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Poisson Point Processes and Their Application to Markov Processes

Kiyosi Itô
Publication Date: 
Number of Pages: 
Springer Briefs in Probability and Mathematical Statistics
[Reviewed by
Richard Durrett
, on

Itô received his Ph.D. in 1945. In the next five years he wrote several papers laying the groundwork for stochastic integration and the associated stochastic calculus, including the formula that bears his name. Throughout his career, Itô made a number of important contributions to stochastic analysis in finite and infinite dimensional spaces. The work under review is not one of them.

The book contains notes of a course given in Japan in 1969. This was soon after the publication (in 1965) of his book with Henry McKean: Diffusion Processes and their Sample Paths. The focus of that book was to understand the structure of one-dimensional Markov processes with continuous paths (diffusion processes). The answer is simple: after a change of variables to make the process a martingale it is a time change of Brownian motion. If one has a diffusion on a half line \([0,\infty)\), there is the question: what can it do when it reaches the boundary? It can (i) be absorbed, (ii) be reflected, and (iii) undergo sticky reflection, so that the time spent on the boundary has positive measure. This work considers a fourth possibility: jumping in from the boundary.

To quote from the description on Springer’s web page:

The problem is to obtain all possible recurrent extensions of a given minimal process (i.e., the process on \(S\setminus \{a\}\) 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. A one-to-one correspondence is established between a recurrent extension and a pair consisting of a positive measure \(k(db)\) on \(S\setminus \{a\}\) (called the jumping-in measure) and a non-negative number \(m<\infty\) (called the stagnancy rate). For this, Itô used, as a fundamental tool, the notion of Poisson point processes formed of all excursions of the process on \(S\setminus \{a\}\).

It is not clear why Springer chose to publish this 43 page booklet (which it sells for $55). On the web site they praise the clarity of Itô’s lecturing and in particular the 18 page description of Poisson processes in Chapter 1. However, there are many other treatments of that topic that are well written and more comprehensive. The solution of the problem described above given in Chapter 2 is much less important than the work Itô did later decomposing the Brownian path which I will now briefly describe.

The set of zeros of a Brownian motion is a closed set with Hausdorf dimension \(1/2\). Like the Cantor set, the complement of the zero set consists of “excursion” intervals of a wide variety of sizes. However, like many aspects of Brownian motion, the excursions are irregular in a very regular way. Define the local time at \(0\) up to time \(t\), \(L_t\), by the requirement that \(|B_t|-L_t\) is a martingale. The number of excursion intervals (that make up the complement of the zero set) up to time \(L_t\) that are of length larger than \(a\) are a Poisson process with rate \(ca^{-1/2}\). This observation has been generalized to a wide variety of Markov processes, providing insights into their structure. A treatment of this result and its extensions would have made a much better book.

Richard Durrett taught at UCLA and Cornell before he came to Duke in 2010. He is a member of the National Academy of Science, who for the last thirty years has used probability to study problems that arise from ecology, genetics, and cancer modeling.

See the table of contents in the publisher's webpage.