Symmetry of the isotropic Ornstein-Uhlenbeck process in a force field

We classify simple symmetries for an Ornstein-Uhlenbeck process, describing a particle in an external force field f(x). It turns out that for sufficiently regular (in a sense to be defined) forces there are nontrivial symmetries only if f(x) is at most linear. We fully discuss the isotropic case, while for the non-isotropic we only deal with a generic situation (defined in detail in the text).