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

TL;DR

This paper proves that certain Langevin stochastic differential equations with unbounded coefficients converge in the small-mass limit, even under weak assumptions, and applies this to physically relevant cases with unbounded growth.

Contribution

It establishes convergence of Langevin equations with unbounded coefficients in the small-mass limit under very weak conditions, extending previous results to more realistic physical models.

Findings

Convergence holds under weak assumptions on coefficients.

Applicable to models with unbounded coefficients at boundaries or infinity.

Provides rigorous justification for small-mass approximations in complex physical systems.

Abstract

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.