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).
36 pages in OCNMP format, 27 pages in RevTex format. To appear in OCNMP (Open Communications in Nonlinear Mathematical Physics), see https://ocnmp.episciences.org