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

Let $x_0$ be a self-adjoint random variable and $c_t$ be a free circular Brownian motion, freely independent from $x_0$. We use the Hamilton--Jacobi method to compute the Brown measure $rho_t$ of $x_0+c_t$. The Brown measure is absolutely continuous with a density that is emph{constant along the vertical direction} in the support of $rho_t$. The support of the Brown measure of $x_0+c_t$ is related to the subordination function of the free additive convolution of $x_0+s_t$, where $s_t$ is the free semicircular Brownian motion, freely independent from $x_0$. Furthermore, the push-forward of $rho_t$ by a natural map is the law of $x_0+s_t$. Let $u$ be a unitary random variable and $b_t$ is the free multiplicative Brownian motion freely independent from $u$, we compute the Brown measure $mu_t$ of the free multiplicative Brownian motion $ub_t$, extending the recent work by Driver--Hall--Kemp. The measure is absolutely continuous with a density of the special form [frac{1}{r^2}w_t(theta)] in polar coordinates in its support. The support of $mu_t$ is related to the subordination function of the free multiplicative convolution of $uu_t$ where $u_t$ is the free unitary Brownian motion free independent from $u$. The push-forward of $mu_t$ by a natural map is the law of $uu_t$. In the special case that $u$ is Haar unitary, the Brown measure $mu_t$ follows the emph{annulus law}. The support of the Brown measure of $ub_t$ is an annulus with inner radius $e^{-t/2}$ and outer radius $e^{t/2}$. The density in polar coordinates is given by [frac{1}{2pi t}frac{1}{r^2}] in its support.