Entrance law, exit system and Lévy system of time changed processes

Abstract

Let $(X, \wh X)$ be a pair of Borel standard processes on a Lusin space $E$ that are in weak duality with respect to some $\sigma$-finite measure $m$ that has full support on $E$. Let $F$ be a finely closed subset of $E$. In this paper, we obtain the characterization of a L\'evy system of the time changed process of $X$ by a positive continuous additive functional (PCAF in abbreviation) of $X$ having support $F$, under the assumption that every $m$-semipolar set of $X$ is $m$-polar for $X$. The characterization of the L\'evy system is in terms of Feller measures, which are intrinsic quantities for the part process of $X$ killed upon leaving $E\setminus F$. Along the way, various relations between the entrance law, exit system, Feller measures and the distribution of the starting and ending point of excursions of $X$ away from $F$ are studied. We also show that the time changed process of $X$ is a special standard process having a weak dual and that the $\mu$-semipolar set of $Y$ is $\mu$-polar for $Y$, where $\mu$ is the Revuz measure for the PCAF used in the time change.