Advective-diffusive motion on large scales from small-scale dynamics with an internal symmetry

We consider coupled diffusions in $n$-dimensional space and on a compact manifold and the resulting effective advective-diffusive motion on large scales in space. The effective drift (advection) and effective diffusion are determined as a solvability conditions in a multi-scale analysis. As an example we consider coupled diffusions in $3$-dimensional space and on the group manifold $SO(3)$ of proper rotations, generalizing results obtained by H. Brenner (1981). We show in detail how the analysis can be conveniently be carried out using local charts and invariance arguments. As a further example we consider coupled diffusions in $2$-dimensional complex space and on the group manifold $SU(2)$. We show that although the local operators may be the same as for $SO(3)$, due to the global nature of the solvability conditions the resulting diffusion will be different, and generally more isotropic.