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Exact weak mean-field identity for empirical particle measures

Statement

Let $x_{1}, \dots, x_{N}$ solve the equal-probability system of Equal-probability mean-field and weighted signed vortex particle systems on $[0, T]$ , and put $μ_{N} (t) = N^{- 1} \sum_{i} δ_{x_{i} (t)}$ . Then for every $φ \in C_{c}^{1} (R^{d})$ , $\int φ d μ_{N} (t) - \int φ d μ_{N} (0) = \int_{0}^{t} \int \nabla φ (x) \cdot (\int K (x, y) d μ_{N} (s) (y)) d μ_{N} (s) (x) d s .$ Thus the empirical measure satisfies the same weak transport equation exactly, with no particle quadrature error in this identity.

Facts & Assumptions

Given: A classical particle solution, the particle equations, and a test function $φ \in C_{c}^{1} (R^{d})$ .

[F1]

The equal-probability particle system is ${\dot{x}}_{i} = N^{- 1} \sum_{j} K (x_{i}, x_{j})$ and its empirical law is $N^{- 1} \sum_{i} δ_{x_{i}}$ (Equal-probability mean-field and weighted signed vortex particle systems).

[F2]

The chain rule differentiates $t \mapsto φ (x_{i} (t))$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

Proof

1.1F1F2algebra

By [F2] and [F1], $\frac{d}{d t} \frac{1}{N} \sum_{i} φ (x_{i} (t)) = \frac{1}{N^{2}} \sum_{i, j} \nabla φ (x_{i} (t)) \cdot K (x_{i} (t), x_{j} (t)) .$ The double sum is finite, so integration in time gives the left side of the Statement as its time integral.

2.1F1step 1.1algebra∎

Expanding both empirical integrals on the right of the Statement gives exactly the same finite double sum as in step 1.1. This proves the identity for every $t$ and includes repeated particle positions without a distinctness assumption.

Depends on

Used by

A globally Lipschitz alignment model with exact mean trajectory Example

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

François Golse, Mean-Field Limits in Statistical Dynamics, Lecture 1 (standard reference, not scraped)