Generators of some non-commutative stochastic processes

Details Generators of some non-commutative stochastic processes Generators of some non-commutative stochastic processes

2011-04-07

arxiv.org/abs/1104.1381v2

Mathematics

Operator Algebras

v2: Theorem 13 added, abstract modified accordingly

Scientific paper

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 Levy 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 Levy 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.

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