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Lévy Systems and Time Changes

  • P.J. FitzsimmonsEmail author
  • R.K. Getoor
Chapter
  • 824 Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 1979)

Abstract

The Lévy system for a Markov process X provides a convenient description of the distribution of the totally inaccessible jumps of the process. We examine the effect of time change (by the inverse of a not necessarily strictly increasing CAF A) on the Lévy system, in a general context. They key to our time-change theorem is a study of the “irregular” exits from the fine support of A that occur at totally inaccessible times. This permits the construction of a partial predictable exit system (à la Maisonneuve).

The second part of the paper is devoted to some implications of the preceding in a (weak, moderate Markov) duality setting. Fixing an excessive measure m (to serve as duality measure) we obtain formulas relating the “killing” and “jump” measures for the time-changed process to the analogous objects for the original process. These formulas extend, to a very general context, recent work of Chen, Fukushima, and Ying. The key to our development is the Kuznetsov process associated with X and m, and the associated moderate Markov dual process \(\hat X\). Using \(\hat X\) and some excursion theory, we exhibit a general method for constructing excessive measures for X from excessive measures for the time-changed process.

Key words and phrases

Lévy system exit system time change Markov process continuous additive functional excessive measure Kuznetsov process 

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of Mathematics, 0112;University of California San Diego 9500 Gilman DriveLa JollUSA

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