Poisson point processes and their application to Markov processes - 1

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< (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.