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Lean gate passed 2026-08-19T06:32:39.895922+00:00 · 7bcd37294df1
Reviewed teaching declaration · langevin.finite-euclidean-generator-coordinate-display

finiteEuclidean_langevinGenerator_coordinateDisplay

compiled Samplinglib leaf Partial explicit smoke test

The finite-dimensional Langevin generator is displayed as a Laplacian term minus the inner product of the potential gradient with the test-function gradient.

Plain-English statement

The finite-dimensional Langevin generator is displayed as a Laplacian term minus the inner product of the potential gradient with the test-function gradient.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Mathematical statement

L phi = laplacian(phi) - <grad V, grad phi> in finite Euclidean coordinates, under the derivative hypotheses stated in Lean.

Intuition

This is the differential expression predicted by Ito's formula. It is an algebraic and calculus identity, not yet a theorem that a stochastic process has this closed generator.

Conditions

  • Finite-dimensional Euclidean coordinates are available.
  • The required first and second Frechet derivatives exist at the point.
  • Coordinate and basis representations use the same PiLp convention.

Why these conditions cannot be dropped

  • Mathlib's fderiv is totalized and has no derivative meaning without differentiability evidence.
  • A displayed differential expression does not identify a semigroup domain.

Proof route

  • Expand the basis form of gradient and Laplacian.
  • Rewrite Euclidean inner products as finite coordinate sums.
  • Assemble the coordinate generator identity.

Lean interface notes

  • EuclideanSpace is a PiLp wrapper, so WithLp coercions are visible.
  • HasFDerivAt hypotheses carry genuine differentiability evidence.
  • The theorem does not prove SDE existence, Ito's formula, or invariance.
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Lean statement

theorem finiteEuclidean_langevinGenerator_coordinateDisplay
    {ι : Type*} [Fintype ι] [DecidableEq ι]
    (V f : EuclideanSpace ℝ ι → ℝ) (x : EuclideanSpace ℝ ι) :
    Laplacian.laplacian f x - inner ℝ (gradient V x) (gradient f x) =
      (∑ i, iteratedFDeriv ℝ 2 f x
        ![(EuclideanSpace.single i (1 : ℝ)), (EuclideanSpace.single i (1 : ℝ))]) -
        ∑ i, (gradient V x) i * (gradient f x) i := by
  simpa [EuclideanSpace.basisFun_apply] using
    finiteEuclidean_langevinGenerator_basisDisplay V f x

/-- Supplied-hypothesis finite-coordinate handoff from weighted-divergence
algebra to the Mathlib pointwise expression `Δ f - <∇V, ∇f>`.

The hypotheses still provide the coordinate product-rule output, the
Gibbs-weight chain-rule output, and the coordinate divergence sum.  This theorem
only replaces the coordinate second-derivative and gradient-product sums by
Mathlib's `Laplacian.laplacian` and `gradient` display. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:257published source at 7bcd37294df1

Common pitfall. A wrapper or display identity is not a new analytic theorem merely because it has its own Lean name. Check the statement, hypotheses, and downstream consumers before interpreting its mathematical contribution.