%ADriver, Bruce%AHall, Brian%AKemp, Todd%BJournal Name: Probability Theory and Related Fields; Journal Volume: 184; Journal Issue: 1-2; Related Information: CHORUS Timestamp: 2022-10-18 08:06:34 %D2022%ISpringer Science + Business Media %JJournal Name: Probability Theory and Related Fields; Journal Volume: 184; Journal Issue: 1-2; Related Information: CHORUS Timestamp: 2022-10-18 08:06:34 %K %MOSTI ID: 10372851 %PMedium: X; Size: p. 209-273 %TThe Brown measure of the free multiplicative Brownian motion %XAbstract

The free multiplicative Brownian motion$$b_{t}$$${b}_{t}$is the large-Nlimit of the Brownian motion on$$\mathsf {GL}(N;\mathbb {C}),$$$\mathrm{GL}\left(N;C\right),$in the sense of$$*$$$\ast$-distributions. The natural candidate for the large-Nlimit of the empirical distribution of eigenvalues is thus the Brown measure of$$b_{t}$$${b}_{t}$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$\Sigma _{t}$$${\Sigma }_{t}$that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$W_{t}$$${W}_{t}$on$$\overline{\Sigma }_{t},$$${\overline{\Sigma }}_{t},$which is strictly positive and real analytic on$$\Sigma _{t}$$${\Sigma }_{t}$. This density has a simple form in polar coordinates:\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$\begin{array}{c}{W}_{t}\left(r,\theta \right)=\frac{1}{{r}^{2}}{w}_{t}\left(\theta \right),\end{array}$where$$w_{t}$$${w}_{t}$is an analytic function determined by the geometry of the region$$\Sigma _{t}$$${\Sigma }_{t}$. We show also that the spectral measure of free unitary Brownian motion$$u_{t}$$${u}_{t}$is a “shadow” of the Brown measure of$$b_{t}$$${b}_{t}$, precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

%0Journal Article