Explicit expressions of the Hua-Pickrell semi-group

TL;DR

This paper derives explicit integral representations of the Hua-Pickrell semi-group density, connecting stationary and non-stationary cases, and extends results to multi-dimensional systems using advanced probabilistic and analytical techniques.

Contribution

It provides novel integral formulas for the Hua-Pickrell semi-group density, addressing a previously open question and extending to multi-dimensional particle systems.

Findings

Unified semi-group density expression via Legendre functions.

Intertwining relations between different parameter Hua-Pickrell diffusions.

Explicit semi-group density formulas for multi-dimensional Hua-Pickrell systems.

Abstract

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 semi-group 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 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 semi-group density, answering a question raised by Alili, Matsumoto and Shiraishi (S\'eminaire de Probabilit\'es, 35, 2001). To this end, we appeal to the semi-group 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 semi-group density of the multi-dimensional Hua-Pickrell particle system introduced by T. Assiotis.