Cyclic-graph covariance formula

Establish the covariance formula for the directed Gaussian-process model on cyclic directed metric graphs when the transfer operator inverse exists, including justification of the stochastic representation and the interchange of the resulting infinite directed-path sums.

Background

The paper derives an explicit covariance kernel for proper global Ornstein–Uhlenbeck processes on finite acyclic directed metric graphs by expanding source and noise contributions over directed paths. For cyclic graphs, the transfer operator is still formally given by the inverse of the identity minus the transfer matrix, and a spectral-radius condition guarantees convergence of the directed-path expansion.

The authors explicitly refrain from asserting that the acyclic covariance formula remains valid on cyclic graphs. A complete result would need to establish both a valid stochastic representation of the cyclic boundary-value problem and the legitimacy of interchanging the infinite path expansion with the stochastic operations used to compute covariances.

References

On a cyclic graph the transfer operator remains $(I-){-1}$ whenever the inverse exists, and $\rho(||)<1$ guarantees absolute convergence of its directed-path expansion. We do not claim eq:cov-kernel in this case, because the stochastic representation and interchange of the infinite path sums also require justification.

— Gaussian Processes on Directed Metric Graphs  (2609.01435 - Bolin et al., 1 Sep 2026) in Remark (cyclic-kernel), Section 4.2, “Covariance functions”