Integration with respect to the non-commutative fractional Brownian motion

We study the issue of integration with respect to the non-commutative fractional Brownian motion, that is the analog of the standard fractional Brownian in a non-commutative probability setting.When the Hurst index $H$ of the process is stricly larger than $1/2$, integration can be handled through the so-called Young procedure. The situation where $H=1/2$ corresponds to the specific free case, for which an It{\^o}-type approach is known to be possible.When $H<1/2$, rough-path-type techniques must come into the picture, which, from a theoretical point of view, involves the use of some a-priori-defined L{\'e}vy area process. We show that such an object can indeed be \enquote{canonically} constructed for any $H\in (\frac14,\frac12)$. Finally, when $H\leq 1/4$, we exhibit a similar non-convergence phenomenon as for the non-diagonal entries of the (classical) L{\'e}vy area above the standard fractional Brownian.