Strong Convergence of Multiplicative Brownian Motions on the General Linear Group

Abstract: We consider the family of multiplicative Brownian motions $G_{\lambda,\tau}$ on the general linear group introduced by Driver-Hall-Kemp. They are parametrized by the real variance $\lambda\in \mathbb{R}$ and the complex covariance $\tau \in \mathbb{C}$ of the underlying elliptic Brownian motion. We show the almost sure strong convergence of the finite-dimensional marginals of $G_{\lambda,\tau}$ 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 $(\lambda,\tau)=(1,0)$ which corresponds to the Brownian motion on the unitary group. Actually, this strong convergence remains valid when the family of multiplicative Brownian motions $G_{\lambda,\tau}$ is considered alongside a family of strongly converging deterministic matrices.