Noncolliding Squared Bessel Processes

TL;DR

This paper studies a noncolliding particle system of squared Bessel processes, proving determinantal properties for finite and infinite particles, and analyzing the long-term relaxation to a stationary process with the extended Bessel kernel.

Contribution

It introduces a new determinantal process for noncolliding squared Bessel particles, including infinite configurations, with explicit correlation kernels and long-term behavior analysis.

Findings

Finite particle system is determinantal with a continuous correlation kernel.

Infinite particle system is well-defined under certain initial configurations.

The process exhibits relaxation to a stationary determinantal process with the extended Bessel kernel.

Abstract

We consider a particle system of the squared Bessel processes with index $\nu > -1$ conditioned never to collide with each other, in which if $-1 < \nu < 0$ 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 $J_{\nu}$ 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.