Publisher's Synopsis
This book is a survey of work on passage times in stable Markov chains with a discrete state space and a continuous time. Passage times have been investigated since early days of probability theory and its applications. The best known example is the first entrance time to a set, which embraces waiting times, busy periods, absorption problems, extinction phenomena, etc. Another example of great interest is the last exit time from a set.;The book presents a unifying treatment of passage times, written in a systematic manner and based on modern developments. The unifying framework is provided by probabilistic potential theory, and the results presented in the text are interpreted from this point of view. In particular, the crucial role of the Direchlet problem and the Poisson equation is stressed.;The first two chapters consider the analytic approach (dealing with transition probabilities and Markov semi-groups), whereas the remaining two chapters treat the measure-theoretic approach. The interdependence of both approaches is stressed to obtain the best results, and the book has been divided into analytic and measure-theoretic parts for convenience of presentation.;Chapter 3 introduces functionals, martingales and discusses transformations of Markov chains (including random time change). Chapter 4 treats several applications, including notions of capacity and energy of a chain; a brief introduction to boundary theory is also included. Illustrative examples are provided throughout the book.;The book is addressed to applied probabilists, and to those who are interested in applications of probabilistic methods in their own areas of interest. The level of presentation is that of a graduate text in applied stochastic processes. Advanced concepts described in the text gain nowadays growing acceptance in applied fields, and it is hoped that this book will serve as a useful introduction.
