Noncolliding Squared Bessel Processes

Makoto Katori, Hideki Tanemura

Research output: Contribution to journalArticlepeer-review

29 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)592-615
Number of pages24
JournalJournal of Statistical Physics
Issue number3
Publication statusPublished - 2011 Feb
Externally publishedYes


  • 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


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