Theory of Probability &Its Applications, Volume 67, Issue 2, Page 208-228, August 2022.
In this paper, we study the one-dimensional Hua--Pickrell diffusion. We start by revisiting the stationary case considered by E. Wong for which we supply omitted details and write down a unified expression of its semigroup density through the associated Legendre function in the cut. Next, we focus on the general (not necessarily stationary) case for which we prove an intertwining relation between Hua--Pickrell diffusions corresponding to different sets of parameters. Using the Cauchy beta integral on the one hand and Girsanov's theorem on the other hand, we discuss the connection between the stationary and general cases. Afterwards, we prove our main result providing novel integral representations of the Hua--Pickrell semigroup density, answering a question raised by Alili, Matsumoto, and Shiraishi [Séminaire de Probabilités XXXV, Lecture Notes in Math. 1755, Springer, 2001, pp. 396--415]. To this end, we appeal to the semigroup density of the Maass Laplacian and extend it to purely imaginary values of the magnetic field. In the last section, we use the Karlin--McGregor formula to derive an expression of the semigroup density of the multidimensional Hua--Pickrell particle system introduced by T. Assiotis.

中文翻译：

---

华的显式表达--Pickrell 半群

概率论及其应用，第 67 卷，第 2 期，第 208-228 页，2022 年 8 月。
在本文中，我们研究了一维Hua--Pickrell扩散。我们首先重新审视 E. Wong 考虑的平稳情况，为此我们提供省略的细节，并通过切割中相关的勒让德函数写下其半群密度的统一表达式。接下来，我们关注一般（不一定是静止的）情况，我们证明了对应于不同参数集的 Hua-Pickrell 扩散之间的相互交织关系。一方面使用柯西 beta 积分，另一方面使用 Girsanov 定理，我们讨论了平稳情况和一般情况之间的联系。之后，我们证明我们的主要结果提供了 Hua-Pickrell 半群密度的新颖积分表示，回答了 Alili、Matsumoto 和 Shiraishi 提出的问题 [Séminaire de Probabilités XXXV，数学讲义。1755，施普林格，2001，第 396--415 页]。为此，我们求助于马斯拉普拉斯算子的半群密度，并将其扩展到磁场的纯虚值。在上一节中，我们使用Karlin--McGregor 公式推导了T. Assiotis 引入的多维Hua--Pickrell 粒子系统的半群密度表达式。