Illinois Journal of Mathematics

Excursion theory revisited

P. J. Fitzsimmons and R. K. Getoor

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Abstract

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 $\widehat X$ arises naturally and plays an important role. This allows the basic quantities of excursion theory such as the Laplace-L\'evy 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 $\widehat X$ are honest, then the resolvent of $X$ may be expressed entirely in terms of quantities that depend only on $X$ and $\widehat X$ killed when they first hit $b$.

Article information

Source
Illinois J. Math., Volume 50, Number 1-4 (2006), 413-437.

Dates
First available in Project Euclid: 12 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258059481

Digital Object Identifier
doi:10.1215/ijm/1258059481

Mathematical Reviews number (MathSciNet)
MR2247835

Zentralblatt MATH identifier
1106.60062

Subjects
Primary: 60J40: Right processes
Secondary: 60G51: Processes with independent increments; Lévy processes 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J55: Local time and additive functionals

Citation

Fitzsimmons, P. J.; Getoor, R. K. Excursion theory revisited. Illinois J. Math. 50 (2006), no. 1-4, 413--437. doi:10.1215/ijm/1258059481. https://projecteuclid.org/euclid.ijm/1258059481


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