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

Blumenthal, R.M.

Hardcover

ISBN 10: 3764335750 ISBN 13: 9783764335755

Publisher: Birkhauser Verlag AG, 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.

Edition details

• Publisher: Birkhauser Verlag AG
• Publication date: 1991
• ISBN 10: 3764335750
• ISBN 13: 9783764335755
• Binding: Hardcover
• Language: English
• Number of pages: 275