Hua-Pickrell diffusions and differential equations related with pseudo-Jacobi polynomials

Abstract

Following Assiotis (2020), we study general $\beta$-Hua-Pickrell diffusions of $N$ particles on $\mathbb{R}$ as solutions of the stochastic differential equations (SDEs) $$d{X}_{j,t}=\sqrt{2(1+X_{j,t}^2)}d{B}_{j,t}+\beta \left[b-a{X}_{j,t}+\sum _{l=1,\dots ,N;l\ne j}\frac{X_{j,t}{X}_{l,t}+1}{X_{j,t}-X_{l,t}}\right]dt,(j=1,\dots ,N)$$ with $\beta \ge 1,a,b\in \mathbb{R}$. These processes form a subclass of the Pearson diffusions which are defined as solutions of algebraic SDEs where the moments of the empirical distributions ${\mu }_t^N:=\frac{1}{N}{\sum }_{j=1}^N{\delta }_{X_{j,t}}$ can be computed inductively. This Pearson class also contains other well known diffusions like Dyson Brownian motions, and multivariate Laguerre and Jacobi processes After the time normalization $t\mapsto t/\beta$, the SDEs above degenerate in the frozen case for $\beta =\infty$ into ordinary differential equations which are related to pseudo-Jacobi polynomials. For $N\to \infty$ and under suitable initial conditions, the empirical distributions ${\mu }_t^N$ converge weakly almost surely for $t>0$ to some limit which is independent from $\beta \in [1,\infty ]$. For $a=-N,b=0$, we describe the limit explicitly via free convolutions. Moreover, if $a=cN$ for some $c>0$, the solutions of our SDEs converge for $t\to \infty$ to stationary distributions, which are Hua-Pickrell (or Cauchy) measures. We thus obtain connections between known results for the empirical distributions of these ensembles and the zeros of the pseudo-Jacobi polynomials. Furthermore, we derive a freezing central limit theorem for $\beta \to \infty$ for the Hua-Pickrell ensembles which is related to these zeros.

Some misprints corrected; extension of the introduction

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