Hua-Pickrell扩散与伪Jacobi多项式相关的微分方程
Hua-Pickrell diffusions and differential equations related with pseudo-Jacobi polynomials
AI总结：
本文研究了β-Hua-Pickrell扩散与伪Jacobi多项式相关的微分方程，探讨了其极限行为和与经验分布的关系。
摘要：
在Assiotis(2020)的基础上，我们研究了N个粒子在实数集上的一般β-Hua-Pickrell扩散作为随机微分方程(SDEs)的解：dX_{j,t}=√{2(1+X_{j,t}^2)}dB_{j,t}+β[b-aX_{j,t}+∑_{l=1,…,N; l≠j}(X_{j,t}X_{l,t}+1)/(X_{j,t}-X_{l,t})]dt，(j=1,…,N)，其中β≥1，a,b∈R。这些过程形成了一种Pearson扩散的子类，这些扩散被定义为满足代数SDEs的解，其中经验分布μ_t^N:=1/N∑_{j=1}^N δ_{X_{j,t}}的矩可以递归计算。这个Pearson类也包含其他著名的扩散过程，如Dyson布朗运动、多变量Laguerre和Jacobi过程。在时间归一化t→t/β后，上述SDEs在β=∞的冻结情况下退化为普通的微分方程，这些方程与伪Jacobi多项式相关。对于N→∞且在合适的初始条件下，经验分布μ_t^N几乎 surely 弱收敛到某个极限，该极限与β∈[1,∞]无关。对于a=-N，b=0，我们通过自由卷积明确描述了极限。此外，如果a=cN对于某个c>0，我们的SDEs的解在t→∞时收敛到平稳分布，这些分布是Hua-Pickrell（或Cauchy）测度。因此，我们获得了这些集合的经验分布已知结果与伪Jacobi多项式零点之间的联系。此外，我们还推导了对于β→∞的Hua-Pickrell集合的冻结中心极限定理，这与这些零点相关。
英文摘要：
Following Assiotis (2020), we study general $β$-Hua-Pickrell diffusions of $N$ particles on $\mathbb R$ as solutions of the stochastic differential equations (SDEs) $$dX_{j,t}=\sqrt{2(1+X_{j,t}^2)}\,dB_{j,t}+β\left[b-a X_{j,t}+\sum_{l=1,\ldots, N; \> l\neq j}\frac{X_{j,t}X_{l,t}+1}{X_{j,t}-X_{l,t}}\right]dt\,,\;\; (j=1,\ldots,N)$$ with $β\ge 1,\> a,b\in\mathbb R$. These processes form a subclass of the Pearson diffusions which are defined as solutions of algebraic SDEs where the moments of the empirical distributions $μ_t^N:=\frac{1}{N}\sum_{j=1}^N δ_{X_{j,t}}$ 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 $t\mapsto t/β$, the SDEs above degenerate in the frozen case for $β=\infty$ into ordinary differential equations which are related to pseudo-Jacobi polynomials. For $N\to\infty$ and under suitable initial conditions, the empirical distributions $μ_t^N$ converge weakly almost surely for $t>0$ to some limit which is independent from $β\in[1,\infty]$. 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 $β\to\infty$ for the Hua-Pickrell ensembles which is related to these zeros.
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