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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.

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