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^{ℓ / 2} over compact time intervals for any ℓ∈ Z⁺. This generalizes prior derivations of the homogenized equation for the position degrees of freedom in the m→ 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.