Two-state free Brownian motions

TL;DR

This paper introduces a class of two-state free Brownian motions within a two-state free probability framework, characterizing their structure and existence conditions, especially in von Neumann algebra settings.

Contribution

It defines algebraic two-state free Brownian motions and proves their classification and existence constraints in von Neumann algebras with normal, faithful states.

Findings

Only a one-parameter family of such processes exists under specified conditions.

These processes generally exist only for finite time, except for the free Brownian motion.

The distribution with respect to the second state is arbitrary unless additional conditions are met.

Abstract

In a two-state free probability space $(A, \phi, \psi)$, we define an algebraic two-state free Brownian motion to be a process with two-state freely independent increments whose two-state free cumulant generating function is quadratic. Note that a priori, the distribution of the process with respect to the second state $\psi$ is arbitrary. We show, however, that if $A$ is a von Neumann algebra, the states $\phi, \psi$ are normal, and $\phi$ is faithful, then there is only a one-parameter family of such processes. Moreover, with the exception of the actual free Brownian motion (corresponding to $\phi = \psi$), these processes only exist for finite time.