Mean-square invariant manifolds for ill-posed stochastic evolution equations driven by nonlinear noise

Zonghao Li, Caibin Zeng, Jianhua Huang

correctmedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper rigorously constructs mean-square random unstable invariant manifolds and mean-square random stable invariant sets for ill-posed stochastic evolution equations via integrated semigroups and Lyapunov–Perron maps that explicitly use conditional expectations E(·|F_t) to ensure adaptedness; see the modified variation-of-constants and the backward/forward fixed-point equations (Theorem 2.1; (28); (67)) and the main results (Theorems 3.1 and 4.1) with the stated smallness conditions. The candidate solution matches the high-level structure and constants but omits the crucial conditional-expectation terms in the Lyapunov–Perron operators, so it does not guarantee Ft-adapted solutions and hence does not rigorously produce a mean-square random dynamical system. This is the central technical point emphasized by the paper.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

A careful and correct contribution extending invariant-manifold theory to mean-square random dynamical systems for ill-posed stochastic equations with non-dense domain. The integrated-semigroup framework and the use of conditional expectations within Lyapunov–Perron maps to ensure adaptedness are well handled. The work is technically solid and fills a niche where pathwise approaches fail. Minor editorial improvements would enhance readability.