Spectral distribution of the free unitary Brownian motion: another approach

Nizar Demni (1) , Taoufik Hmidi (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We revisit the description provided by Ph. Biane of the spectral measure of the free unitary Brownian motion. We actually construct for any $t \in (0,4)$ a Jordan curve $\gamma_t$ around the origin, not intersecting the semi-axis $[1,\infty[$ and whose image under some meromorphic function $h_t$ lies in the circle. Our construction is naturally suggested by a residue-type integral representation of the moments and $h_t$ 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.

Domaines: Algèbres d'opérateurs [math.OA]; Analyse classique [math.CA]

Nizar Demni, Taoufik Hmidi. Spectral distribution of the free unitary Brownian motion: another approach. Catherine Donati-Martin, Antoine Lejay, Alain Rouault. Séminaire de Probabilités XLIV, Springer, pp.191-206, 2012, Lecture Notes in Mathematics 2046, 978-3-642-27460-2. ⟨10.1007/978-3-642-27461-9_9⟩. ⟨hal-00578761⟩