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Field name	Details
Dewey Class	519.2
Title	Poisson Point Processes and Their Application to Markov Processes ([EBook]) / by Kiyosi Itô.
Author	Itō, Kiyoshi , 1915-2008
Other name(s)	SpringerLink (Online service)
Edition statement	1st ed. 2015.
Publication	Singapore : Springer Singapore , 2015.
Physical Details	XI, 43 pages, 3 illus. in color. : online resource.
Series	SpringerBriefs in Probability and Mathematical Statistics 2365-4333
ISBN	9789811002724
Summary Note	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.:
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users).
Internet Site	http://dx.doi.org/10.1007/978-981-10-0272-4
Links to Related Works	Subject References: Functional Analysis . Mathematics . Measure and Integration . Measure theory . Probabilities . Probability theory and stochastic processes . Authors: Itō, Kiyoshi 1915-2008 . Itô, Kiyoshi, 1915-2008. . Corporate Authors: SpringerLink (Online service) . Series: SpringerBriefs in Probability and Mathematical Statistics . Classification: 519.2 .

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Poisson Point Processes and Their Application to Markov Processes

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