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.
Highlights
We begin this introduction, by recalling Bougerol’s celebrated identity, first established in [6] in his study of convolution powers of probabilities on certain solvable groups
We prove a Hermitian matrix version of Bougerol’s identity
If we denote by βt(−ν); t ≥ 0 and γt(−μ); t ≥ 0 two independent standard Brownian motions with drifts −ν and −μ respectively, the law of the functional, for ν > 0
Summary
By recalling Bougerol’s celebrated identity, first established in [6] in his study of convolution powers of probabilities on certain solvable groups. If we denote by βt(−ν); t ≥ 0 and γt(−μ); t ≥ 0 two independent standard Brownian motions with drifts −ν and −μ respectively, the law of the functional, for ν > 0, The purpose of this note is to obtain the Hermitian matrix analogues of these results.
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