  We consider a multidimensional time-homogeneous dynamical system and add a
randomly perturbed time-dependent deterministic signal to some of its
components, giving rise to a high-dimensional system of stochastic differential
equations, which is driven by possibly very low-dimensional noise. Equations of
this type commonly occur in biology when modelling neurons or in statistical
mechanics for certain Hamiltonian systems. We provide verifiable conditions on
the original deterministic dynamical system under which the solution to the
respective stochastic system features a point in the interior of its state
space, which can be proved to be attainable by deterministic control arguments,
and at which a local H\"ormander condition holds. Together with a Lyapunov
condition, it follows that the corresponding process is positive Harris
recurrent.
