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.
A hard copy of this manuscript is available from the
author upon request.
The manuscript can also be downloaded in
dvi form (119K)
and pdf form (251K).
(Version of April 30, 2007)
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April 30, 2007
