04838nam 22007095 450 991030024860332120200704083756.0981-10-0272-X10.1007/978-981-10-0272-4(CKB)3710000000541916(EBL)4217715(SSID)ssj0001597392(PQKBManifestationID)16297883(PQKBTitleCode)TC0001597392(PQKBWorkID)14886282(PQKB)10402461(DE-He213)978-981-10-0272-4(MiAaPQ)EBC4217715(PPN)258870583(PPN)190884479(EXLCZ)99371000000054191620151224d2015 u| 0engur|n|---|||||txtccrPoisson Point Processes and Their Application to Markov Processes /by Kiyosi Itô1st ed. 2015.Singapore :Springer Singapore :Imprint: Springer,2015.1 online resource (54 p.)SpringerBriefs in Probability and Mathematical Statistics,2365-4333Description based upon print version of record.981-10-0271-1 Includes bibliographical references at the end of each chapters.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 RateAn 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.SpringerBriefs in Probability and Mathematical Statistics,2365-4333ProbabilitiesMeasure theoryFunctional analysisProbability Theory and Stochastic Processeshttps://scigraph.springernature.com/ontologies/product-market-codes/M27004Measure and Integrationhttps://scigraph.springernature.com/ontologies/product-market-codes/M12120Functional Analysishttps://scigraph.springernature.com/ontologies/product-market-codes/M12066Probabilities.Measure theory.Functional analysis.Probability Theory and Stochastic Processes.Measure and Integration.Functional Analysis.519.23Itô Kiyosiauthttp://id.loc.gov/vocabulary/relators/aut344545Watanabe Shinzo1935-Shigekawa Ichirō1953-Bernoulli Society for Mathematical Statistics and Probability.MiAaPQMiAaPQMiAaPQBOOK9910300248603321Poisson Point Processes and Their Application to Markov Processes2498776UNINA