On the two Green's function procedures for transport noise

The noise spectra of y(r, t) governed by (∂/∂t + λ)y(r, t) = ξ(r, t), where ξ is a volume Langevin source, are expressed with the aid of the Fourier-Laplace transformed Green's function G. We give the response form (quadratic in G) and the correlation form (linear in G plus quadratic surface term). In the latter we need the covariance function Γ(r, r'), which is converted to Sξ by the λ-theorem. We show that the correlation form is invariant against the choice of the particular solution ĝG of the λ-theorem. In uniform systems one can usually choose ĝG such that the surface contribution vanishes.