Proposition: Colored transfer

Equation (eq:colored) is the exact added covariance for classical Gaussian support noise independent of the initial quantum state, under the stated quadratic dynamics. It includes correlations between different curved stages. The white limit reduces to

Proof. The force-driven linear Hamilton equations give the stated $\delta R$. Its mean is zero. Take the expectation of $\delta R\delta R^{\top }$ and use the displacement covariance. Independence removes initial-state/noise cross terms. The delta correlation reduces the double integral to the single one.