Smoluchowski-Kramers approximation for stochastic differential equations under discretization

The Smoluchowski-Kramers approximation of discrete time dynamical system is considered, where the system is described by the motion of a particle in a force field. It is shown that the Smoluchowski-Kramers approximation holds when a drift-implicit Euler-Maruyama scheme is used for the discretization, moreover the convergence rate is obtained. In particular, the solution of discretized system converges to the solution of the first order equation in mean square sense, which does not depend on the order of mass \mu and step size h tend to zero.