Poisson Point Processes and Their Application to Markov Processes

byKiyosi ItForeword byShinzo Watanabe, Ichiro Shigekawa

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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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