Noncolliding Squared Bessel Processes

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.

💡 Research Summary

The paper studies a system of interacting diffusion particles whose individual dynamics are given by squared Bessel processes with index ν > −1, conditioned never to collide. When −1 < ν < 0 the origin is reflecting, otherwise it is an entrance/exit boundary. The authors first treat the case of a finite number N of particles. By applying a Doob h‑transform with the Vandermonde determinant multiplied by the single‑particle transition density, they obtain a non‑colliding diffusion whose transition density can be written in the Karlin–McGregor form as an N × N determinant. Consequently every multitime correlation function is expressed as a determinant built from a continuous kernel K(t,x;s,y). This kernel admits an explicit representation in terms of modified Bessel functions Iν and the ordinary Bessel function Jν; it can also be expanded as a series of orthogonal eigenfunctions of the squared Bessel generator, each term decaying exponentially with the eigenvalue.

The second part of the work extends the construction to an infinite number of particles. The key idea is to encode the initial configuration {a_k} (a_k > 0) into an entire function  f(z)=∏_{k=1}^{∞}(1−z/a_k), which is a Weierstrass canonical product. Under natural growth restrictions on f (order ≤ 1, finite type), the infinite Vandermonde determinant and the associated h‑transform converge, guaranteeing that the infinite‑particle process is well defined. The authors prove that the infinite system remains a determinantal point process. Its correlation kernel is now expressed through the logarithmic derivative f′/f together with the Bessel kernel, and can be written as an integral over the continuous spectrum of the squared Bessel operator:  K(s,x; t,y)=∫_0^∞ e^{−λ(t−s)} Φ_λ(x) Ψ_λ(y) dμ(λ), where Φ_λ, Ψ_λ are generalized eigenfunctions built from Jν and Iν, and μ is a measure determined by the zero set {a_k} of f.

A particularly illuminating example is provided by choosing the initial positions to be the squares of the positive zeros of Jν, i.e. a_k = j_{ν,k}² where j_{ν,k} denotes the k‑th positive zero of Jν. This “Bessel‑zero lattice” satisfies the growth conditions, and the corresponding entire function is essentially the Bessel function itself. Starting from this configuration, the authors show that the process exhibits a relaxation phenomenon: as time t→∞ the correlation kernel converges to the extended Bessel kernel, which is the universal hard‑edge scaling limit appearing in random matrix theory (e.g., the Laguerre unitary ensemble). In this limit the system becomes stationary and remains determinantal with the well‑known Bessel kernel.

Overall, the paper makes three major contributions. First, it establishes that finite non‑colliding squared Bessel systems are determinantal with an explicitly computable kernel. Second, it provides a general framework for constructing infinite‑particle non‑colliding squared Bessel processes via Weierstrass products, together with sufficient conditions for existence and determinantal structure. Third, it identifies a concrete infinite initial configuration (the Bessel‑zero lattice) that relaxes to the universal extended Bessel kernel, thereby linking the dynamics of non‑colliding diffusions to the hard‑edge universality class of random matrix theory. These results enrich the interplay between stochastic analysis, integrable probability, and mathematical physics, and open avenues for further study of boundary‑condition effects, scaling limits, and connections with other integrable particle systems.