A matrix Bougerol identity and the Hua-Pickrell measures

TL;DR

This paper extends Bougerol's identity to Hermitian matrices and constructs Hua-Pickrell measures using stochastic integrals involving matrix-valued Brownian motions and exponential conjugations.

Contribution

It introduces a matrix version of Bougerol's identity and provides a novel stochastic integral representation of Hua-Pickrell measures.

Findings

Established a Hermitian matrix Bougerol identity

Constructed Hua-Pickrell measures via stochastic integrals

Linked matrix Brownian motion with complex exponential conjugation

Abstract

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.