Fluctuation Analysis for Particle-Based Stochastic Reaction-Diffusion Models

The paper proves a fluctuation CLT for the PBSRD model (Theorem 5.6) by: (i) a prelimit semimartingale decomposition for test functions f in W^{2+Γ,a}_0; (ii) centering at the mean-field μ̄ and expanding reaction drifts to first order to obtain a linearized operator Δ(ℓ)[μ̄,·]; (iii) tightness in weighted negative Sobolev spaces via uniform moment bounds and Aldous-type estimates; (iv) identification of the Gaussian limit by convergence of predictable covariations and vanishing jumps; and (v) uniqueness of the limiting linear SPIDE in W^{−(2+Γ),a}. All of these steps appear explicitly in Sections 8.1–8.5 and the statement/covariance in (5.2)–(5.3) of the paper, including the key assumption √γ η→0 to kill placement-mollifier errors and the restriction to uni- and bimolecular reactions via Definition 5.1. The candidate solution follows the same semimartingale/characteristics program with the same function spaces, assumptions, and limit characterization; differences are presentational only (e.g., the exact Hilbert–Schmidt embedding used to implement compact containment), not substantive. Hence both are correct and essentially the same proof. Key correspondences: prelimit decomposition (paper Eq. (8.46)) matches the candidate’s Step 1; linearization and √γ η→0 remainders (paper’s I/II separation) match Step 2; tightness via weighted Sobolev embeddings and Aldous conditions matches Step 3; identification via brackets and a martingale CLT (paper Lemma 8.14 and Theorem 8.15) matches Step 4–5; uniqueness (paper Theorem 8.16) matches Step 6. Citations: semimartingale decomposition and scaling (paper 8.1) ; assumptions and spaces (Assumption 5.1; Section 6) ; main result and covariance (Theorem 5.6, (5.2)–(5.3)) ; tightness and Aldous lemmas (Theorem 8.12) ; jump sizes O(γ^{-1/2}) and continuity of limits (Lemma 8.13) ; identification of the limit (Eq. (8.45)) ; control of placement error using √γ η→0 (the I/II split) .