Population Dynamics under Demographic and Environmental Stochasticity

The paper’s Theorem A establishes the dichotomy precisely: as ε→0, the QSD μ_ε concentrates at 0 in the sense ∫φ dμ_ε→0 for φ∈C_b with φ(0)=0 when Λ0<0, and μ_ε⇒μ_0 with u_ε→u_0 in C^2_loc when Λ0>0 . The candidate’s write-up correctly restates this dichotomy and much of the setup (generator L_ε, Gibbs profile u^G_ε, existence/uniqueness of QSD via the principal eigenpair) in line with Lemma 3.4 and the spectral framework in Section 3 . However, the candidate also asserts a boundary-layer result in the Λ0<0 regime that implies ε^2 λ_ε→b′(0). This directly contradicts the paper’s quantitative bounds: for Λ0<0 the first eigenvalue satisfies only a logarithmic bound λ_ε ≳ C/|ln ε| (and the authors indicate this scale is sharp) rather than λ_ε being of order 1/ε^2 . In addition, the candidate characterizes 0 as an “exit boundary” for X^0 when Λ0<0, but the paper shows X^0_t>0 a.s. for all t>0 starting from x>0, i.e., 0 is unattainable in finite time; δ0 is stationary but not reached in finite time (Proposition 3.1) . The paper’s proof of Theorem A proceeds via tightness near ∞ (Lyapunov function; Proposition 4.1) and near 0 (a uniform x^{-k} bound; Proposition 4.2), then identification/limit arguments—an approach that does not require the candidate’s incorrect boundary-layer scaling .