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) $$dX_{j,t}=\sqrt{2(1+X_{j,t}^2)}\,dB_{j,t}+\beta\left[b-a X_{j,t}+\sum_{l=1,\ldots, N; \> l\neq j}\frac{X_{j,t}X_{l,t}+1}{X_{j,t}-X_{l,t}}\right]dt\,,\;\; (j=1,\ldots,N)$$ 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.