4 Dobrushin stability and the mean-field limit

Dobrushin’s 1979 theorem: two Lagrangian solutions separate at most exponentially in \(W_1\), at the explicit rate \(C = 2\max (1,L)\). The proof is his own coupling argument, followed piece for piece — couple the data optimally, push the coupling forward along the pair of flows, bound the force difference by the coupling cost, and close with a Gronwall inequality in mild form. Applied with one solution the empirical measure of the Newton dynamics — an exact weak solution by Chapter 1 — the estimate yields the mean-field limit: propagation of chaos in quantitative \(W_1\) form.

Theorem 33 Dobrushin stability (1979) — Theorem 1.5 of the paper

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Let \(W\) satisfy Assumption 1. Any two Lagrangian solutions \(f, g\) with finite first moment at every time are stable in \(W_1\) at an exponential rate:

\[ W_1(f_t, g_t) \; \le \; e^{C t}\, W_1(f_0, g_0), \qquad t \ge 0, \]

with the explicit rate \(C = 2\max (1, L)\).

Proof ▼

Dobrushin’s own coupling argument, followed piece for piece. Couple the data \(\varepsilon \)-optimally and push the coupling forward along the pair of characteristic flows; the force difference at coupled points is bounded by the coupling cost (the force-versus-metric estimate), the mild integral form of the trajectories turns this into a Gronwall inequality for the transported cost, and integrating gives the exponential estimate. Kantorovich–Rubinstein duality converts between the coupling cost and the \(W_1\) distance at both ends.

Definition 34

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The stability estimate packaged as a reusable predicate on a pair of measure curves: \(W_1(f_t, g_t) \le e^{Ct}\, W_1(f_0, g_0)\) for all \(t \ge 0\).

Theorem 35 Mean-field limit — Corollary 1.6 of the paper

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Let \(f\) be the Vlasov solution with datum \(f_0\) and let \(\mu ^N_t\) be the empirical measures of \(N\) particles evolving by Newton’s equations. Assume the Dobrushin estimate holds for every pair \((\mu ^N, f)\) with one constant \(C\) — Theorem 1.5 supplies it, each empirical curve being a Lagrangian solution with the rate uniform in \(N\), and the Lean takes it as an explicit hypothesis rather than re-deriving it. If the initial empirical measures converge, \(W_1(\mu ^N_0, f_0) \to 0\), then for every \(T {\gt} 0\)

\[ \sup _{t \in [0,T]} W_1(\mu ^N_t, f_t) \; \longrightarrow \; 0 \qquad (N \to \infty ). \]

Propagation of chaos for the Vlasov equation, in its quantitative \(W_1\) form.

Proof ▼

For each \(N\) the stability estimate lifts to the window supremum, \(\sup _{t \le T} W_1(\mu ^N_t, f_t) \le e^{CT}\, W_1(\mu ^N_0, f_0)\), and the right-hand side tends to \(0\) by the assumed initial convergence.