论文标题
两个自由非偶聚物随机变量的棕色量度，其中之一是R-Diagonal
The Brown measure of a sum of two free nonselfadjoint random variables, one of which is R-diagonal
论文作者
论文摘要
假设$ x_ {1} $和$ x_ {2} $是两个$*$ - 免费（通常是无界的）随机变量，带有棕色测量$μ__{x_ {1}} $和$μ__{x_ {2}}} $。使用经典免费添加卷积的属性，我们开发了一种计算$μ__{x_ {1}+x_ {2}} $的方法时，当$ x_ {2} $是$ r $ -diagonal。此方法确定了相对于lebesgue度量的密度，其闭合包含$μ_{x_ {1}+x_ {2}} $的支持。在重要情况下，有效的计算是可能的。 Biane和Lehner是第一个在我们考虑的问题上取得重大进展的人，即使在某些情况下，$ x_ {1} $也不是$ x_ {2} $是$ r $ -diagonal。我们的示例与他们的示例重叠，但我们强调使用从属功能。当$ x_ {2} $是圆形的时，$μ_{x_ {1}+x_ {2}}} $是使用两种不同的方法，一种涉及汉密尔顿 - 雅各比方程，另一种使用标准的免费概率技术。我们的工作扩展了第二种方法。
Suppose that $X_{1}$ and $X_{2}$ are two $*$-free (generally unbounded) random variables with Brown measures $μ_{X_{1}}$ and $μ_{X_{2}}$, respectively. Using properties of classical free additive convolutions, we develop a method for calculating $μ_{X_{1}+X_{2}}$when $X_{2}$ is $R$-diagonal. This method determines a density relative to Lebesgue measure on an open set whose closure contains the support of $μ_{X_{1}+X_{2}}$. Effective calculations are possible in important cases. Biane and Lehner were the first to make significant progress on the problem we consider, even in some cases in which neither $X_{1}$ nor $X_{2}$ is $R$-diagonal. Our examples overlap with theirs, but we emphasize the use of subordination functions. When $X_{2}$ is circular, $μ_{X_{1}+X_{2}}$ was studied earlier using two different approaches, one involving Hamilton-Jacobi equations, and another using standard free probability techniques. Our work extends the second approach.
