TY - GEN

T1 - Spectral distribution of the free unitary Brownian motion

T2 - Another approach

AU - Demni, Nizar

AU - Hmidi, Taoufik

PY - 2012

Y1 - 2012

N2 - We revisit the description provided by Ph. Biane of the spectral measure of the free unitary Brownian motion. We actually construct for any τ ∈ (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 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.

AB - We revisit the description provided by Ph. Biane of the spectral measure of the free unitary Brownian motion. We actually construct for any τ ∈ (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 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.

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U2 - 10.1007/978-3-642-27461-9_9

DO - 10.1007/978-3-642-27461-9_9

M3 - Conference contribution

AN - SCOPUS:84861837362

SN - 9783642274602

T3 - Lecture Notes in Mathematics

SP - 191

EP - 206

BT - Seminaire de Probabilites XLIV

PB - Springer Verlag

ER -