Journal of Operator Theory

Volume 55, Issue 2, Spring 2006 pp. 373-392.

A free Girsanov property for free Brownian motions

Authors: Jocelyne Bion-Nadal
Author institution: Centre de Math\'ematiques Appliqu\'ees (CMAP), CNRS UMR 7641, Ecole Polytechnique, F-91128 Palaiseau Cedex, France

Summary: A ``free Girsanov'' property is proved for free Brownian motions. It is reminiscent of the classical Girsanov theorem in probability theory. In the free probability context, we prove that if $(\sigma_s)_{s\in \R^+}$ is a free Brownian motion in $(M,\tau)$, if $x$ is a process free from the $\sigma_s$, if $\widetil de{\sigma_s}=\sigma_s+\int \limits_{0}^{s}\ x(u)\mathrm du$, then there is a trace $\widetilde{\tau}$ such that$(\widetilde{\sigma_s})_{s\in \R^+}$ is a free Brownian motion for $\widetilde \tau$ and the two traces are ``asymptotically equivalent''. This means that $\tau$ respectively $\widetilde\tau$ are asymptotic limits of states $ \Psi_n$ respectively $\widetilde\Psi_n$ and that for each $n$ $\widetilde\Psi_n$ is obtained from $\Psi_n$ by a change of probability given by an exponential density.

Keywords: Free probability theory, free products of $C^*$ algebras, free Brownian motion, Girsanov theorem.

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