Two-state free Brownian motions

Abstract

In a two-state free probability space $(A,\varphi ,\psi )$, 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 $\varphi ,\psi$ are normal, and $\varphi$ 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 $\varphi =\psi$), these processes only exist for finite time.

21 pages

References in corpus (2)

• Limit theorems for additive c-free convolution
• Bochner-Pearson-type characterization of the free Meixner class