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Levy area for the free Brownian motion: existence and non-existence. (English) Zbl 1062.46055
The author shows that the Lévy area of the piecewise linear path which coincides with \((X_s)_{0\leq s\leq 1}\) (free Brownian motions) at the points of a subdivision \(D\) tends to the Lévy area of the free Brownian motion when the mesh size of \(D\) tends to \(0\). To this end, the author proves a Burkholder-Davis-Gundy type inequality. Then he shows that there exists no multiplicative functional above the free Brownian motion, that is, in other words, there does not exist a Lévy area in the projective tensor product.

MSC:
46L54 Free probability and free operator algebras
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References:
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