Excursions of Markov Processes

by Robert M. Blumenthal

Overview

Let {Xti t ~ O} be a Markov process in Rl, and break up the path X t into (random) component pieces consisting of the zero set ({ tlX = O}) and t the "excursions away from 0," that is pieces of path X. : T ::5 s ::5 t, with Xr- = X = 0, but X. 1= 0 for T < s < t. When one measures the time in t the zero set appropriately (in terms of the local time) the excursions acquire a measure theoretic structure practically identical to that of processes with stationary independent increments, except the values of the process are paths rather than real numbers. And there is a measure on path space that helps describe the measure theoretic properties of the excursions in the same way that the Levy measure describes the jumps of a process with independent increments. The entire circle of ideas is called excursion theory. There are many attractive things about the subject: it is an area where one can use to advantage general probabilistic potential theory to make quite specific calculations, it provides a natural setting for apply­ ing esoteric things like David Williams' path decomposition, it provides a method for constructing processes whose description in terms of an in­ finitesimal generator or some such analytic object would be complicated. And the ideas seem to be closely related to a good deal of current research in probability.

Product Details

ISBN-13:	9781468494143
Publisher:	Birkh�user Boston
Publication date:	06/02/2012
Series:	Probability and Its Applications
Edition description:	Softcover reprint of the original 1st ed. 1992
Pages:	276
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

I Markov Processes.- 0. Introduction.- 1. Basic terminology.- 2. Stationary transition functions.- 3. Time homogeneous Markov processes.- 4. The strong Markov property.- 5. Hitting times.- 6. Standard processes.- 7. Killed and stopped processes.- 8. Canonical realizations.- 9. Potential operators and resolvents.- II Examples.- 1. Examples.- 2. Brownian motion.- 3. Feller Brownian motions and related examples.- III Point Processes of Excursions.- 1. Additive processes.- 2. Poisson point processes.- 3. Poisson point processes of excursions.- IV Brownian Excursion.- 1. Brownian excursion.- 2. Path decomposition.- 3. The non-recurrent case.- 4. Feller Brownian motions.- 5. Reflecting Brownian motion.- V Itô’s Synthesis Theorem.- 1. Introduction.- 2. Construction.- 3. Examples and complements.- 4. Existence and uniqueness.- 5. A counter-example.- 6. Integral representation.- VI Excursions and Local Time.- 1. Introduction.- 2. Ray’s local time theorem.- 3. Trotter’s theorem.- 4. Super Brownian motion.- VII Excursions Away From a Set.- 1. Introduction.- 2. Additive functionals and Lévy systems.- 3. Exit systems.- 4. Motoo Theory.- Notation Index.