Strong convergence with error estimates for a stochastic compartmental model of electrophysiology

The paper’s Theorem 1 states and proves the almost-sure strong error bound sup_{t∈[0,T]}(max_k|V^(k)_t−v(t,hk)| + max_k|Z^(k)_{i,j}(t)−z_{i,j}(t,hk)|) ≤ C_{ω,T,ρ}(h^p ∨ h^{1/2−p/2} ∨ h^{ρ−p})|log h^{-1}| for p∈[0,1), ρ∈(p,1) (definitions of the spatial local averages and N_{h,p} are given around (37)–(40), and the bound is displayed as (41) in the text) . The proof uses a homogenization-style corrector χ and a uniform-in-time Poisson-sum concentration lemma (Lemma 3) to control stochastic fluctuations, together with a discrete-to-continuum Laplacian consistency estimate (62) that carries the p-dependence via the averaging scale . The candidate’s solution establishes the same rate with a different route: direct ℓ∞-stability of the discrete heat semigroup for V, Lipschitz control of reaction terms, and Freedman-type martingale bounds for the block-averaged Poisson noise, yielding the same h^{1/2−p/2}|log h| contribution and closing with Grönwall. One minor issue in the paper is a typographical inconsistency in the definition of the averaging window (the displayed N_{h,p}:=2[hp−1/2]+1 in (39) and Fig. 8 reads as if it could degenerate for p≥1/2), yet the analysis explicitly uses N_{h,p}≈n^{1−p}, consistent with averaging over a physical radius ∼h^p and ensuring vanishing fluctuations as h→0 . With that clarification, both arguments yield the same conclusion.