Poisson Point Processes and Their Application to Markov Processes
Authors: Itô, Kiyosi
Free Preview- Gives a beautiful elementary treatment of general Poisson point processes in Chapter 1, especially recommended for beginners
- Shows how the notion of Poisson point processes with values in a function space of paths called excursions plays a key role in an extension problem of Markov processes in Chapter 2
- Demonstrates how the general theory in Chapter 2 can answer completely the extension problem for the minimal diffusion on [0, ∞) with an exit boundary 0
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- About this book
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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.
- Reviews
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“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)
- Table of contents (2 chapters)
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Poisson Point Processes
Pages 1-18
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Application to Markov Processes
Pages 19-43
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Table of contents (2 chapters)
- Download Preface 1 PDF (78.7 KB)
- Download Sample pages 2 PDF (292.2 KB)
- Download Table of contents PDF (51.3 KB)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- Poisson Point Processes and Their Application to Markov Processes
- Authors
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- Kiyosi Itô
- Series Title
- SpringerBriefs in Probability and Mathematical Statistics
- Copyright
- 2015
- Publisher
- Springer Singapore
- Copyright Holder
- The Author(s)
- eBook ISBN
- 978-981-10-0272-4
- DOI
- 10.1007/978-981-10-0272-4
- Softcover ISBN
- 978-981-10-0271-7
- Series ISSN
- 2365-4333
- Edition Number
- 1
- Number of Pages
- XI, 43
- Number of Illustrations
- 3 b/w illustrations
- Topics
