Electronic Journal of Probability

Homogeneous Random Measures and Strongly Supermedian Kernels of a Markov Process

Patrick Fitzsimmons and Ronald Getoor

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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 $[0,\infty[$. 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 $m$.

Article information

Electron. J. Probab., Volume 8 (2003), paper no. 10, 54 p.

First available in Project Euclid: 23 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J55: Local time and additive functionals
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J40: Right processes

Homogeneous random measure additive functional Kuznetsov measure potential kernel characteristic measure strongly supermedian smooth measure


Fitzsimmons, Patrick; Getoor, Ronald. Homogeneous Random Measures and Strongly Supermedian Kernels of a Markov Process. Electron. J. Probab. 8 (2003), paper no. 10, 54 p. doi:10.1214/EJP.v8-142. https://projecteuclid.org/euclid.ejp/1464037583

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