Uncertainty Propagation Through Green’s Kernels and Gaussian Process Inference Dynamics

We study uncertainty propagation under Gaussian process priors whose covariance is induced by Green or resolvent operators associated with latent generators. Rather than specifying non-stationary dependence through input warping or other coordinate deformations, we construct covariance from discounted propagation under an underlying semigroup. This yields a separation between covariance geometry and transport: the self-adjoint dissipative part of the generator provides the operator from which prior covariance is constructed, while the skew-adjoint part contributes transport within the full propagation without being encoded directly as covariance. We derive observation, posterior, and propagation identities in operator form, and compare this framework with pullback, stationary, and Matérn baselines. Experiments are organised as a four-stage ladder ascending from three-dimensional electrostatics through a frequency-domain Helmholtz proxy and time-domain semigroup propagation to an operator-mismatch ablation. At each stage, the Green prior is realised as a Gaussian Markov random field (GMRF) whose precision is the discretised PDE operator; after assembly, the posterior mean is obtained by a single sparse linear solve. Across the reported experiments, Green’s kernel priors often outperform tuned Matérn-$3/2$ and RBF baselines on boundary-aware and PDE-residual metrics, with larger gains under heterogeneous material coefficients and semigroup propagation. In the late-time propagation setting the normalised RMSE gap exceeds $4{\times}$. The central claim is that, when an informative governing operator is available, covariance induced by that operator yields boundary-respecting and dynamically interpretable uncertainty—especially on operator-sensitive diagnostics—under the reported synthetic conditions.