# Excursions of Markov Processes

• Robert M. Blumenthal
Textbook

Part of the Probability and Its Applications book series (PA)

1. Front Matter
Pages i-xi
2. Robert M. Blumenthal
Pages 1-37
3. Robert M. Blumenthal
Pages 38-73
4. Robert M. Blumenthal
Pages 74-109
5. Robert M. Blumenthal
Pages 110-131
6. Robert M. Blumenthal
Pages 132-182
7. Robert M. Blumenthal
Pages 183-219
8. Robert M. Blumenthal
Pages 220-269
9. Back Matter
Pages 270-276

### Introduction

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.

### Keywords

Generator Markov process local time path space probability

#### Authors and affiliations

• Robert M. Blumenthal
• 1
1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4684-9412-9
• Copyright Information Birkhäuser Boston 1992
• Publisher Name Birkhäuser Boston
• eBook Packages
• Print ISBN 978-1-4684-9414-3
• Online ISBN 978-1-4684-9412-9
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