Remarks on multi-dimensional noncommutative generalized Brownian motions

We consider certain questions pertaining to noncommutative generalized Brownian motions with multiple processes. We establish a framework for generalized Brownian motion with multiple processes similar to that defined by Guta and prove multi-dimensional analogs of some results of Guta and Maassen. We then consider examples of processes indexed by a two-element set and characterize the function on I-indexed pair partitions associated via the I-indexed generalized Brownian motion construction to certain pairs of representations connected to certain spherical representations of infinite symmetric groups. In doing so, we generalize the notion (introduced by Bozejko and Guta) of the cycle decomposition of a pair partition. We then generalize Guta's q-product of generalized Brownian motions to a product corresponding to a (possibly infinite) matrix $(q_{ij})$ and show that this $q_{ij}$-product satisfies a central limit theorem.