Stochastic functional partial differential equations with monotone coefficients: Poisson stability measures, exponential mixing and limit theorems

The paper’s Theorem 5.2 proves uniform exponential mixing in the bounded Lipschitz (a Wasserstein-type) metric via: (i) an L2-contractivity estimate for the segment solution map under synchronous coupling (their eq. (3.15)), (ii) the Lipschitz test-function bound coupled with Cauchy–Schwarz to pass to second moments (their eq. (5.13)), and (iii) a uniform second-moment bound for the invariant measure obtained from time averages (their eq. (5.14)). The candidate solution reproduces exactly these steps: synchronous coupling/L2-contraction, Lipschitz test-function/Kantorovich bound, and the second-moment control for µ*. Differences are only in minor constant handling (absorbing √2 into L7) and presentation; no substantive gap. See Theorem 5.2 and its proof, the contractivity estimate (3.15), and the BL metric definition in Section 2 for precise alignment .