Interacting diffusions on positive definite matrices

We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to $K$-Bessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda chain.