Strong Convergence of Multiplicative Brownian Motions on the General Linear Group
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arXiv
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| Format: | Preprint |
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2025
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| author | Banna, Marwa Capitaine, Mireille Cébron, Guillaume |
| author_facet | Banna, Marwa Capitaine, Mireille Cébron, Guillaume |
| contents | We consider the family of multiplicative Brownian motions $G_{λ,τ}$ on the general linear group introduced by Driver-Hall-Kemp. They are parametrized by the real variance $λ\in \mathbb{R}$ and the complex covariance $τ\in \mathbb{C}$ of the underlying elliptic Brownian motion. We show the almost sure strong convergence of the finite-dimensional marginals of $G_{λ,τ}$ to the corresponding free multiplicative Brownian motion introduced by Hall-Ho: as the dimension tends to infinity, not only does the noncommutative distribution converge almost surely, but the operator norm does as well. This result generalizes the work of Collins-Dahlqvist-Kemp for the special case $(λ,τ)=(1,0)$ which corresponds to the Brownian motion on the unitary group. Actually, this strong convergence remains valid when the family of multiplicative Brownian motions $G_{λ,τ}$ is considered alongside a family of strongly converging deterministic matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13922 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong Convergence of Multiplicative Brownian Motions on the General Linear Group Banna, Marwa Capitaine, Mireille Cébron, Guillaume Probability Mathematical Physics We consider the family of multiplicative Brownian motions $G_{λ,τ}$ on the general linear group introduced by Driver-Hall-Kemp. They are parametrized by the real variance $λ\in \mathbb{R}$ and the complex covariance $τ\in \mathbb{C}$ of the underlying elliptic Brownian motion. We show the almost sure strong convergence of the finite-dimensional marginals of $G_{λ,τ}$ to the corresponding free multiplicative Brownian motion introduced by Hall-Ho: as the dimension tends to infinity, not only does the noncommutative distribution converge almost surely, but the operator norm does as well. This result generalizes the work of Collins-Dahlqvist-Kemp for the special case $(λ,τ)=(1,0)$ which corresponds to the Brownian motion on the unitary group. Actually, this strong convergence remains valid when the family of multiplicative Brownian motions $G_{λ,τ}$ is considered alongside a family of strongly converging deterministic matrices. |
| title | Strong Convergence of Multiplicative Brownian Motions on the General Linear Group |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2507.13922 |