Limit theorems for multivariate Bessel processes in the freezing regime

TL;DR

This paper investigates the asymptotic behavior of multivariate Bessel processes in the freezing regime, revealing new limit laws and central limit theorems for specific initial conditions as multiplicities grow large.

Contribution

It derives novel limit laws for multivariate Bessel processes with scaled initial points using SDEs, extending previous results to a broader class of starting conditions.

Findings

Limit laws described by zeros of Hermite and Laguerre polynomials.

Almost sure local uniform convergence in time.

Associated central limit theorems established.

Abstract

Multivariate Bessel processes describe the stochastic dynamics of interacting particle systems of Calogero-Moser-Sutherland type and are related with $\beta$-Hermite and Laguerre ensembles. It was shown by Andraus, Katori, and Miyashita that for fixed starting points, these processes admit interesting limit laws when the multiplicities $k$ tend to $\infty$, where in some cases the limits are described by the zeros of classical Hermite and Laguerre polynomials. In this paper we use SDEs to derive corresponding limit laws for starting points of the form $\sqrt k\cdot x$ for $k\to\infty$ with $x$ in the interior of the corresponding Weyl chambers. Our limit results are a.s. locally uniform in time. Moreover, in some cases we present associated central limit theorems.