Abstract
The additive monotone (respectively boolean) unitary Brownian motion is a non-commutative stochastic process with monotone (respectively boolean) independent and stationary increments which are distributed according to the arcsine law (respectively Bernoulli law). We introduce the monotone and boolean unitary Brownian motions and 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.