DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite-action continuity equation in Euclidean space

Definition

Fix $T > 0$ and an integer $d \geq 1$ . Let $t \mapsto μ_{t}$ be a narrowly continuous curve in $P_{2} (R^{d})$ (so $t \mapsto \int f d μ_{t}$ is continuous for every bounded continuous $f$ ), let $M$ be its space-time measure $M (d t, d x) = d t μ_{t} (d x)$ , and let $v : [0, T] \times R^{d} \to R^{d}$ be Borel. Its squared action is $A (μ, v) = \int ∣ v (t, x) ∣^{2} d M (t, x)$ . If $A (μ, v) < \infty$ , say that $v$ solves the continuity equation for $μ$ when $\int_{0}^{T} ⁣ \int_{R^{d}} (\partial_{t} φ (t, x) + \nabla_{x} φ (t, x) \cdot v (t, x)) d μ_{t} (x) d t = 0$ for every $φ \in C_{c}^{\infty} ((0, T) \times R^{d})$ . This is the distributional meaning of $\partial_{t} μ_{t} + \nabla \cdot (v_{t} μ_{t}) = 0$ ; the continuous representative fixes the endpoint laws.

Definition references

def wasserstein two space and couplings

lem spacetime law of a narrow probability curve

Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Theorem 1.1.2 and Chapter 8

Filippo Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, §§5.1–5.3

Depends on

Used by

Dependency tree · two levels

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Sources

Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Theorem 1.1.2 and Chapter 8 (standard reference, not scraped)
Filippo Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, §§5.1–5.3 (standard reference, not scraped)