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 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.

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