Identifiability of Multivariate Linear Systems Driven by Ornstein–Uhlenbeck Forcing

We consider a stable multivariate linear system driven by unobserved Ornstein-Uhlenbeck forcing with a common, unknown relaxation rate and positive-definite diffusion covariance. We characterize all parameter triples producing the same stationary law of the fully observed state. The parameters are identifiable whenever the dynamical matrix is not a scalar multiple of the identity. The only distinct equivalent triples occur when that matrix is scalar: the deterministic and forcing relaxation rates can then be exchanged. This scalar ambiguity is known; the present note provides its complete multivariate classification within the stated model class. A derivative-augmented representation yields a closed reconstruction formula from covariance derivatives and a deterministic perturbation bound. The results clarify population identifiability for colored linear inverse modeling while leaving finite-sample estimation and incomplete observation as separate questions.