Free multiplicative convolution with an arbitrary measure on the real line
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910144338591744 |
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| author | Arizmendi, Octavio Hasebe, Takahiro Kitagawa, Yu |
| author_facet | Arizmendi, Octavio Hasebe, Takahiro Kitagawa, Yu |
| contents | We develop analytic tools for studying the free multiplicative convolution of any measure on the real line and any measure on the nonnegative real line. More precisely, we construct the subordination functions and the $S$-transform of an arbitrary probability measure. The important multiplicativity of $S$-transform is proved with the help of subordination functions. We then apply the $S$-transform to establish convolution identities for stable laws, which had been considered in the literature only for the positive and symmetric cases. Subordination functions are also used in order to extend Belinschi--Nica's semigroup of homomorphisms, and to establish regularity properties of free multiplicative convolution, in particular, the absence of singular continuous part and analyticity of the density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14992 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free multiplicative convolution with an arbitrary measure on the real line Arizmendi, Octavio Hasebe, Takahiro Kitagawa, Yu Probability Operator Algebras 46L54 We develop analytic tools for studying the free multiplicative convolution of any measure on the real line and any measure on the nonnegative real line. More precisely, we construct the subordination functions and the $S$-transform of an arbitrary probability measure. The important multiplicativity of $S$-transform is proved with the help of subordination functions. We then apply the $S$-transform to establish convolution identities for stable laws, which had been considered in the literature only for the positive and symmetric cases. Subordination functions are also used in order to extend Belinschi--Nica's semigroup of homomorphisms, and to establish regularity properties of free multiplicative convolution, in particular, the absence of singular continuous part and analyticity of the density. |
| title | Free multiplicative convolution with an arbitrary measure on the real line |
| topic | Probability Operator Algebras 46L54 |
| url | https://arxiv.org/abs/2503.14992 |