A HARRIS THEOREM FOR ENHANCED DISSIPATION, AND AN EXAMPLE OF PIERREHUMBERT

The paper establishes (1) almost-sure exponential mixing for the stochastic flow X^κ via Harris theory for the κ>0 two-point chain (Lemmas 3.1–3.3) and (2) the L∞ decay and mixing-time bounds (Theorems 1.1 and 1.3) with the sharp κ-exponent κ^{-(d/2+α)} and a random constant D_κ independent of W with finite moments, see (1.3), (1.4), and (1.7) in the text . The model’s outline matches the final statements (rates, κ-scaling, and dependence of D_κ), but its central reduction step is invalid: it claims that under synchronous coupling the difference X^κ_n(x)−X^κ_n(x′) is exactly the κ=0 difference, leading to a contraction controlled purely by the κ=0 two-point chain. This is false—the difference dynamics depend on the random base path and cannot be replaced by the deterministic κ=0 flow. The paper, by contrast, proves stability of the Harris drift/minorization to establish geometric ergodicity directly for the κ>0 two-point chain (Sections 4–6), then deduces almost-sure exponential mixing (Lemma 3.3) and finally the L∞ decay via parabolic regularity (proof of Theorem 1.1 and (3.5)) . In fact, the paper carefully shows only an approximate relation between κ>0 and κ=0 flows (e.g., Lemma 4.3), not an equality as asserted by the model .