Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions

Let Z_N be a Ginibre ensemble and let A_N be a Hermitian random matrix independent of Z_N such that A_N converges in distribution to a self-adjoint random variable x_0 in a W^* -probability space (A,τ) . For each t>0 , the random matrix A_N+√t\,Z_N converges in -distribution to x_0+c_t , where c_t is a circular variable of variance t , freely independent of x_0 . We use the Hamilton–Jacobi method to compute the Brown measure ρ_t of x_0+c_t . The Brown measure has a density that is constant along the vertical direction inside the support. 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 a semicircular variable of variance t , freely independent of x_0 . Furthermore, the push-forward of ρ_t by a natural map is the law of x_0+s_t . Let G_N(t) be the Brownian motion on the general linear group and let U_N be a unitary random matrix independent of G_N(t) such that U_N converges in distribution to a unitary random variable u in (A,τ) . The random matrix U_NG_N(t) converges in -distribution to ub_t where b_t is the free multiplicative Brownian motion, freely independent of u . We compute the Brown measure μ_t of ub_t , extending the recent work by Driver–Hall–Kemp, which corresponds to the case u=I . The measure has a density of the special form 1/r^2w_t(θ) in polar coordinates in its support. The support of μ_t is related to the subordination function of the free multiplicative convolution of uu_t where u_t is the free unitary Brownian motion, freely independent of u . The push-forward of μ_t by a natural map is the law of uu_t . In the special case that u is Haar unitary, the Brown measure μ_t follows the annulus law . The support of the Brown measure of ub_t is an annulus with inner radius e-t/2 and outer radius et/2