The Brown measure of the free multiplicative Brownian motion
The Brown measure of the free multiplicative Brownian motion
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DOI：
10.1007/s00440-022-01142-z
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发表时间：
2019-03
影响因子：
2
通讯作者：
B. Driver;B. Hall;Todd Kemp
中科院分区：
文献类型：
--
作者：
B. Driver;B. Hall;Todd Kemp
The free multiplicative Brownian motionis the large-Nlimit of the Brownian motion onin the sense of-distributions. The natural candidate for the large-Nlimit 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 regionthat appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous densityonwhich is strictly positive and real analytic on. This density has a simple form in polar coordinates: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$\end{document}whereis an analytic function determined by the geometry of the region. We show also that the spectral measure of free unitary Brownian motionis a “shadow” of the Brown measure of, 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.
The free multiplicative Brownian motionis the large-Nlimit of the Brownian motion onin the sense of-distributions. The natural candidate for the large-Nlimit 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 regionthat appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous densityonwhich is strictly positive and real analytic on. This density has a simple form in polar coordinates: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$\end{document}whereis an analytic function determined by the geometry of the region. We show also that the spectral measure of free unitary Brownian motionis a “shadow” of the Brown measure of, 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.