Empirical approximation to invariant measures for McKean–Vlasov processes: mean-field interaction vs self-interaction

The candidate solution proves the same rates as the paper’s Theorems 2.1–2.2—namely, E W2(E^$_t[X], μ*)^2 and E W2(E^$_t[Y], μ*)^2 decay like t^{-(ε1 ∧ κ ε2)} with κ=ρ/((d+2)(ρ+2))—under the same Assumption 1.1 and weight classes Π1(ε1), Π2(ε2) (the paper’s statements and definitions appear explicitly in Theorem 2.1–2.2 and (2.2) ). Both approaches compare the target dynamics to the frozen Markovian reference X̂ solving dX̂_t=b(X̂_t,μ*)dt+σ(X̂_t,μ*)dW_t, then use a polynomial stabilization lemma (Lemma 4.1) for Volterra-type inequalities (the paper’s Lemma 4.1 and its use in the proofs of Theorems 2.1–2.2 are shown in Section 4 ; see also the displayed steps in the proofs ). Where they differ is in establishing the sampling term for the stationary reference: the paper proves Proposition 3.1 via a density-coupling/smoothing argument to obtain E W2(E^$_t[X̂], μ*)^2 ≤ C t^{-κ ε2} (Section 3, Proposition 3.1 and its proof outline ), whereas the model leverages covariance decay plus a Fournier–Guillin-style smoothing/variance proxy to reach the same exponent. These are methodologically distinct but yield the same rate. Minor differences (e.g., the paper’s use of a β̂>β in the comparison step) do not affect the conclusion; the model’s direct convexity step avoids that technicality while staying within the paper’s assumptions.