K-averaging agent-based model: propagation of chaos and convergence to equilibrium

The paper’s Theorem 2 proves that the discrete-time law ρ_n contracts toward the Gaussian fixed point ρ_∞ in W2 with factor 1/K by constructing an optimal W2 coupling for one pair, taking K i.i.d. copies, and using a common Gaussian noise; iterating yields geometric decay, using that ρ_∞ is a fixed point of T. This is exactly the coupling-and-common-noise argument the model solution gives. One nuance: the paper momentarily states the auxiliary contraction W2^2(T(µ), T(ν)) ≤ (1/K) W2^2(µ, ν) “for each µ, ν ∈ P(R^d)”, but that inequality only holds when µ and ν share the same mean (e.g., under the mean-zero normalization (2.12) used for ρ_n and ρ_∞). With the normalization in place, the paper’s steps are correct and coincide with the model’s solution. See Theorem 2 statement and proof sketch, including the coupling and common-noise construction, and the fixed point property of ρ_∞ in Lemma 2.2 .