Michael Anshelevich
Generators of some Non-Commutative Stochastic Processes
Preprint series: ESI preprints

MSC:

46L50 Noncommutative measure, integration and probability

60J25 Markov processes with continuous parameter

47D06 One-parameter semigroups and linear evolution equations

Abstract: A fundamental result of Biane (1998) states that a process with
freely independent increments has the Markov property, but that
there are two kinds of free L{\'e}vy 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 L{\'e}vy 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.