Spectral distribution of the free unitary Brownian motion: another approach
نویسندگان
چکیده
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 absolutecontinuity and its support. Our approach has the merit to be an easy yet technical exercise from real analysis. 1. Reminder and Motivation In his pioneering paper [1], Ph. Biane defined and studied the so-called free unitary or multiplicative Brownian motion. It is a unitary operator-valued Lévy process with respect to the free multiplicative convolution of probability measures on the unit circle T (or equivalently the multiplication of unitary operators that are free in some non commutative probability space). Besides, the spectral distribution μt at any time t ≥ 0 is characterized by its moments
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