Quantum Langevin Equation

It is shown (by means of a perturbation series) that for a class of potentials $V(x)$ the stationary distribution of the solution $x(t)$ of the quantum Langevin equation approaches in the weak-coupling limit (f→0) the quantum mechanical canonical distribution of the displacement of the oscillator, subject to the potential $V(x)$, if and only if $E(t)$ is the operator version of the purely random Gaussian process so that, in particular, higher symmetrized averages 〈E(t₁)⋯E(tₙ)〉ₛ are expressible in terms of pair correlations, in the usual way.