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Excursions of Markov Processes

  • Robert M. Blumenthal

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

Table of contents

  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

About this book

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