The Brown measure of the free multiplicative Brownian motion

The free multiplicative Brownian motion $b_t$ is the large-$N$ limit of the Brownian motion on $\mathsf{G}\mathsf{L}(N;\mathbb{C}),$ in the sense of $\ast$-distributions. The natural candidate for the large-$N$ limit of the empirical distribution of eigenvalues is thus the Brown measure of $b_t$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region ${\mathrm{\Sigma }}_t$ that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $W_t$ on ${\bar{\mathrm{\Sigma }}}_t,$ which is strictly positive and real analytic on ${\mathrm{\Sigma }}_t$. This density has a simple form in polar coordinates: $$W_t(r,\theta )=\frac{1}{r^2}{w}_t(\theta ),$$ where $w_t$ is an analytic function determined by the geometry of the region ${\mathrm{\Sigma }}_t$. We show also that the spectral measure of free unitary Brownian motion $u_t$ is a “shadow” of the Brown measure of $b_t$, precisely mirroring the relationship between Wigner’s semicircle law and Ginibre’s circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.