A Note on the Smoluchowski-Kramers Approximation for the Langevin Equation with Reflection

TL;DR

This paper investigates the Smoluchowski-Kramers approximation for Langevin equations with elastic reflection, extending classical results to include boundary interactions and demonstrating convergence as the mass parameter approaches zero.

Contribution

It extends the Smoluchowski-Kramers approximation to Langevin processes with boundary reflection, providing new convergence results for such systems.

Findings

Established convergence of reflected Langevin processes to their overdamped limits.

Extended classical approximation results to systems with boundary reflection.

Provided mathematical framework for analyzing stochastic dynamics with boundary conditions.

Abstract

According to the Smoluchowski-Kramers approximation, the solution of the equation ${\mu}\ddot{q}^{\mu}_t=b(q^{\mu}_t)-\dot{q}^{\mu}_t+{\Sigma}(q^{\mu}_t)\dot{W}_t, q^{\mu}_0=q, \dot{q}^{\mu}_0=p$ converges to the solution of the equation $\dot{q}_t=b(q_t)+{\Sigma}(q_t)\dot{W}_t, q_0=q$ as {\mu}->0. We consider here a similar result for the Langevin process with elastic reflection on the boundary.