Two-state free Brownian motions

In a two-state free probability space (A,ϕ,ψ), 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 Rϕ,ψ(z) is quadratic. Note that a priori, the distribution of the process with respect to the second state ψ is arbitrary. We show, however, that if A is a von Neumann algebra, the states ϕ, ψ are normal, and ϕ 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 ϕ = ψ), these processes only exist for finite time. © 2010 Elsevier Inc. All rights reserved.