  We consider hypo-elliptic diffusion and convection-diffusion on $\mathbb{R}^3
\rtimes S^2$, the quotient of the Lie group of rigid body motions SE(3) in
which group elements are equivalent if they are equal up to a rotation around
the reference axis. We show that we can derive expressions for the convolution
kernels in terms of eigenfunctions of the PDE, by extending the approach for
the SE(2) case. This goes via application of the Fourier transform of the PDE
in the spatial variables, yielding a second order differential operator. We
show that the eigenfunctions of this operator can be expressed as (generalized)
spheroidal wave functions. The same exact formulas are derived via the Fourier
transform on SE(3). We solve both the evolution itself, as well as the
time-integrated process that corresponds to the resolvent operator.
Furthermore, we have extended a standard numerical procedure from SE(2) to
SE(3) for the computation of the solution kernels that is directly related to
the exact solutions. Finally, we provide a novel analytic approximation of the
kernels that we briefly compare to the exact kernels.
