  The additive monotone (resp. boolean) unitary Brownian motion is a
non-commutative stochastic process with monotone (resp. boolean) independent
and stationary increments which are distributed according to the arcsine law
(resp. Bernoulli law) . We introduce the monotone and booleen unitary Brownian
motions and we derive a closed formula for their associated moments. This
provides a description of their spectral measures. We prove that, in the
monotone case, the multiplicative analog of the arcsine distribution is
absolutely continuous with respect to the Haar measure on the unit circle,
whereas in the boolean case the multiplicative analog of the Bernoulli
distribution is discrete. Finally, we use quantum stochastic calculus to
provide a realization of these processes as the stochastic exponential of the
correspending additive Brownian motions.
