Prove genericity of the spectral and rank nondegeneracy conditions

Establish that the spectral and rank nondegeneracy conditions imposed in the identifiability proofs for multivariate Ornstein–Uhlenbeck processes hold generically over the relevant admissible parameter spaces.

Background

The identifiability results for recovering the drift matrix, input vector, and diagonal diffusion matrix of a multivariate Ornstein–Uhlenbeck process rely on spectral and rank nondegeneracy assumptions involving moment-derived linear systems. The paper verifies these conditions numerically and provides additional justification for directed-cycle SCCs, but does not establish that the conditions hold generically for arbitrary strongly connected drift graphs.

A general proof of genericity would strengthen the theoretical guarantees by showing that the exceptional parameter sets violating the nondegeneracy assumptions are negligible rather than merely observing numerical nondegeneracy in sampled systems.

References

We do not prove their genericity in this work; instead, we evaluate them numerically in Appendix~\ref{app:Spec_assumption}. The precise statements of the assumptions and their roles in the proofs are deferred to Appendix~\ref{app:proofs} and Appendix~\ref{app:spec_assumption_role}, respectively. Establishing genericity of these conditions alone is left as a direction for future work.

— One Intervention per Component is Enough: Towards Identifiability in Linear Stochastic Dynamics from Steady State  (2609.19955 - Salehkaleybar, 17 Sep 2026) in Remark following Theorem 1 (Single SCC), Section 3.1; see also Section 7, Conclusions and Future Work