The starting point for the standard derivation of a Langevin equation is the identity 1 = P + Q , where Q projects onto the fast subspace. Consider discrete small time steps τ τ with evolution operator U ≅ ≅ 1 + i τ τ L , where L is the Liouville operator. The goal is to express U n in terms of U k P and Q ( U Q ) m . The motivation is that U k P is a functional of slow variables and that Q ( U Q ) m generates expressions which are fast variables at every time step. The expectation is that fast variables isolated in this way can be represented by some model data, for instance by a Gaussian white noise. The decomposition is achieved by multiplying 1 = P + Q from the left with U , except for the last term, which is multiplied with U = P U + Q U . Iteration gives