The Brown measure of the free multiplicative Brownian motion

Abstract

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 *-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 the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density W_{t} on overline{Sigma }_{t}, which is strictly positive and real analytic on Sigma _{t}. This density has a simple form in polar coordinates: Wt(r,θ)=1r2wt(θ),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} W_{t}(r,\ heta )=\\frac{1}{r^{2}}w_{t}(\ heta ), \\end{aligned}$$\\end{document}where w_{t} is an analytic function determined by the geometry of the region 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 the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.