Levy area for the free Brownian motion: existence and non-existence. (Q1428452)

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.