The Large-N Limits of Brownian Motions on GLN

We introduce a two-parameter family of diffusion processes (B r,s(t))t≥0, r, s > 0, on the general linear group GLN 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 Itô processes (br,s(t))t≥0 in a faithful, tracial W ∗-probability space, and we prove that the full process (B r,s(t))t≥0 converges to (br,s(t))t≥0 in noncommutative distribution as N → ∞ for each r, s > 0. The processes (br,s(t))t≥0 interpolate between the free unitary Brownian motion when (r, s) = (1, 0), and the free multiplicative Brownian motion when r = s = 12 ; we thus resolve the open problem of convergence of the Brownian motion on GLN posed by Biane in [2].