Fine Densities for Excessive Measures and the Revuz Correspondence

Abstract

Suppose that U is the resolvent of a Borel right process on a Lusin space X. If ξ is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure η with η≪ξ the Radon–Nikodym derivative dη/dξ possesses a finely continuous version. (Fitzsimmons and Fitzsimmons and Getoor gave a probabilistic approach for this result.) We extend essentially a technique initiated by Mokobodzki and deepened by Feyel. The result allows us to establish a Revuz type formula involving the fine versions, and to study the Revuz correspondence between the σ-finite measures charging no set that is both ξ-polar and ρ-negligible (ρ○U being the potential component of ξ) and the strongly supermedian kernels on X. This is an analytic version of a result of Azema, Fitzsimmons and Dellacherie, Maisonneuve and Meyer, in terms of additive functionals or homogeneous random measures. Finally we give an application to the context of the semi-Dirichlet forms, covering a recent result of Fitzsimmons.