  We study the twirling semigroups of (super)operators, namely, certain quantum
dynamical semigroups that are associated, in a natural way, with the pairs
formed by a projective representation of a locally compact group and a
convolution semigroup of probability measures on this group. The link
connecting this class of semigroups of operators with (classical) Brownian
motion is clarified. It turns out that every twirling semigroup associated with
a finite-dimensional representation is a random unitary semigroup, and,
conversely, every random unitary semigroup arises as a twirling semigroup.
Using standard tools of the theory of convolution semigroups of measures and of
convex analysis, we provide a complete characterization of the infinitesimal
generator of a twirling semigroup associated with a finite-dimensional unitary
representation of a Lie group.
