HRMs and Strongly Supermedian Kernels
Homogeneous Random Measures and
Strongly Supermedian Kernels of a Markov Process
P.J. Fitzsimmons and R.K. Getoor
(To appear in
Electronic Journal of Probability)
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.
A hard copy of this manuscript is available from the
first-named author upon request.
The manuscript can also be downloaded in
dvi form (216K),
postscript form (764K),
and pdf form (637K).
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August 27, 2002; revised July 1, 2003