  In this work we investigate the well-posedness for difussion equations
associated to subelliptic pseudo-differential operators on compact Lie groups.
The diffusion by strongly elliptic operators is considered as a special case
and in particular the fractional diffusion with respect to the Laplacian. The
general case is studied within the H\"ormander classes associated to a
sub-Riemannian structure on the group (encoded by a H\"ormander system of
vector fields). Applications to diffusion equations for fractional
sub-Laplacians, fractional powers of more general subelliptic operators, and
the corresponding quasi-geostrophic model with drift $D$ are investigated.
Examples on SU(2) for diffusion problems with fractional diffusion are
analysed.
