Infinite systems of noncolliding generalized meanders and Riemann-Liouville differintegrals

• Published in: Probability theory and related fields Vol. 138; no. 1-2; pp. 113 - 156
• Main Authors: KATORI, Makoto, TANEMURA, Hideki
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01-05-2007; Berlin Springer Nature B.V; New York, NY
• Subjects: Bessel functions; Brownian motion; Eigenvalues; Exact sciences and technology; Fractional calculus; General topics; Growth models; Inhomogeneity; Mathematics; Matrix theory; Physics; Probability; Probability and statistics; Probability theory and stochastic processes; Quaternions; Sciences and techniques of general use; Stochastic models; Stochastic processes; Bessel function; Random matrix; Random matrix theory; Noncolliding generalized meanders; Probability theory; 15A52; 26A33; 60G55; Infinite system; Fractional calculus; Kernel method; Bessel processes; Kernels; Fredholm Pfaffian and determinant; Particle system; Correlation function; Riemann-Liouville differintegrals; 60J60
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-006-0015-4

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 . 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.