## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 2 (1994), 546-565.

### Ito Excursion Theory for Self-Similar Markov Processes

#### Abstract

Let $X_t$ be an $\alpha$-self-similar Markov process on $(0, \infty)$ killed when hitting 0. $\alpha$-self-similar extensions of $X(t)$ to $\lbrack 0, \infty)$ are studied via Ito execusion theory (entrance laws). We give a condition that guarantees the existence of an extension, which either leaves 0 continuously (a.s.) or (a.s.) jumps from 0 to $(0, \infty)$ according to the "jumping in" measure $\eta(dx) = dx/x^{\beta+1}$. Two applications are given: the diffusion case and the "reflecting barrier process" of S. Watanabe.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 2 (1994), 546-565.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988721

**Digital Object Identifier**

doi:10.1214/aop/1176988721

**Mathematical Reviews number (MathSciNet)**

MR1288123

**Zentralblatt MATH identifier**

0810.60067

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J25: Continuous-time Markov processes on general state spaces

**Keywords**

Self-similar Markov processes entrance laws

#### Citation

Vuolle-Apiala, J. Ito Excursion Theory for Self-Similar Markov Processes. Ann. Probab. 22 (1994), no. 2, 546--565. doi:10.1214/aop/1176988721. https://projecteuclid.org/euclid.aop/1176988721