Poisson Point Processes and Their Application to Markov Processes

Itô, Kiyosi; Watanabe, Shinzo (FRW); Shigekawa, Ichiro (FRW)

ISBN 10: 9811002711 / ISBN 13: 9789811002717

Verlag: Springer, 2016

Beschreibung

Bestandsnummer des Verkäufers 25079976-n

Bibliografische Details

• Titel: Poisson Point Processes and Their Application to Markov Processes
• Verlag: Springer
• Erscheinungsjahr: 2016
• Sprache: Englisch
• ISBN-10: 9811002711
• ISBN-13: 9789811002717
• Einband: Softcover
• Zustand: New

Über diesen Titel

Inhaltsangabe

An extension problem (often called a boundary problem) of Markov processes has been studied, particularly in the case of one-dimensional diffusion processes, by W. Feller, K. Itô, and H. P. McKean, among others. In this book, Itô discussed a case of a general Markov process with state space S and a specified point a ∈ S called a boundary. The problem is to obtain all possible recurrent extensions of a given minimal process (i.e., the process on S {a} which is absorbed on reaching the boundary a). The study in this lecture is restricted to a simpler case of the boundary a being a discontinuous entrance point, leaving a more general case of a continuous entrance point to future works. He established a one-to-one correspondence between a recurrent extension and a pair of a positive measure k(db) on S {a} (called the jumping-in measure and a non-negative number m< (called the stagnancy rate). The necessary and sufficient conditions for a pair k, m was obtained so that the correspondence is precisely described. For this, Itô used, as a fundamental tool, the notion of Poisson point processes formed of all excursions of the process on S {a}. This theory of Itô’s of Poisson point processes of excursions is indeed a breakthrough. It has been expanded and applied to more general extension problems by many succeeding researchers. Thus we may say that this lecture note by Itô is really a memorial work in the extension problems of Markov processes. Especially in Chapter 1 of this note, a general theory of Poisson point processes is given that reminds us of Itô’s beautiful and impressive lectures in his day.

Críticas

“The main idea of this volume has had a profound influence on the boundary theory of Markov processes. This volume is beautifully written and it is a pleasure to read.” (Ren Ming Song, Mathematical Reviews, December, 2016)