标题： 乘法布朗运动在一般线性群上的强收敛性 显示英文标题

标题： Strong Convergence of Multiplicative Brownian Motions on the General Linear Group

摘要： 我们考虑由Driver-Hall-Kemp引入的广义线性群上的乘法布朗运动族$G_{\lambda,\tau}$。 它们由底层椭圆布朗运动的实方差$\lambda\in \mathbb{R}$和复协方差$\tau \in \mathbb{C}$参数化。 我们证明了$G_{\lambda,\tau}$的有限维边缘几乎必然强收敛于Hall-Ho引入的相应自由乘法布朗运动：当维度趋于无穷时，不仅非交换分布几乎必然收敛，算子范数也如此。 这个结果推广了Collins-Dahlqvist-Kemp针对特殊情形$(\lambda,\tau)=(1,0)$的工作，该情形对应于酉群上的布朗运动。 实际上，当考虑乘法布朗运动族$G_{\lambda,\tau}$与一组几乎必然收敛的确定性矩阵一起时，这种强收敛仍然成立。 显示英文摘要

摘要： 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.