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 *U*_{A}. 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