The Brown measure of a sum of two free nonselfadjoint random variables, one of which is R-diagonal
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| Format: | Preprint |
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2022
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| _version_ | 1866909473795211264 |
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| author | Bercovici, Hari Zhong, Ping |
| author_facet | Bercovici, Hari Zhong, Ping |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_12379 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Brown measure of a sum of two free nonselfadjoint random variables, one of which is R-diagonal Bercovici, Hari Zhong, Ping Probability Mathematical Physics Functional Analysis Operator Algebras 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. |
| title | The Brown measure of a sum of two free nonselfadjoint random variables, one of which is R-diagonal |
| topic | Probability Mathematical Physics Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2209.12379 |