Excursions of Markov Processes (Probability and Its Applications) - Hardcover

Blumenthal, R.M.

Hardcover

ISBN 10: 0817635750 ISBN 13: 9780817635756

Publisher: Birkhauser Boston Inc, 1991

Synopsis

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.

About this title

Synopsis

An introduction for graduate students and research mathematicians to the excursion theory of analyzing and constructing Markov processes. The explanation of the approach's theory, techniques, and applications emphasizes making calculations and proving the concepts of probabilistic potential theory. Also suitable as a textbook for a graduate course

Edition details

• Publisher: Birkhauser Boston Inc
• Publication date: 1991
• Language: English
• ISBN 10: 0817635750
• ISBN 13: 9780817635756
• Binding: Hardcover
• Number of pages: 288