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Strongly supermedian kernels and Revuz measures. (English) Zbl 1025.60031
Let \(X\) be a Lusin space, \({\mathcal U}=(U_\alpha)_{\alpha>0}\) a proper sub-Markov resolvent on \(X\) and \(\xi\) an \({\mathcal U}\)-excessive measure. For a strongly supermedian kernel \(V\) on \(X\), define the Revuz measure \(\nu^\xi_V\) by \(\nu^\xi_V(f)=\sup\{\nu(Vf)\); \(\nu\circ U\leq \xi\}\). Further, \(V\) is said regular if, \(Vf \leq s\) on \(\{f>0\}\) for a strongly supermedian function \(s\), then \(Vf \leq s\) everywhere. Analytic characterizations of regular strongly supermedian kernels including Motoo-Mokobodzki property are given. For the excessive measure \(\xi=\rho\circ U+h\) with a harmonic measure \(h\), it is proved that any \(\sigma\)-finite measure charging no set that is both \(\xi\)-polar and \(\rho\)-negligible is a Revuz measure of a regular strongly supermedian kernel. An excessive kernel \(V\) is said natural if, for any Ray open set \(G\) and a function \(f\) vanishing outside of \(G\), \(B^GVf=Vf\), where \(B^GVf\) is the excessive regularization of the réduite of \(Vf\) on \(G\). A characterization of natural excessive kernels by means of the non-charging property of the associated Revuz measure is given. Furthermore, a characterization of the hypothesis (B) of Hunt is given in terms of Revuz measures.

60J40 Right processes
31D05 Axiomatic potential theory
60J45 Probabilistic potential theory
31C15 Potentials and capacities on other spaces
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