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. The present work generalizes prior derivations of the homogenized equation for the position degrees of freedom in the m → 0 limit. Specifically, we develop a hierarchy of approximate equations for the position degrees of freedom that achieves accuracy of order m^ℓ/2 over compact time intervals for any ℓ ∈ Z⁺. The results cover bounded forces, for which we prove convergence in L^p norms, and unbounded forces, in which case we prove convergence in probability.