Kinetic equation for Brownian particles

In an approach inspired by studies on elastic chains and similarity to diffusion in phase space, a stochastic kinetic equation is derived for the velocity and position dependent probability distribution. By restricting the particle trajectory to those paths which do not deviate strongly from the initial direction, the coupling terms for diffusion in velocity and position of freely propagating particles and for particles in an external field are found. For the field free case, the relation between diffusion and friction as well as the solution for short and long time behaviour are given.