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

14 pages, 2 figures