Homogeneous Random Measures and Strongly Supermedian Kernels of a Markov Process
Patrick J. Fitzsimmons, UCSD
Ronald K. Getoor, UCSD
Abstract
The potential kernel of a positive left
additive functional (of a strong Markov process X) maps positive
functions to strongly supermedian functions and satisfies a variant of the
classical domination principle of potential theory. Such a kernel
V is called a regular strongly supermedian kernel in recent work
of L. Beznea and N. Boboc. We establish the converse: Every regular strongly
supermedian kernel V is the potential kernel of a
random measure homogeneous on [0, infinity). Under additional
finiteness conditions such random measures give rise to left additive
functionals. We investigate such random measures, their potential
kernels, and their associated characteristic measures.
Given a left additive functional A (not necessarily continuous),
we give
an explicit construction of a simple Markov process Z whose resolvent
has initial kernel equal to the potential kernel UA.
The theory we
develop is the probabilistic counterpart of the work of Beznea and Boboc.
Our main tool is the Kuznetsov process associated with X and a given
excessive measure m.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.