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

Abstract

According to the Smoluchowski-Kramers approximation, the solution of the equation $μ\ddot{q}^μ_t=b(q^μ_t)-\dot{q}^μ_t+Σ(q^μ_t)\dot{W}_t, q^μ_0=q, \dot{q}^μ_0=p$ converges to the solution of the equation $\dot{q}_t=b(q_t)+Σ(q_t)\dot{W}_t, q_0=q$ as μ->0. We consider here a similar result for the Langevin process with elastic reflection on the boundary.