## Osaka Journal of Mathematics

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

Seiji Hiraba

#### 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

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

#### References

• S.N. Ethier and T.G. Kurtz: Markov Processes, Wiley, New York, 1986.
• S. Hiraba: Infinite Markov particle systems with singular immigration; martingale problems and limit theorems, Osaka J. Math. 33 (1996), 145–187.
• S. Hiraba: Asymptotic behavior of hitting rates for absorbing stable motions in a half space, Osaka J. Math. 34 (1997), 905–921.
• S. Hiraba: Independent infinite Markov particle systems with jumps, Theory Stoch. Process. 18 (2012), 65–85.
• R.Sh. Liptser and A.N. Shiryayev: Theory of Martingales, Kluwer Acad. Publ., Dordrecht, 1989.
• T. Shiga and Y. Takahashi: Ergodic properties of the equilibrium process associated with infinitely many Markovian particles, Publ. Res. Inst. Math. Sci. 9 (1973/74), 505–516.