Probability and Its Applications

# Excursions of Markov Processes

Authors: Blumenthal, Robert M.

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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.

• Markov Processes

Blumenthal, Robert M.

Pages 1-37

$29.95 • Examples Blumenthal, Robert M. Pages 38-73$29.95
• Point Processes of Excursions

Blumenthal, Robert M.

Pages 74-109

$29.95 • Brownian Excursion Blumenthal, Robert M. Pages 110-131$29.95
• Itô’s Synthesis Theorem

Blumenthal, Robert M.

Pages 132-182

$29.95 ### Buy this book eBook$74.99
price for USA (gross)
• ISBN 978-1-4684-9412-9
• Digitally watermarked, DRM-free
• Included format: PDF
• ebooks can be used on all reading devices
Softcover \$99.00
price for USA
• ISBN 978-1-4684-9414-3
• Free shipping for individuals worldwide
• Usually dispatched within 3 to 5 business days.

## Bibliographic Information

Bibliographic Information
Book Title
Excursions of Markov Processes
Authors
Series Title
Probability and Its Applications
1992
Publisher
Birkhäuser Basel