Brown Measure and the Free Multiplicative Brownian Motion

The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of Brownian motion $B_t^N$ on the general linear group $\mathrm{GL}(N,\mathbb{C})$. We prove that the Brown measure for $b_{t}$---which is an analog of the empirical eigenvalue distribution for matrices---is supported on the closure of certain domain $\Sigma_{t}$ in the plane, introduced by Biane in the context of the large-$N$ limit of the Segal--Bargmann transform associated to $\mathrm{GL}(N,\mathbb{C})$. We also consider a two-parameter version, $b_{s,t}$: the large-$N$ limit of a related family of diffusion processes on $\mathrm{GL}(N,\mathbb{C})$ introduced by the second author. We show that the Brown measure of $b_{s,t}$ is supported on the closure of a certain complex domain $\Sigma_{s,t}$ generalizing $\Sigma_t$, introduced by Ho.