The Small-Mass Limit for Langevin Dynamics with Unbounded Coefficients and positive friction

A class of Langevin stochastic differential equations is shown to converge in the small-mass limit under very weak assumptions on the coefficients defining the equation. The convergence result is applied to physically realizable examples where the coefficients defining the Langevin equation grow unboundedly either at a boundary, such as a wall, and/or at the point at infinity.