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Poisson Point Processes and Their Application to Markov Processes / by Kiyosi Itô
(SpringerBriefs in Probability and Mathematical Statistics. ISSN:23654341)
版 | 1st ed. 2015. |
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出版者 | Singapore : Springer Nature Singapore : Imprint: Springer |
出版年 | 2015 |
大きさ | XI, 43 p. 3 illus : online resource |
著者標目 | *Itô, Kiyosi author SpringerLink (Online service) |
件 名 | LCSH:Probabilities LCSH:Measure theory LCSH:Functional analysis FREE:Probability Theory FREE:Measure and Integration FREE:Functional Analysis |
一般注記 | 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 HTTP:URL=https://doi.org/10.1007/978-981-10-0272-4 |
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電子ブック | オンライン | 電子ブック |
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Springer eBooks | 9789811002724 |
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EB00207715 |
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データ種別 | 電子ブック |
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分 類 | LCC:QA273.A1-274.9 DC23:519.2 |
書誌ID | 4000116841 |
ISBN | 9789811002724 |
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※2017年9月4日以降