## Osaka Journal of Mathematics

- Osaka J. Math.
- Volume 51, Number 2 (2014), 337-359.

### Infinite Markov particle systems associated with absorbing stable motion on a half space

#### Abstract

In general, for a Markov process which does not have an invariant measure, it is possible to realize a stationary Markov process with the same transition probability by extending the probability space and by adding new paths which are born at random times. The distribution (which may not be a probability measure) is called a
*Kuznetsov measure*. By using this measure we can construct a stationary Markov particle system, which is called an
*equilibrium process with immigration*. This particle system can be decomposed as a sum of the original part and the immigration part (see [2]). In the present paper, we consider an
*absorbing stable motion* on a half space $H$, i.e., a time-changed absorbing Brownian motion on $H$ by an increasing strictly stable process. We first give the martingale characterization of the particle system. Secondly, we discuss the finiteness of the number of particles near the boundary of the immigration part. (cf. [2], [3], [4].)

#### Article information

**Source**

Osaka J. Math., Volume 51, Number 2 (2014), 337-359.

**Dates**

First available in Project Euclid: 8 April 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ojm/1396966252

**Mathematical Reviews number (MathSciNet)**

MR3192545

**Zentralblatt MATH identifier**

1305.60036

**Subjects**

Primary: 60G57: Random measures

Secondary: 60G52: Stable processes

#### Citation

Hiraba, Seiji. Infinite Markov particle systems associated with absorbing stable motion on a half space. Osaka J. Math. 51 (2014), no. 2, 337--359. https://projecteuclid.org/euclid.ojm/1396966252