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Brownian motion on the complex general group and the non-compact Jacobi process

  • Martin Auer
  • Nizar Demni
  • Nicolas Gilliers
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Abstract

In this paper, we introduce and study a new non Hermitian matrix-valued process built from truncations of Brownian motion on complex positive-definite matrices and of its inverse. We prove that its eigenvalue process is valued in (1,+∞) and that it forms a non-compact Jacobi particle system. Surprisingly, the same particle system occurs in \cite{BDW} in connection with Brownian motion on a non-compact complex Grassmann manifold. In this respect, we supply evidences showing that this occurrence holds true only at the spectral level and not on the corresponding matrix diffusions. We then study the large-size limit of our matrix model and prove results parallel to those obtained by the second author for the free Jacobi process.

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Cite

BibTeXRIS
BibTeX
@misc{auer2026brownian,
  title = {{Brownian motion on the complex general group and the non-compact Jacobi process}},
  author = {Martin Auer and Nizar Demni and Nicolas Gilliers},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16708},
  eprint = {2609.16708},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16708},
}
RIS
TY  - UNPB
TI  - Brownian motion on the complex general group and the non-compact Jacobi process
AU  - Martin Auer
AU  - Nizar Demni
AU  - Nicolas Gilliers
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.16708
UR  - https://arxiv.org/abs/2609.16708
AB  - In this paper, we introduce and study a new non Hermitian matrix-valued process built from truncations of Brownian motion on complex positive-definite matrices and of its inverse. We prove that its eigenvalue process is valued in (1,+∞) and that it forms a non-compact Jacobi particle system. Surprisingly, the same particle system occurs in \cite{BDW} in connection with Brownian motion on a non-compact complex Grassmann manifold. In this respect, we supply evidences showing that this occurrence holds true only at the spectral level and not on the corresponding matrix diffusions. We then study the large-size limit of our matrix model and prove results parallel to those obtained by the second author for the free Jacobi process.
ER  -