Noise-induced periodicity in a frustrated network of interacting diffusions

The paper’s Theorem 4.2 states that for the two-population McKean–Vlasov SDE (2) there exists a Gaussian Markov pair (x̃,ỹ) with independent components whose first two moments satisfy the moment ODEs (7) for p=1,2, and that the small-noise error satisfies E sup_{t≤T}(|x−x̃|+|y−ỹ|) ≤ C_T σ^2. The statement and its construction appear in the main text and Appendix D, where the authors derive (6)-(7) via Itô, build linear Gauss–Markov auxiliaries z1,z2 in (27) and set x̃=m1+σz1, ỹ=m2+σz2 so that (m1,m2,v1,v2) solve (8) and properties (1)-(2) follow, with the σ^2 error bound obtained via a variation-of-constants estimate (30)-(32) and moment bounds for Gaussians . The candidate model solution follows the same three-step structure: (A) well-posedness and derivation of (7); (B) constructing an independent Gaussian Markov pair whose first two moments match (7) for p=1,2; and (C) a σ-closure estimate via a Grönwall argument using Gaussian moment remainders. Differences are mostly expository: the paper uses an explicit linear SDE for z and a variation-of-constants bound, while the model uses an affine-coefficient linear SDE for (~x,~y) and a Lipschitz/closure decomposition with Grönwall. Both yield the same order-σ^2 estimate and the same moment equations, hence the proofs are substantially the same in spirit and outcome. The paper’s appendices also establish well-posedness for (2) (Appendix A), aligning with the model’s preliminary step .