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

TL;DR

This paper classifies symmetries of the Ornstein-Uhlenbeck process in force fields, showing nontrivial symmetries exist mainly for linear forces, with a detailed focus on the isotropic case.

Contribution

It provides a complete classification of symmetries for the Ornstein-Uhlenbeck process under various force fields, especially highlighting the isotropic case.

Findings

Nontrivial symmetries occur mainly when the force is at most linear.

Full classification of symmetries for the isotropic Ornstein-Uhlenbeck process.

Partial results for non-isotropic cases in generic situations.

Abstract

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).