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