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We introduce a two-parameter family of diffusion processes$(B_{r,s}^N(t))_{t\ge 0}$, $r,s>0$, on the general linear group $\mathbb{GL}_N$that are Brownian motions with respect to certain natural metrics on the group.At the same time, we introduce a two-parameter family of free It\^o processes$(b_{r,s}(t))_{t\ge 0}$ in a faithful, tracial $W^\ast$-probability space, andwe prove that the full process $(B^N_{r,s}(t))_{t\ge 0}$ converges to$(b_{r,s}(t))_{t\ge 0}$ in noncommutative distribution as $N\to\infty$ for each$r,s>0$. The processes $(b_{r,s}(t))_{t\ge 0}$ interpolate between the freeunitary Brownian motion when $(r,s)=(1,0)$, and the free multiplicativeBrownian motion when $r=s=\frac12$; we thus resolve the open problem ofconvergence of the Brownian motion on $\mathbb{GL}_N$ posed by Biane in 1997.

The Large-$N$ Limits of Brownian Motions on $\mathbb{GL}_N$