Two-state free Brownian motions

Details Two-state free Brownian motions Two-state free Brownian motions

2010-06-06

arxiv.org/abs/1006.1132v1

J. Funct. Anal. 260 (2011), 541-565

Mathematics

Operator Algebras

21 pages

Scientific paper

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

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