Langevin equations in the small-mass limit: Higher-order approximations

We study the small-mass (overdamped) limit of Langevin equations for a particle in a potential and/or magnetic field with matrix-valued and state-dependent drift and diffusion. We utilize a bootstrapping argument to derive a hierarchy of approximate equations for the position degrees of freedom that are able to achieve accuracy of order $m^{ell/2}$ over compact time intervals for any $ellinmathbb{Z}^+$. This generalizes prior derivations of the homogenized equation for the position degrees of freedom in the $mto 0$ limit, which result in order $m^{1/2}$ approximations. Our results cover bounded forces, for which we prove convergence in $L^p$ norms, and unbounded forces, in which case we prove convergence in probability.