Spectral distribution of the free unitary Brownian motion: another approach

الأصل · EN

We revisit the description provided by Ph. Biane of the spectral measure of the free unitary Brownian motion. We actually construct for any t ∈ (0,4) a Jordan curve γₜ around the origin, not intersecting the semi-axis [1,∞[and whose image under some meromorphic function hₜ lies in the circle. Our construction is naturally suggested by a residue-type integral representation of the moments and hₜ is up to a Möbius transformation the main ingredient used in the original proof. Once we did, the spectral measure is described as the push-forward of a complex measure under a local diffeomorphism yielding its absolute-continuity and its support. Our approach has the merit to be an easy yet technical exercise from real analysis.