Coarse-grain description of the distribution of solutions of the Langevin equation

The phenomenological Langevin equation for a Brownian particle with a random force having exponentially fast correlation weakening is considered. It is shown that at times much longer than the time of correlation dampening of the random force the momentum distribution function $f(p,t)$ of the Brownian particle can be approximated in the weak sense with exponential accuracy by a coarse-grain distribution function satisfying a self-consistent evolution equation.