Quantitative coarse-graining of Markov chains

The paper’s Theorem 3.1 establishes, for finite-state continuous-time Markov chains with coarse map ξ and effective generator N, the bound H(μ̂t|ηt) ≤ H(μ̂0|η0) + c sqrt(2/αLSI) [H(μ0|ρ)−H(μt|ρ)]^{1/2}, with a time-independent constant c depending on L, N, ρ and the initial macroscopic data (μ̂0, η0) (Theorem 3.1) . The core identity d/dt H(μ̂t|ηt) = −RN(μ̂t|ηt) + mismatch follows from Lemma 3.4 (eq. (23)) and the paper controls the mismatch via an L2-in-time bound on a forcing gt and blockwise total-variation distances, which are then estimated using CKP and a log-Sobolev inequality to recover (19) (proof of Lemma 3.4 and Theorem 3.1) . The constant c arises from ∥g∥L2(0,∞), which is shown finite using exponential convergence of μ̂t and ηt to ξ#ρ (Lemma 3.5) . By contrast, the model’s proof invokes a blockwise Poisson equation −Ly φ = g with mean-zero condition relative to ρ(·|y). In general, ρ(·|y) is not stationary for Ly (the paper explicitly discusses that (Ly)T ρ(·|y) ≠ 0 in the nonreversible setting), so the Poisson problem need not be solvable with that zero-mean condition; this is a missing hypothesis in the model’s argument and breaks the logic except in the reversible case . In addition, the model equates −⟨φ, Ly φ⟩ρ to the symmetric Dirichlet form ½∑ ρ(x)Ly(x,x′)(φ(x′)−φ(x))², which is only valid under detailed balance; without reversibility one would need the symmetrized form. The paper’s proof avoids these pitfalls and is correct within its stated assumptions, whereas the model proof requires unspoken reversibility/solvability conditions and is therefore flawed in general.