We prove a Hermitian matrix version of Bougerol’s identity. Moreover, we construct the Hua-Pickrell measures on Hermitian matrices, as stochastic integrals with respect to a drifting Hermitian Brownian motion and with an integrand involving a conjugation by an independent, matrix analogue of the exponential of a complex Brownian motion with drift.
"A matrix Bougerol identity and the Hua-Pickrell measures." Electron. Commun. Probab. 23 1 - 11, 2018. https://doi.org/10.1214/18-ECP107