior conditioned to not being killed.

Recenzje

From the reviews:

"This book puts together several important contributions of the authors to the field. ... the writing is pedagogical and provides a timely and varied introduction to a subject which is increasingly attracting its due attention. ... it should serve as a useful basis for further research." (Laurent Miclo, Mathematical Reviews, September, 2013)

"The authors of the book have made remarkable contributions to this topic of research, and this book cultivates their work to date. ... The authors of the book are leading experts in the area, and they made all the effort to make the book accessible to everyone, from graduate students to researchers, working in this area. ... I have never seen a book which is devoted only to the study of QSD and which covers both Markov process and dynamical systems." (Wael Bahsoun, SIAM Review, Vol. 55 (4), 2013)

"The monograph under review is a thorough study of QSDs and related concepts for Markov chains, diffusions and dynamical systems. ... The exposition of the chosen material is well-organized and proofs are mostly given in complete detail. Historical references are scattered throughout the text. The book will be very useful to researchers and graduate students who want to learn more about the subject, as well as to the experts in the field." (Zoran Vondracek, Zentralblatt MATH, Vol. 1261, 2013)

Spis treści

1.Introduction.- 2.Quasi-stationary Distributions: General Results.- 3.Markov Chains on Finite Spaces.- 4.Markov Chains on Countable Spaces.- 5.Birth and Death Chains.- 6.Regular Diffusions on [0,∞).- 7.Infinity as Entrance Boundary.- 8.Dynamical Systems.- References.- Index.- Table of Notations.- Citations Index. 

Blurb

Main concepts of quasi-stationary distributions (QSDs) for killed processes are the focus of the present volume. For diffusions, the killing is at the boundary and for dynamical systems there is a trap. The authors present the QSDs as the ones that allow describing the long-term behavior conditioned to not being killed. Studies in this research area started with Kolmogorov and Yaglom and in the last few decades have received a great deal of attention. The authors provide the exponential distribution property of the killing time for QSDs, present the more general result on their existence and study the process of trajectories that survive forever. For birth-and-death chains and diffusions, the existence of a single or a continuum of QSDs is described. They study the convergence to the extremal QSD and give the classification of the survival process. In this monograph, the authors discuss Gibbs QSDs for symbolic systems and absolutely continuous QSDs for repellers.

The findings described are relevant to researchers in the fields of Markov chains, diffusions, potential theory, dynamical systems, and in areas where extinction is a central concept. The theory is illustrated with numerous examples. The volume uniquely presents the distribution behavior of individuals who survive in a decaying population for a very long time. It also provides the background for applications in mathematical ecology, statistical physics, computer sciences, and economics.