## Abstract

In this paper, we introduce and study a unitary matrix-valued process which is closely related to the Hermitian matrix-Jacobi process. It is precisely defined as the product of a deterministic self-adjoint symmetry and a randomly-rotated one by a unitary Brownian motion. Using stochastic calculus and the action of the symmetric group on tensor powers, we derive an ordinary differential equation for the moments of its fixed-time marginals. Next, we derive an expression of these moments which involves a unitary bridge between our unitary process and another independent unitary Brownian motion. This bridge motivates and allows to write a second direct proof of the obtained moment expression.

Original language | English (US) |
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Article number | 2150002 |

Journal | Infinite Dimensional Analysis, Quantum Probability and Related Topics |

Volume | 24 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2021 |

## Keywords

- Brownian motion in the unitary group
- free unitary Brownian motion
- Hermitian matrix-Jacobi process
- Schur-Weyl duality
- self-adjoint symmetries

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Mathematical Physics
- Applied Mathematics