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 $\mathsf{GL}(N;\mathbb{C})$ (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 $\Sigma_{t}$ that appeared in Biane's earlier work. In the present paper, we compute the Brown measure completely. It has a continuous density which is strictly positive and real analytic on $\Sigma_{t}$. Moreover, its density $W_{t}$ 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 $\Sigma_{t}$. The density is also closely linked to the distribution of free unitary Brownian motion. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.