2026/02/28 by Martin Auer, Michael Voit
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Mathematical functions and polynomials #Random Matrices and Applications #Stochastic processes and financial applications #math-ph #math.CA #math.MP #math.PR #msc:33C45 #msc:60B20 #msc:60F05 #msc:60F15 #msc:60K35 #msc:70F10 #msc:82C22
paper · pdf · doi:10.1063/5.0328924
Some misprints corrected; extension of the introduction
openalex publication_date 2026/07/01 · openalex created_date 2026/07/29 · openalex updated_date 2026/07/29 · arxiv created 2026/08/01 · arxiv updated 2026/08/04
Following Assiotis [Ann. Inst. Henri Poincare B 56, 1251–1283 (2020)], we study general β-Hua–Pickrell diffusions of N particles on R as solutions of the stochastic differential equations (SDEs) dXj,t=2(1+Xj,t2)dBj,t+βb−aXj,t+∑l=1,…,N;l≠jXj,tXl,t+1Xj,t−Xl,tdt,(j=1,…,N) with β≥1,a,b∈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 μtN≔1N∑j=1NδXj,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 ↦ t/β, the SDEs above degenerate in the frozen case for β = ∞ into ordinary differential equations which are related to pseudo-Jacobi polynomials. For N → ∞ and under suitable initial conditions, the empirical distributions μtN converge weakly almost surely for t > 0 to some limit which is independent from β ∈ [1, ∞]. 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 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 β → ∞ for the Hua–Pickrell ensembles which is related to these zeros.