### Abstract

We consider a particle system of the squared Bessel processes with index ν > -1 conditioned never to collide with each other, in which if -1 < ν < 0 the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function J_{ν} is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.

Original language | English |
---|---|

Pages (from-to) | 592-615 |

Number of pages | 24 |

Journal | Journal of Statistical Physics |

Volume | 142 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 Feb 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Entire functions
- Fredholm determinants
- Infinite particle systems
- Noncolliding diffusion process
- Squared Bessel process
- Weierstrass canonical products

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*142*(3), 592-615. https://doi.org/10.1007/s10955-011-0117-y

**Noncolliding Squared Bessel Processes.** / Katori, Makoto; Tanemura, Hideki.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 142, no. 3, pp. 592-615. https://doi.org/10.1007/s10955-011-0117-y

}

TY - JOUR

T1 - Noncolliding Squared Bessel Processes

AU - Katori, Makoto

AU - Tanemura, Hideki

PY - 2011/2/1

Y1 - 2011/2/1

N2 - We consider a particle system of the squared Bessel processes with index ν > -1 conditioned never to collide with each other, in which if -1 < ν < 0 the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function Jν is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.

AB - We consider a particle system of the squared Bessel processes with index ν > -1 conditioned never to collide with each other, in which if -1 < ν < 0 the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function Jν is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.

KW - Entire functions

KW - Fredholm determinants

KW - Infinite particle systems

KW - Noncolliding diffusion process

KW - Squared Bessel process

KW - Weierstrass canonical products

UR - http://www.scopus.com/inward/record.url?scp=79551685987&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79551685987&partnerID=8YFLogxK

U2 - 10.1007/s10955-011-0117-y

DO - 10.1007/s10955-011-0117-y

M3 - Article

AN - SCOPUS:79551685987

VL - 142

SP - 592

EP - 615

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -