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arXiv 1302.4844math.PRmath.AP
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自由酉布朗运动下投影的一般位置及其像

General position of a projection and its image under a free unitary Brownian motion

Nizar Demni, Taoufik Hmidi

备注：
The letter `a' was used to denote two different objects: an operator and a real number. The operator is now denoted `S' referring to `symmetry'
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摘要：

给定$W^{\star}$ -非交换概率空间中的正交投影$P$和自由酉布朗运动$Y = (Y_t)_{t \geq 0}$，使得$P$和$P$在Voiculescu意义下是$\star$ -自由的，本文的主要结果是：$P$和$Y_tPY_t^{\star}$在任意时刻$t$处于一般位置.为此，我们研究了酉算子$SY_tSY_t^{\star}$的动力学，其中$S = 2P-1$。更确切地说，我们推导出一个偏微分方程的赫格洛茨变换的光谱分布，说$μ_t$。然后，我们提供了一个流动的间隔$[-1,1]$在这样一种方式组成的$μ_t$的Herglotz变换由初始（$t=0$）和固定（$t = \infty)$分布）的Herglotz变换。这一事实允许计算$μ_t$分配给$z=1$的权重，从而得到主要结果。作为副产品，自由Jacobi过程的谱分布分配给$x=1$的权重在归一化之后。在本文的最后一部分，我们利用非交叉分拆的组合数学方法，分析了$SY_tSY_t^{\star}$的$n$阶矩展开式中对应于指数衰减$e^{-nt}$的项。

英文摘要：

Given an orthogonal projection $P$ and a free unitary Brownian motion $Y = (Y_t)_{t \geq 0}$ in a $W^{\star}$-non commutative probability space such that $Y$ and $P$ are $\star$-free in Voiculescu's sense, the main result of this paper states that $P$ and $Y_tPY_t^{\star}$ are in general position at any time $t$. To this end, we study the dynamics of the unitary operator $SY_tSY_t^{\star}$ where $S = 2P-1$. More precisely, we derive a partial differential equation for the Herglotz transform of its spectral distribution, say $μ_t$. Then, we provide a flow on the interval $[-1,1]$ in such a way that the Herglotz transform of $μ_t$ composed with this flow is governed by both the Herglotz transforms of the initial ($t=0$) and the stationary ($t = \infty)$ distributions. This fact allows to compute the weight that $μ_t$ assigns to $z=1$ leading to the main result. As a by-product, the weight that the spectral distribution of the free Jacobi process assigns to $x=1$ follows after a normalization. In the last part of the paper, we use combinatorics of non crossing partitions in order to analyze the term corresponding to the exponential decay $e^{-nt}$ in the expansion of the $n$-th moment of $SY_tSY_t^{\star}$.

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