2409.15532
A theory of generalised coordinates for stochastic differential equations
Lancelot Da Costa, Nathaël Da Costa, Conor Heins, Johan Medrano, Grigorios A. Pavliotis, Thomas Parr, Ajith Anil Meera, Karl Friston
correctmedium confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves that the unique solution of x'(t)=Ax(t)+w(t) with mean-zero stationary Gaussian w whose autocovariance κ is analytic (radius 2R) and with a.s. analytic sample paths is a Gaussian process on T=(−R/λ,R/λ), with mean e^{At}z and covariance given by a locally uniformly convergent double series with coefficients involving A-powers and κ-derivatives; see Proposition 2.2.5 (series and domain), Theorem 2.2.10 (Gaussianity and uniqueness), and Proposition 2.1.7 (derivative identities) . The candidate solves the same problem directly by variation of constants, derives the same mean, an equivalent covariance integral and the same Taylor–series coefficients, and identifies an analytic domain that contains T×T. Hence both are correct, with substantially different proof strategies.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The paper provides a coherent and largely self-contained framework for analyzing linear SDEs with colored Gaussian noise via generalised coordinates, culminating in correct, explicit mean/covariance formulas and convergence guarantees. Assumptions are transparent, proofs are sound, and the results have methodological and practical interest. One acknowledged gap—linking analytic kernels to a.s. analytic sample paths—could be tightened with an added reference or discussion. Overall, correctness and clarity are high with minor points to refine.