A limit theorem for stochastic acceleration

We consider the motion of a particle in a weak mean zero random force fieldF, which depends on the position,x(t), and the velocity,v(t)=(t). The equation of motion is(t)=F(x(t),v(t), ), wherex(·) andv(·) take values in ^d,d3, and ranges over some probability space. We show, under suitable mixing and moment conditions onF, that as 0,v(t)v(t/²) converges weakly to a diffusion Markov processv(t), and ²x(t) converges weakly to, wherex=lim ²x(0).