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Homogeneous Random Measures and Strongly Supermedian Kernels of a Markov Process
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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.
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Full text: PDF
Pages: 1-54
Published on: July 3, 2003
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