Generators of some non-commutative stochastic processes

A fundamental result of Biane (Math Z 227:143–174, 1998) states that a process with freely independent increments has the Markov property, but that there are two kinds of free Lévy processes: the first kind has stationary increments, while the second kind has stationary transition operators. We show that a process of the first kind (with mean zero and finite variance) has the same transition operators as the free Brownian motion with appropriate initial conditions, while a process of the second kind has the same transition operators as a monotone Lévy process. We compute an explicit formula for the generators of these families of transition operators, in terms of singular integral operators, and prove that this formula holds on a fairly large domain. We also compute the generators for the $q$ -Brownian motion, and for the two-state free Brownian motions.