# Poisson Point Processes and Their Application to Markov Processes

Part of the SpringerBriefs in Probability and Mathematical Statistics book series (SBPMS )

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Part of the SpringerBriefs in Probability and Mathematical Statistics book series (SBPMS )

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.

Poisson point process Poisson point process of excursions characteristic measure jumping-in measure and stagnancy rate discontinuous and continuous entrance points

- DOI https://doi.org/10.1007/978-981-10-0272-4
- Copyright Information The Author(s) 2015
- Publisher Name Springer, Singapore
- eBook Packages Mathematics and Statistics
- Print ISBN 978-981-10-0271-7
- Online ISBN 978-981-10-0272-4
- Series Print ISSN 2365-4333
- Series Online ISSN 2365-4341
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