Lévy Systems and Time Changes
### Lévy Systems and Time Changes

#### P.J. Fitzsimmons and R.K. Getoor

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. Our basic hypothesis (beyond the "right"
Markov property) is that the "irregular" exits from the fine support of *A* occur at
totally inaccessible times. This condition permits the construction of a predictable
exit system (à la Maisonneuve), the key tool for our time change theorem.
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 *Xhat*. Using *Xhat* and some
excursion theory, we exhibit a general method for construction excessive measures
for *X* from excessive measures for the time-changed process.

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(Version of April 30, 2007)

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April 30, 2007