This chapter gives an analytic approach for the Revuz correspondence between classes of strongly supermedian kernels and their associated measures, completing the probabilistic works of Revuz, Azéma, Getoor-Sharpe, Fitzsimmons, Fitzsimmons-Getoor, Dellacherie-Maisonneuve-Meyer. In Section 6.1 we extend the energy functional to every pair (ξ, s) where ξ is a u-excessive measure and s is a strongly supermedian function, u being the given sub-Markovian resolvent. We establish the Revuz correspondence between the regular strongly supermedian kernels and the σ-finite measures charging no set that is both ξ-polar and ρ-negligible, ρ o u being the potential component of ξ. It turns out that for these kernels and measures a Revuz type formula (involving the ξ-fine versions) holds. Consequently it is obtained the Revuz correspondence between the semiregular excessive kernels and the σ-finite measures charging no ξ-polar set. Section 6.2 discusses the hypothesis (B) of Hunt with respect to ξ and a given natural topology T, in connection with the identification of the semiregular and natural (with respect to T) excessive kernels. Section 6.3 exposes the Revuz correspondence between the smooth measures and the suitable class of regular strongly supermedian kernels. In Section 6.4 we present the measure perturbation of the sub-Markovian resolvents.
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