Velocity-Verlet-like algorithm for simulations of stochastic dynamics

Molecular simulations of many particles which move rather according to a brownian than a newtonian type of dynamics, nevertheless, can be performed by means of a "velocity-Verlet-like" algorithm. The derivation of this algorithm requires the "Ito formula" of stochastic calculus which usually is not part of a scientist's education. Therefore, it is going to be shown, how this formula can be motivated and applied in order to find that algorithm. Furthermore, it is demonstrated, why it is sufficient to use uniformly distributed random numbers within that algorithm, thus avoiding gaussian distributed ones, although gaussian distributed white noise forces are assumed to model the brownian-like motion of the particles. Finally, a solution to the problem is presented that the linear total momentum is not conserved due to the presence of stochastic forces in the equations of motion of the particles.