## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 50, Number 1-4 (2006), 269-312.

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

Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying

#### 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.

#### Article information

**Source**

Illinois J. Math., Volume 50, Number 1-4 (2006), 269-312.

**Dates**

First available in Project Euclid: 12 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258059476

**Digital Object Identifier**

doi:10.1215/ijm/1258059476

**Mathematical Reviews number (MathSciNet)**

MR2247830

**Zentralblatt MATH identifier**

1098.60076

**Subjects**

Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Secondary: 60J60: Diffusion processes [See also 58J65]

#### Citation

Chen, Zhen-Qing; Fukushima, Masatoshi; Ying, Jiangang. Entrance law, exit system and Lévy system of time changed processes. Illinois J. Math. 50 (2006), no. 1-4, 269--312. doi:10.1215/ijm/1258059476. https://projecteuclid.org/euclid.ijm/1258059476