Dependencies

Let \(W : \mathbb {R}^d \to \mathbb {R}\) be an even function of class \(C^{1,1}\): differentiable, with a globally Lipschitz gradient of constant \(L := \mathrm{Lip}(\nabla W) {\lt} \infty \). This assumption is in force throughout. Evenness gives \(\nabla W(0) = 0\) — the cancellation that makes the empirical measure an exact weak solution below.

In addition to Assumption 1, let \(\nabla W \in C^1\) (equivalently \(W \in C^2\)). One degree of regularity beyond Assumption 1, spent to make the phase-space field \(C^1\) in space — exactly what the variational equation for the flow’s derivative in its initial point requires. Only the superposition principle carries this assumption.

Under Assumption 1 the remainder vanishes identically: \(W\) even forces \(\nabla W(0) = 0\), so the empirical measure of every Newton solution satisfies the distributional Vlasov equation exactly — not merely up to an \(O(1/N)\) error. This is the sense in which the particle system already solves the limit equation.

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 \(T {\gt} 0\)

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

A characteristic flow for the field driven by \(\rho \): a map \(\Phi = (X,V)\) solving the mean-field ODE

the system along which the Vlasov equation transports mass.

The convolution of a vector field \(k : \mathbb {R}^d \to \mathbb {R}^d\) with a finite measure \(\rho \): \((k * \rho )(x) := \int k(x - y)\, \mathrm{d}\rho (y)\). The Vlasov force is \(\nabla W * \rho _t\), with \(\rho _t\) the spatial marginal of \(f_t\).

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 \(t \ge 0\).

The empirical measure of a configuration \((X,V) \in (\mathbb {R}^d \times \mathbb {R}^d)^N\):

a probability measure on phase space.

The curve of empirical measures \(t \mapsto \mu ^N_t\) carried by a solution of Newton’s equations.

The canonical Dyson-series solution \(M(t)\) of the matrix variational equation \(\dot M = A(s)\, M\), \(M(0) = I\), continuous jointly in time and the parameter. It provides the candidate derivative of the flow with respect to its initial point — constructed explicitly rather than chosen, so that its regularity in the parameter is provable.

The mean-field Hamiltonian of \(N\) identical unit-mass particles,

The \(1/N\) scaling keeps the kinetic and potential energies of the same order as \(N \to \infty \).

A Lagrangian solution is a weak solution \(f\) for which there exists a characteristic flow \(\Phi \) driven by its own marginal \(\rho ^f\) with \(f_t = (\Phi _t)_\# f_0\) for every \(t\) — the solution is transported by the flow it generates. Every Lagrangian solution is weak by definition; the converse is the superposition principle (Theorem 1.7 of the paper, the final chapter).

The window-localized Lagrangian-solution predicate on \([0,T]\): a weak solution on the window together with a characteristic-flow witness transporting the initial datum.

Newton’s equations for \(N\) particles under the mean-field force:

A solution is a curve \((X,V) : \mathbb {R} \to (\mathbb {R}^d \times \mathbb {R}^d)^N\) satisfying both systems at every time.

The phase-space velocity field \(b(t,x,v) = \big(v,\; -(\nabla W * \rho _t)(x)\big)\) generating the characteristic flow. Under Assumption 1 it is Lipschitz in the phase variable, with constant governed by \(\max (1, L)\).

A weak solution of the Vlasov equation: a curve \(t \mapsto f_t\) of measures such that for every \(\varphi \in C_c^\infty (\mathbb {R}^d \times \mathbb {R}^d)\) the map \(t \mapsto \langle f_t, \varphi \rangle \) is differentiable at every \(t \in \mathbb {R}\) with

The predicate states exactly this identity; membership in \(\mathcal{P}_1\), windows, and continuity in time enter as explicit hypotheses of the theorems that consume it (cf. the definition in Section 1.3 of the paper, whose prose imposes them up front).

The window-localized weak-solution predicate: the distributional Vlasov equation holds on the open interval \((0,T)\). The predicate carries only the PDE; continuity in time, where needed, is a separate hypothesis (as in Theorem 1.7 of the paper). Local existence lives on windows; the global theorem glues them.

A coupling \(\pi \) of \(\mu \) and \(\nu \): a probability measure on the product space whose marginals are \(\mu \) and \(\nu \). The set of couplings is the domain of the primal face of \(W_1\).

The hard direction of duality: the optimal coupling cost is at most the dual value.

For probability measures \(\mu , \nu \) with finite first moment,

the dual face of Definition 1.4 of the paper. The primal (coupling) face and the equality between them are the duality theorem closing this chapter. Neither optimum is attained anywhere in the development — every bound is \(\varepsilon \)-optimal.

Symmetry of \(W_1\).

\(W_1\) expressed via the infimum over couplings — the primal face of Definition 1.4.

