Generators of some non-commutative stochastic processes

TL;DR

This paper characterizes the generators of certain non-commutative stochastic processes, including free, monotone, and q-Brownian motions, revealing their transition operators and explicit generator formulas.

Contribution

It provides explicit formulas for the generators of free, monotone, and q-Brownian motions, extending understanding of their transition operators and Markov properties.

Findings

Derived explicit generator formulas using singular integral operators.

Identified the transition operators for free and monotone Levy processes.

Extended formulas to q-Brownian and two-state free Brownian motions.

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 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.