The free multiplicative Brownian motion
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Abstract is the large$$b_{t}$$ ${b}_{t}$N limit of the Brownian motion on in the sense of$$\mathsf {GL}(N;\mathbb {C}),$$ $\mathrm{GL}(N\u037eC),$ distributions. The natural candidate for the large$$*$$ $\ast $N limit of the empirical distribution of eigenvalues is thus the Brown measure of . In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$b_{t}$$ ${b}_{t}$ that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$\Sigma _{t}$$ ${\Sigma}_{t}$ on$$W_{t}$$ ${W}_{t}$ which is strictly positive and real analytic on$$\overline{\Sigma }_{t},$$ ${\overline{\Sigma}}_{t},$ . This density has a simple form in polar coordinates:$$\Sigma _{t}$$ ${\Sigma}_{t}$ where$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ $\begin{array}{c}{W}_{t}(r,\theta )=\frac{1}{{r}^{2}}{w}_{t}\left(\theta \right),\end{array}$ is an analytic function determined by the geometry of the region$$w_{t}$$ ${w}_{t}$ . We show also that the spectral measure of free unitary Brownian motion$$\Sigma _{t}$$ ${\Sigma}_{t}$ is a “shadow” of the Brown measure of$$u_{t}$$ ${u}_{t}$ , 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.$$b_{t}$$ ${b}_{t}$