Averaging Principle for Stochastic Complex Ginzburg–Landau Equations

The paper proves three averaging results for the stochastic complex Ginzburg–Landau (CGL) equation on Td: (i) finite-time averaging (first Bogolyubov), (ii) a second Bogolyubov theorem via recurrent solutions, and (iii) a global averaging principle in a probability-measure space; see the problem setup (1.3) with |β| ≤ √3 and the assumptions (H1)–(H3), (G1)–(G2) and the cocycle/attractor framework in Pr2(L2) with the hull H(F) . The candidate solution reproduces the finite-time and global (measure-attractor) averaging under essentially the same structural hypotheses, but via a different route: it leverages synchronous coupling and a uniform W2-contraction under the dissipativity condition λ*−λf−Lg^2/2>0 (which the paper also imposes for stability) to build a uniform attractor and to pass to the averaged singleton attractor, using a Khasminskii time discretization with δf(·), δg(·) as in (G1)–(G2). This aligns with the paper’s estimates and attractor convergence (Theorem 5.6), although the model does not address the paper’s second Bogolyubov theorem (recurrent solutions). Technically, both analyses depend on the same key ingredients: monotonicity/dissipativity of the cubic with |β| ≤ √3, uniform moment bounds and well-posedness, and the δf, δg averaging hypotheses; see (H1)–(H3), Lemma-type energy bounds, and the statements of the first/global averaging results . Net: for the overlapping parts (finite-time and global averaging toward the measure attractor), both are correct but via different proofs; the paper additionally establishes the second Bogolyubov theorem, which the model does not cover.