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[null,"1.","Record Nr.",null,"UNINA9910300248603321"]
[null,null,"Autore",null,"Itô Kiyosi"]
[null,null,"Titolo",null,"Poisson Point Processes and Their Application to Markov Processes / / by Kiyosi Itô"]
[null,null,"Pubbl/distr/stampa",null,"Singapore : , : Springer Singapore : , : Imprint : Springer, , 2015"]
[null,null,"ISBN",null,"981-10-0272-X"]
[null,null,"Edizione",null,"[1st ed. 2015.]"]
[null,null,"Descrizione fisica",null,"1 online resource (54 p.)"]
[null,null,"Collana",null,"SpringerBriefs in Probability and Mathematical Statistics, , 2365-4333"]
[null,null,"Disciplina",null,"519.23"]
[null,null,"Soggetti",null,"Probabilities"]
[{"$nulls":4},"Measure theory"]
[{"$nulls":4},"Functional analysis"]
[{"$nulls":4},"Probability Theory and Stochastic Processes"]
[{"$nulls":4},"Measure and Integration"]
[{"$nulls":4},"Functional Analysis"]
[null,null,"Lingua di pubblicazione",null,"Inglese"]
[null,null,"Formato",null,"Materiale a stampa"]
[null,null,null,"Livello bibliografico","Monografia"]
[null,null,null,"Note generali","Description based upon print version of record."]
[null,null,null,"Nota di bibliografia","Includes bibliographical references at the end of each chapters."]
[null,null,"Nota di contenuto",null,"Foreword; Preface; References; Contents; 1 Poisson Point Processes; 1.1 Point Functions; 1.2 Point Processes; 1.3 Poisson Point Processes; 1.4 The Structure of Poisson Point Processes  (1) the Discrete Case; 1.5 The Structure of Poisson Point Processes  (2) the General Case; 1.6 Transformation of Poisson Point Processes; 1.7 Summable Point Processes; 1.8 The Strong Renewal Property of Poisson Point Processes; References; 2 Application to Markov Processes; 2.1 Problem; 2.2 The Poisson Point Process Attached to a Markov Process at a State a; 2.3 The Jumping-In Measure and the Stagnancy Rate"]
[null,null,null,"Sommario/riassunto","An extension problem (often called a boundary problem) of Markov processes has been studied, particularly in the case of one-dimensional diffusion processes, by W. Feller, K. Itô, and H. P. McKean, among others. In this book, Itô discussed a case of a general Markov process with state space S and a specified point a ∈ S called a boundary. The problem is to obtain all possible recurrent extensions of a given minimal process (i.e., the process on S \\ {a} which is absorbed on reaching the boundary a). The study in this lecture is restricted to a simpler case of the boundary a being a discontinuous entrance point, leaving a more general case of a continuous entrance point to future works. He established a one-to-one correspondence between a recurrent extension and a pair of a positive measure k(db) "]
[{"$nulls":4},"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."]