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Probability W1 stability under bounded force defects

Statement

Let $K$ have two-variable global Lipschitz constant $L$ as in Globally Lipschitz mean-field interaction kernel. Let $μ_{t} = (X_{t})_{#} μ_{0}$ and $ν_{t} = (Y_{t})_{#} ν_{0}$ be probability curves from finite-first-moment initial laws, with absolutely continuous characteristic paths satisfying ${\dot{X}}_{t} (x) = \int K (X_{t} (x), X_{t} (z)) d μ_{0} (z) + r_{μ} (t, x), {\dot{Y}}_{t} (y) = \int K (Y_{t} (y), Y_{t} (w)) d ν_{0} (w) + r_{ν} (t, y)$ for almost every time and almost every label, with $X_{0} (x) = x$ and $Y_{0} (y) = y$ . Assume additionally the weighted path integrability conditions $\int \sup_{s \leq T} ∣ X_{s} ∣ d μ_{0} < \infty$ and $\int \sup_{s \leq T} ∣ Y_{s} ∣ d ν_{0} < \infty$ . Assume the measurable label defects satisfy $∣ r_{μ} ∣ \leq η_{μ}$ and $∣ r_{ν} ∣ \leq η_{ν}$ on $[0, T]$ . For every prescribed initial coupling $π_{0}$ of cost $D_{0}$ , its transport by the two characteristic paths has cost at time $t$ at most $e^{2 L t} (D_{0} + (η_{μ} + η_{ν}) t) .$ Consequently $W_{1} (μ_{t}, ν_{t}) \leq e^{2 L t} (W_{1} (μ_{0}, ν_{0}) + (η_{μ} + η_{ν}) t)$ . The estimate is first proved for each coupling; no optimal plan is assumed.

Facts & Assumptions

Given: Two characteristic flows for the same kernel, finite-first-moment probability initial laws, bounded label defects, and a fixed initial coupling.

[F1]

The two-variable kernel difference is at most $L (∣ x - x^{'} ∣ + ∣ y - y^{'} ∣)$ (Globally Lipschitz mean-field interaction kernel).

[F2]

The jointly Borel path and pushforward conventions are those in Characteristic solution of a probability mean-field equation; the force-defect equations are assumptions of the Statement, replacing the unforced equation in that definition; the coordinatewise Borel pair maps define pushforwards in the Euclidean product Borel space (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}).

[F3]

Product coupling shows the initial $W_{1}$ infimum is finite (The coupling infimum is finite for finite-first-moment probabilities).

[F4]

Gronwall gives $u (t) \leq a (t) e^{b t}$ when $u (t) \leq a (t) + b \int_{0}^{t} u (s) d s$ and $a$ is nondecreasing (Gronwall's integral inequality with variable and constant coefficients).

[F5]

Tonelli and Fubini justify integration of the coupled force bounds over the two label measures and time (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product).

[F6]

Dominated convergence gives continuity of the transported cost from the integrable path suprema (Dominated convergence).

Proof

1.1F2F3F6construct

Transport the prescribed coupling by the two flows: $π_{t} = (X_{t}, Y_{t})_{#} π_{0} .$ Its marginals are $μ_{t}, ν_{t}$ . Put $D (t) = \int ∣ X_{t} (x) - Y_{t} (y) ∣ d π_{0} (x, y)$ . This is finite and continuous by the first moments, integrable path suprema, and [F6].

2.1F1F2F4F5F6step 1.1

Subtract the two characteristic equations and apply [F1, F5]. Integrating over the outer coupling, whose mass is one, gives $D (t) \leq D_{0} + 2 L \int_{0}^{t} D (s) d s + (η_{μ} + η_{ν}) t .$ The inhomogeneous term is nondecreasing, so [F4] yields $D (t) \leq e^{2 L t} (D_{0} + (η_{μ} + η_{ν}) t)$ . Since $π_{t}$ is a coupling, this is the asserted transported-coupling bound.

3.1F3step 2.1∎

The bound in step 2.1 holds for every initial coupling. Taking the infimum over their costs gives the claimed $W_{1}$ inequality by [F3] and the definition of $W_{1}$ .

Depends on

Used by

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Sources

François Golse, Mean-Field Limits in Statistical Dynamics, Lecture 1 (standard reference, not scraped)
Simon Mayer, The Vlasov Equation and Mean Field Limit of a Particle System (standard reference, not scraped)