Brown Measures of Free Circular and Multiplicative Brownian Motions with Self-Adjoint and Unitary Initial Conditions

Abstract

Given a selfadjoint random variable x 0 and a unitary random variable u, different from Haar unitary, free from the free circular Brownian motion c t and the free multiplicative Brownian motion b t , we use the Hamilton-Jacobi method to compute the Brown measures of free circular Brownian motion x 0 + c t and the free multiplicative Brownian motion ub t with probabilistic initial point, extending the recent work [13] by Driver-Hall-Kemp. We find that the supports of the Brown measures are related to the subordination functions of the free additive and multiplicative convolutions. The density of the Brown measure of x 0 + c t is constant along the vertical direction in the additive case, and the density of the Brown measure of ub t has the form, in polar coordinates, of (1/r 2 )w t (θ). The densities of the Brown measures are closely related to the laws of x 0 + s t and uu t , where s t is the free semicircular Brownian motion and u t is the free unitary Brownian motion, both freely independent of x 0 and u.

Brown measure support and the free multiplicative Brownian motion

“…The paper [13] showed that the pushforward of the Brown measure µ s,s of the "standard" free multiplicative Brownian motion b s by a certain map Φ s is equal to the law of the free unitary Brownian motion u s . This result was extended in [23] to relate the Brown measure of ub s to the law of uu s , where u is an arbitrary unitary element freely independent of b s and of u s . The map Φ s in [13,23] is simply the limiting case of the maps Γ τ1,τ2 s with τ 1 = s and τ 2 tending to zero.…”

“…This result was extended in [23] to relate the Brown measure of ub s to the law of uu s , where u is an arbitrary unitary element freely independent of b s and of u s . The map Φ s in [13,23] is simply the limiting case of the maps Γ τ1,τ2 s with τ 1 = s and τ 2 tending to zero. We see, then, that the map Φ s is simply a limiting case of the whole family of maps Γ τ1,τ2 s relating all the Brown measures µ s,τ with s fixed and τ varying.…”

“…It is also related to the law ν t of the free unitary Brownian motion. The results of [13] were then extended by Ho and Zhong [23] to compute the Brown measure of ub t , where u is an arbitrary unitary element freely independent of b t .…”

“…where the answer will depend on the choice of the unitary initial condition u 0 in (1.4). The case τ = s corresponds to the Brown measure computed in [13,23].…”

“…In the notation of the present paper, [13] corresponds to taking τ = s and u 0 = 1. The method in [13] was then extended by the second author and Zhong [23] to analyze the standard free multiplicative Brownian motion with arbitrary unitary initial condition (τ = s and u 0 is a arbitrary). The method has also been used in [10] and [20] and discussed in the physics literature in [16].…”