Explicit expressions of the Hua–Pickrell semigroup

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.