The Limit Of Small-Mass And White-Noise Of A General Langevin Equation

This thesis derives the small mass and white noise limit of a generalized Langevin equation.In our case,the generalized Langevin equation is a stochastic differential equation which describes the motion of a particle with small mass in some liquid.The mass of the particle and the scale of the fluctuation are both described by a small parameter ? > 0.We show that if the fluctuation is fast enough,the motion of the particle converges to Brownian motion which show the effectiveness of the classical Langevin equation.Here we follow a tightness discussion and a martingale approximation.First we split the solution into two parts,and build the bounded estimates and some continuous property for each part which yield the tightness of the solutions;Then we construct a continuous martingale which converges to Brownian motion as ? ? 0.