Excursion Theory Revisited
### Excursion Theory Revisited

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

Excursions from a fixed point *b* are studied in the
framework of a general Borel right process *X*, with a fixed excessive
measure *m* serving as background measure; such a measure always exists
if *b* is accessible from every point of the state space of *X*. In this
context the left-continuous moderate Markov dual process
*Xhat* arises naturally and plays an important role. This allows
the basic quantities of excursion theory such as the Laplace-Lévy
exponent of the inverse local time at *b* and the Laplace transform of
the entrance law for the excursion process to be expressed as inner
products involving simple hitting probabilities and expectations. In
particular if *X* and
*Xhat* are honest, then the resolvent of *X* may be expressed
entirely in terms of quantities that depend only on *X* and
*Xhat* killed when they first hit *b*.

A hard copy of this manuscript is available from the
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May 10, 2005