Unitary Brownian motions are linearizable

Introduction The most celebrated and useful random process surely is the standard Brownian motion in R (Wiener process). It is Markovian and Gaussian. Its increments are independent and stationary. Its continuous sample paths are bizarre, but many associated probability distributions are smooth, and connected by wonderful formulas. The multidimensional standard Brownian motion can produce a lot of random processes by means of stochastic differential equations. Especially, it can produce its close relatives, well-known during half a century, --- Brownian motions in Lie groups and other topological groups. A Brownian motion in a Lie group G could be defined constructively, by means of its generator, an invariant differential operator of second order on G, or descriptively, as a continuous G-valued random process with stationary independent increments. The former (constructive) definition stipulates smoothness of th