Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions
Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions
复制标题
具有自伴和酉初始条件的自由圆和乘法布朗运动的布朗测度
DOI：
10.4171/jems/1233
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发表时间：
2019
影响因子：
2.6
通讯作者：
Ping Zhong
中科院分区：
文献类型：
--
作者：
Ching;Ping Zhong
Let $x_0$ be a self-adjoint random variable and $c_t$ be a free circular Brownian motion, freely independent from $x_0$. We use the Hamilton--Jacobi method to compute the Brown measure $\rho_t$ of $x_0+c_t$. The Brown measure is absolutely continuous with a density that is \emph{constant along the vertical direction} in the support of $\rho_t$. The support of the Brown measure of $x_0+c_t$ is related to the subordination function of the free additive convolution of $x_0+s_t$, where $s_t$ is the free semicircular Brownian motion, freely independent from $x_0$. Furthermore, the push-forward of $\rho_t$ by a natural map is the law of $x_0+s_t$.
Let $u$ be a unitary random variable and $b_t$ is the free multiplicative Brownian motion freely independent from $u$, we compute the Brown measure $\mu_t$ of the free multiplicative Brownian motion $ub_t$, extending the recent work by Driver--Hall--Kemp. The measure is absolutely continuous with a density of the special form
\[\frac{1}{r^2}w_t(\theta)\]
in polar coordinates in its support. The support of $\mu_t$ is related to the subordination function of the free multiplicative convolution of $uu_t$ where $u_t$ is the free unitary Brownian motion free independent from $u$. The push-forward of $\mu_t$ by a natural map is the law of $uu_t$.
In the special case that $u$ is Haar unitary, the Brown measure $\mu_t$ follows the \emph{annulus law}. The support of the Brown measure of $ub_t$ is an annulus with inner radius $e^{-t/2}$ and outer radius $e^{t/2}$. The density in polar coordinates is given by
\[\frac{1}{2\pi t}\frac{1}{r^2}\]
in its support.
Let $x_0$ be a self-adjoint random variable and $c_t$ be a free circular Brownian motion, freely independent from $x_0$. We use the Hamilton--Jacobi method to compute the Brown measure $\rho_t$ of $x_0+c_t$. The Brown measure is absolutely continuous with a density that is \emph{constant along the vertical direction} in the support of $\rho_t$. The support of the Brown measure of $x_0+c_t$ is related to the subordination function of the free additive convolution of $x_0+s_t$, where $s_t$ is the free semicircular Brownian motion, freely independent from $x_0$. Furthermore, the push-forward of $\rho_t$ by a natural map is the law of $x_0+s_t$.
Let $u$ be a unitary random variable and $b_t$ is the free multiplicative Brownian motion freely independent from $u$, we compute the Brown measure $\mu_t$ of the free multiplicative Brownian motion $ub_t$, extending the recent work by Driver--Hall--Kemp. The measure is absolutely continuous with a density of the special form
\[\frac{1}{r^2}w_t(\theta)\]
in polar coordinates in its support. The support of $\mu_t$ is related to the subordination function of the free multiplicative convolution of $uu_t$ where $u_t$ is the free unitary Brownian motion free independent from $u$. The push-forward of $\mu_t$ by a natural map is the law of $uu_t$.
In the special case that $u$ is Haar unitary, the Brown measure $\mu_t$ follows the \emph{annulus law}. The support of the Brown measure of $ub_t$ is an annulus with inner radius $e^{-t/2}$ and outer radius $e^{t/2}$. The density in polar coordinates is given by
\[\frac{1}{2\pi t}\frac{1}{r^2}\]
in its support.
影响因子：
1.7
作者：
B. Hall;Todd Kemp
通讯作者：
B. Hall;Todd Kemp