## The Annals of Probability

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

J. Vuolle-Apiala

#### 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: 19 April 2007

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

JSTOR

Digital Object Identifier
doi:10.1214/aop/1176988721

Mathematical Reviews number (MathSciNet)
MR1288123

Zentralblatt MATH identifier
0810.60067

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

#### Citation

Vuolle-Apiala, J. Ito Excursion Theory for Self-Similar Markov Processes. The Annals of Probability 22 (1994), no. 2, 546--565. doi:10.1214/aop/1176988721. http://projecteuclid.org/euclid.aop/1176988721.