Bifurcations and Early-Warning Signs for SPDEs with Spatial Heterogeneity

The paper establishes (i) an a.s. upper bound L_k(t;ω) ≤ ∑_{j=1}^k(α−λ_j) for all t>0 (Theorem 2.1), (ii) a finite-time lower bound with positive probability L_k(t;ω) ≥ ∑_{j=1}^k(α−λ_j) − η on [0,T] (Theorem 2.4), via an indefinite inner-product Q^{(k)}_δ construction and a smallness event for the random equilibrium a, and (iii) covariance/pointwise-variance early-warning divergences for the linearized problem (Proposition 3.1 and its corollaries) . The candidate’s (A) and (C) match the paper’s results. However, for (B) the model invokes a Danskin-type equality for d/dt log‖∧^kU‖ and integrates a pointwise sup_E tr-bound to obtain a lower bound for ‖∧^kU‖; this step is not justified and ignores the key δ–gap constraints and the Q^{(k)}_δ Lyapunov functional the paper needs to control the (time-varying) maximizing k-subspace. Hence (B) is flawed, while the paper’s argument is correct.