Center Manifolds for Rough Partial Differential Equations

The paper rigorously constructs a local center manifold for rough PDEs by (i) truncating the nonlinearities to achieve small Lipschitz constants in the controlled rough-path setting, (ii) working in a sequence-valued, exponentially weighted space BC_η(D), and (iii) using a discretized Lyapunov–Perron transform J_d. A contraction is proved under the spectral gap condition (6.8) (Theorem 6.5), and one then recovers a continuous-time invariant manifold with the stated graph formula (Theorem 6.8) and invariance in the RDS sense, with tangency at the origin built into the assumptions F(0)=DF(0)=0 and G(0)=DG(0)=D^2G(0)=0 (e.g., , , ). In contrast, the candidate solution asserts a direct continuous-time Lyapunov–Perron contraction on (-∞,0] without the discretization step that the paper explicitly deems necessary to control the Hölder norms of the rough input on each unit interval, and it does not implement the path-dependent cut-off that the paper uses to make the map contractive (see the explicit need to discretize (6.6) and the use of FR, GR and R(W) in Section 4, ; ; ). The candidate also quotes differentiability of the Lyapunov–Perron map with respect to the integrand without justification beyond Theorem 3.5, which in the paper provides stability/mapping estimates for rough convolutions but not C^1-smoothness (). Hence, key steps in the model solution are unproven or rely on assumptions the paper avoids by a different, rigorous route. The paper’s construction, invariance, and the hc(ξ,W) formula are correct and carefully justified; the model’s proof is incomplete and does not meet the rough-PDE-specific obstacles identified by the authors.