Two-state free Brownian motions Academic Article uri icon


  • 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.

author list (cited authors)

  • Anshelevich, M.

citation count

  • 7

publication date

  • January 2011