### Abstract

Yor's generalized meander is a temporally inhomogeneous modification of the 2(ν + 1)-dimensional Bessel process with ν > -1, in which the inhomogeneity is indexed by κ ∈ [0,2(ν + 1)]. We introduce the noncolliding particle systems of the generalized meanders and prove that they are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann-Liouville differintegrals of functions comprising the Bessel functions J_{ν} used in the fractional calculus, where orders of differintegration are determined by ν - κ. As special cases of the two parameters (ν, κ), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.

Original language | English |
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Pages (from-to) | 113-156 |

Number of pages | 44 |

Journal | Probability Theory and Related Fields |

Volume | 138 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2007 May 1 |

Externally published | Yes |

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### Keywords

- Bessel processes
- Fredholm Pfaffian and determinant
- Noncolliding generalized meanders
- Random matrix theory
- Riemann-Liouville differintegrals

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability
- Analysis

### Cite this

**Infinite systems of noncolliding generalized meanders and Riemann-Liouville differintegrals.** / Katori, Makoto; Tanemura, Hideki.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 138, no. 1-2, pp. 113-156. https://doi.org/10.1007/s00440-006-0015-4

}

TY - JOUR

T1 - Infinite systems of noncolliding generalized meanders and Riemann-Liouville differintegrals

AU - Katori, Makoto

AU - Tanemura, Hideki

PY - 2007/5/1

Y1 - 2007/5/1

N2 - Yor's generalized meander is a temporally inhomogeneous modification of the 2(ν + 1)-dimensional Bessel process with ν > -1, in which the inhomogeneity is indexed by κ ∈ [0,2(ν + 1)]. We introduce the noncolliding particle systems of the generalized meanders and prove that they are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann-Liouville differintegrals of functions comprising the Bessel functions Jν used in the fractional calculus, where orders of differintegration are determined by ν - κ. As special cases of the two parameters (ν, κ), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.

AB - Yor's generalized meander is a temporally inhomogeneous modification of the 2(ν + 1)-dimensional Bessel process with ν > -1, in which the inhomogeneity is indexed by κ ∈ [0,2(ν + 1)]. We introduce the noncolliding particle systems of the generalized meanders and prove that they are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann-Liouville differintegrals of functions comprising the Bessel functions Jν used in the fractional calculus, where orders of differintegration are determined by ν - κ. As special cases of the two parameters (ν, κ), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.

KW - Bessel processes

KW - Fredholm Pfaffian and determinant

KW - Noncolliding generalized meanders

KW - Random matrix theory

KW - Riemann-Liouville differintegrals

UR - http://www.scopus.com/inward/record.url?scp=33847680999&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847680999&partnerID=8YFLogxK

U2 - 10.1007/s00440-006-0015-4

DO - 10.1007/s00440-006-0015-4

M3 - Article

VL - 138

SP - 113

EP - 156

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 1-2

ER -