Final published version
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Explicit Expressions of the Hua--Pickrell Semigroup
AU - Arista, J.
AU - Demni, N.
PY - 2022/8/31
Y1 - 2022/8/31
N2 - 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.
AB - 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.
U2 - 10.1137/s0040585x97t990885
DO - 10.1137/s0040585x97t990885
M3 - Journal article
VL - 67
SP - 208
EP - 228
JO - Theory of Probability & Its Applications
JF - Theory of Probability & Its Applications
SN - 0040-585X
IS - 2
ER -