Langevin Equations in the Small-Mass Limit: Higher-Order Approximations

Abstract

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 $$\ell \in \mathbb {Z}^+$$. This generalizes prior derivations of the homogenized equation for the position degrees of freedom in the $$m\rightarrow 0$$ 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.