Connection between perturbation theory, projection-operator techniques, and statistical linearization for nonlinear systems

We employ the equivalence between Zwanzig's projection-operator formalism and perturbation theory to demonstrate that the approximate-solution technique of statistical linearization for nonlinear stochastic differential equations corresponds to the lowest-order $\beta$ truncation in both the consolidated perturbation expansions and in the "mass operator" of a renormalized Green's function equation. Other consolidated equations can be obtained by selectively modifying this mass operator. We particularize the results of this paper to the Duffing anharmonic oscillator equation.