The Large-NLimits of Brownian Motions on 𝔾𝕃N

Abstract

We introduce a two-parameter family of diffusion processes (B N 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 Itˆ o processes (br;s(t))t 0 in a faithful, tracial W -probability space, and we prove that the process (B N (t))t 0 converges to (br;s(t))t 0 in noncommutative distribution as N ! 1 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 whenr = s = 1 ; we thus resolve the open problem of convergence of the Brownian motion on GLN posed by Philippe Biane in 1997.