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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
##### Keywords:
Revuz measure; supermedian function; potential kernel
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