Invariant Manifolds for Random Parabolic Evolution Equations with Almost Sectorial Operators

The paper proves a local stable-manifold theorem (Theorem 5.5) for the SPDE (1.1) with non-dense domain by: (a) constructing an RDS via an integrated-semigroup decomposition, (b) establishing Fréchet differentiability and compactness of the derivative cocycle under compact resolvent (Assumption 4.6), (c) verifying the MET integrability, and (d) invoking an abstract invariant-manifold theorem to obtain properties (i)–(v) for S_loc^ν(ω) . The candidate solution follows the same architecture: reduce to a random semilinear evolution on H0, ensure compactness of the linearized cocycle and MET applicability, and construct the manifold via a random graph transform, recovering exactly items (i)–(v). The only substantive difference is technical: the paper proves compactness of Dφ̃_t via the integrated-semigroup/resolvent representation (Proposition 4.8), whereas the model argues from analytic semigroup compactness; both are consistent with the paper’s Assumption 4.6 and Remark 4.7 . Hence both are correct and substantially the same proof.