Poisson Point Processes and Their Application to Markov Processes by Kiyosi ItPoisson Point Processes and Their Application to Markov Processes by Kiyosi It

Poisson Point Processes and Their Application to Markov Processes

byKiyosi ItForeword byShinzo Watanabe, Ichiro Shigekawa

Paperback | February 1, 2016

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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<_20_28_called20_the20_stagnancy20_rate29_.20_the20_necessary20_and20_sufficient20_conditions20_for20_a20_pair20_k2c_20_m20_was20_obtained20_so20_that20_the20_correspondence20_is20_precisely20_described.20_for20_this2c_20_itc3b4_20_used2c_20_c2a0_as20_a20_fundamental20_tool2c_20_the20_notion20_of20_poisson20_point20_processes20_formed20_of20_all20_excursions20_of20_c2a0_the20_process20_onc2a0_sc2a0_5c_c2a0_7b_a7d_.20_this20_theory20_of20_itc3b4_27_27_s20_of20_poisson20_point20_processes20_of20_excursions20_is20_indeed20_a20_breakthrough.20_it20_has20_been20_expanded20_and20_applied20_to20_more20_general20_extension20_problems20_by20_many20_succeeding20_researchers.20_thus20_we20_may20_say20_that20_this20_lecture20_note20_by20_itc3b4_20_is20_really20_a20_memorial20_work20_in20_the20_extension20_problems20_of20_markov20_processes.20_especially20_in20_chapter20_120_of20_this20_note2c_20_a20_general20_theory20_of20_poisson20_point20_processes20_is20_given20_that20_reminds20_us20_of20_itc3b4_27_27_s20_beautiful20_and20_impressive20_lectures20_in20_his20_day. _28_called="" the="" stagnancy="" _rate29_.="" necessary="" and="" sufficient="" conditions="" for="" a="" pair="" _k2c_="" m="" was="" obtained="" so="" that="" correspondence="" is="" precisely="" described.="" _this2c_="" _itc3b4_="" _used2c_="" _c2a0_as="" fundamental="" _tool2c_="" notion="" of="" poisson="" point="" processes="" formed="" all="" excursions="" _c2a0_the="" process="" _onc2a0_sc2a0_5c_c2a0_7b_a7d_.="" this="" theory="" _itc3b4_27_27_s="" indeed="" breakthrough.="" it="" has="" been="" expanded="" applied="" to="" more="" general="" extension="" problems="" by="" many="" succeeding="" researchers.="" thus="" we="" may="" say="" lecture="" note="" really="" memorial="" work="" in="" markov="" processes.="" especially="" chapter="" 1="" _note2c_="" given="" reminds="" us="" beautiful="" impressive="" lectures="" his="">
Title:Poisson Point Processes and Their Application to Markov ProcessesFormat:PaperbackDimensions:43 pagesPublished:February 1, 2016Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9811002711

ISBN - 13:9789811002717

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Table of Contents

1. Poisson point processes.- 2. Application to Markov Process.

Editorial Reviews

"The main idea of this volume has had a profound influence on the boundary theory of Markov processes. This volume is beautifully written and it is a pleasure to read." (Ren Ming Song, Mathematical Reviews, December, 2016)