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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. Ito, and H.
P. McKean, among others. In this book, Ito 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, Ito used, as a fundamental tool, the notion of Poisson point
processes formed of all excursions of the process on S \ {a}. This
theory of Ito'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 Ito 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 Ito's beautiful and impressive lectures in his
day.
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