At the metric ground cost, for probability measures with finite first moment on a Polish-type space, the dual and primal faces agree: \(W_1(\mu ,\nu ) = \inf _\pi \int \mathrm{dist}\, \mathrm{d}\pi \). This is the bridge on which the Dobrushin argument crosses between the two faces of Definition 1.4.

Finiteness: if \(\mu \) and \(\nu \) have finite first moment then \(W_1(\mu ,\nu ) {\lt} \infty \).

The easy direction of duality: the dual value is at most any coupling cost.

Non-expansion of \(W_1\) under a \(1\)-Lipschitz pushforward — the workhorse estimate for transporting mass along flows.

\(W_1(\mu ,\mu ) = 0\).

The triangle inequality for \(W_1\).

The cost-generic Wasserstein functional \(W_c(\mu ,\nu ) := \sup \{ \int \varphi \, \mathrm{d}\mu - \int \psi \, \mathrm{d}\nu \} \) over the \(c\)-admissible dual pairs (\(\varphi (x) - \psi (y) \le c(x,y)\)). The \(W_1\) distance is its specialization to the metric ground cost \(c = \mathrm{dist}\).

Symmetry: \(W_c(\mu ,\nu ) = W_c(\nu ,\mu )\) for a symmetric cost.

The Monge–Kantorovich (primal) cost: \(\inf _\pi \int \mathrm{dist}\, \mathrm{d}\pi \) over couplings \(\pi \) of \(\mu \) and \(\nu \).

Non-expansion of \(W_c\) under a \(1\)-Lipschitz pushforward: \(W_c(T_\# \mu , T_\# \nu ) \le W_c(\mu ,\nu )\).

\(W_c(\mu ,\mu ) = 0\).

The triangle inequality for \(W_c\).

Along any Newton solution, for every test function \(\varphi \in C_c^\infty (\mathbb {R}^d \times \mathbb {R}^d)\) the pairing \(t \mapsto \langle \mu ^N_t, \varphi \rangle \) is differentiable and

where the remainder is the diagonal correction \(R_N = \frac{1}{N^2} \sum _i \nabla W(0) \cdot \nabla _v \varphi (x_i, v_i)\), of size \(|R_N| \le \frac{1}{N} \| \nabla W\| _\infty \| \nabla _v \varphi \| _\infty \).

The flow’s Fréchet derivative in its initial point is the Dyson-series fundamental matrix \(M(t)\) evaluated along the trajectory.

The two-time flow \((s,z) \mapsto \Phi _{s \to t}(z)\) and its inverse are jointly \(C^1\) on the window — the jointly-smooth change of variables used to transport test functions along characteristics.

\(\dfrac {\mathrm{d}}{\mathrm{d}s} \int \psi _s\, \mathrm{d}f_s = 0\) for \(s \in (0,T)\): differentiating through both the moving test function and the moving measure, the transport identity and the weak PDE cancel exactly.

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:

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

\(\int \varphi \, \mathrm{d}f_T = \int \varphi \, \mathrm{d}g_T\) for every \(C_c^\infty \) test function \(\varphi \), where \(g = (\Phi )_\# f_0\) is the pushforward of the initial datum along the frozen-field flow. Hence \(f_T = g_T\) by measure extensionality.

The weak-solution test class is enlarged from \(C_c^\infty \) to \(C^1_c\). Needed because the transported test \(\psi _s = \varphi \circ \Phi _{s \to t}\) is only \(C^1_c\) — the flow is once, not infinitely, differentiable in space.

The dual transport identity: the transported test \(\psi _s = \varphi \circ \Phi _{s \to t}\) satisfies \(\partial _s \psi _s + D\psi _s \cdot b_s = 0\) along the flow — the dual of the pushforward chain rule.

The map \(z \mapsto \Phi _t(z)\) is Fréchet differentiable in the initial point, with derivative solving the linearized (variational) equation \(\dot M = (D_z b)\, M\), \(M(0)=I\), and depending continuously on \(z\). The research-grade core of the bridge.

Let \(W\) satisfy Assumption 1 and let \(f_0\) be a probability measure with finite first moment. The forward Cauchy problem is well-posed: there exists a single curve \(f : [0,\infty ) \to \mathcal{P}_1(\mathbb {R}^d \times \mathbb {R}^d)\) with datum \(f(0) = f_0\) and finite first moment at every \(t \ge 0\), which is a Lagrangian solution on every window \([0,T]\); and on each window it is the unique Lagrangian solution with that datum.

Let \(W\) satisfy Assumption 2 and let \(T {\gt} 0\) satisfy \(L\, T^2 {\lt} 1\). Every weak solution on \([0,T]\) whose first moments are uniformly bounded on the window and whose force field is regular — \((\nabla W * \rho _t)(x)\) continuous in \(t\) for each \(x\), with jointly continuous spatial derivative — is Lagrangian on \([0,T]\): it coincides with the pushforward of its initial datum along the characteristic flow it generates. Weak solutions cannot branch away from the Lagrangian one on the window.