  We study the global dynamics of free massive scalar fields on general,
globally stationary, asymptotically AdS black hole backgrounds with Dirichlet-,
Neumann- or Robin- boundary conditions imposed on $\psi$ at infinity. This
class includes the regular Kerr-AdS black holes satisfying the Hawking Reall
bound $r_+^2 > |a|l$. We establish a suitable criterion for linear stability
(in the sense of uniform boundedness) of $\psi$ and demonstrate how the issue
of stability can depend on the boundary condition prescribed. In particular, in
the slowly rotating Kerr-AdS case, we obtain the existence of linear scalar
hair (i.e. non-trivial stationary solutions) for suitably chosen Robin boundary
conditions.
