integral_expNeg_langevinGenerator_rhs_eq_zero_of_contDiff_of_hasCompactSupport

Plain-English statement

For every compactly supported twice continuously differentiable test function, the unnormalized Gibbs-weighted Langevin generator has integral zero.

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

If V is C1 and f is C2 with compact support, then integral exp(-V(x)) (Delta f(x) - <grad V(x), grad f(x)>) dx = 0.

Intuition

The displayed integrand is the divergence of exp(-V) grad f. Since f is compactly supported, its derivative and the weighted vector field vanish near infinity, so no boundary flux remains.

Conditions

The potential V is globally C1.
The test function f is globally C2.
The test function f has compact topological support.

Why these conditions cannot be dropped

C1 regularity gives the Gibbs-weight chain rule and a continuous gradient of V on the relevant compact region.
C2 regularity gives the differentiable gradient field and Laplacian appearing in the generator.
Compact support removes the boundary term without requiring finite total Gibbs mass or a globally bounded potential gradient.

Proof route

Build the raw finite-Pi field exp(-V) times the coordinate derivatives of f.
Prove that field is C1 and supported in the transported topological support of f.
Apply the generic compact-support whole-space divergence theorem.
Use the compiled trace identity to identify the divergence with the Gibbs-weighted Langevin display pointwise.
Transport the integral through the volume-preserving PiLp equivalence.

Lean interface notes

The result is the analytic core integral Lf d pi = 0 before normalization; multiplying by the finite normalizing constant is downstream algebra.
It proves no closed-generator membership or derivative of a semigroup pairing.
The invariant-law milestone remains red until those operator and semigroup contracts compile.

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Lean statement

theorem integral_expNeg_langevinGenerator_rhs_eq_zero_of_contDiff_of_hasCompactSupport
    {n : ℕ}
    {V f : EuclideanSpace ℝ (Fin (n + 1)) → ℝ}
    (hV : ContDiff ℝ 1 V)
    (hf : ContDiff ℝ 2 f)
    (hf_support : HasCompactSupport f) :
    ∫ y : EuclideanSpace ℝ (Fin (n + 1)),
        Real.exp (-V y) *
          (Laplacian.laplacian f y - inner ℝ (gradient V y) (gradient f y)) = 0 := by
  let e : EuclideanSpace ℝ (Fin (n + 1)) ≃L[ℝ] (Fin (n + 1) → ℝ) :=
    PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin (n + 1) => ℝ)
  let F : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ :=
    fun z => fun i =>
      Real.exp (-V (e.symm z)) *
        fderiv ℝ f (e.symm z) (EuclideanSpace.single i (1 : ℝ))
  have hF : ContDiff ℝ 1 F := by
    apply contDiff_pi.2
    intro i
    exact (hV.comp e.symm.contDiff).neg.exp.mul
      (((hf.fderiv_right (m := 1) (by norm_num)).comp e.symm.contDiff).clm_apply
        contDiff_const)
  have hF_support : HasCompactSupport F := by
    refine HasCompactSupport.of_support_subset_isCompact
      (hf_support.isCompact.image e.continuous) ?_
    intro x hx
    have hxe : e.symm x ∈ tsupport f := by
      by_contra hxe
      have hzero : fderiv ℝ f (e.symm x) = 0 :=
        fderiv_of_notMem_tsupport ℝ hxe
      exact hx (by
        funext i
        simp [F, hzero])
    exact ⟨e.symm x, hxe, e.apply_symm_apply x⟩
  have hdiv :=
    _root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_wrapped_eq_zero_of_contDiff_of_hasCompactSupport
      F hF hF_support
  have hraw :
      ∫ x : Fin (n + 1) → ℝ,
          Real.exp (-V (e.symm x)) *
            (Laplacian.laplacian f (e.symm x) -
              inner ℝ (gradient V (e.symm x)) (gradient f (e.symm x))) = 0 := by
    calc
      ∫ x : Fin (n + 1) → ℝ,
          Real.exp (-V (e.symm x)) *
            (Laplacian.laplacian f (e.symm x) -
              inner ℝ (gradient V (e.symm x)) (gradient f (e.symm x))) =
          ∫ x : Fin (n + 1) → ℝ,
            _root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence
              (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
                (WithLp.toLp 2 (F (WithLp.ofLp y)) :
                  EuclideanSpace ℝ (Fin (n + 1))))
              (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) := by
        apply integral_congr_ae
        exact Filter.Eventually.of_forall fun x => by
          have htrace :=
            trace_expNeg_fderivCoordinateField_langevinGenerator_display_of_hasFDerivAt
              (V := V) (f := f) (x := x) (F' := fderiv ℝ F x)
              (by simpa [F, e] using (hF.differentiable one_ne_zero x).hasFDerivAt)
              (hV.differentiable one_ne_zero _)
              ((hf.fderiv_right (m := 1) (by norm_num)).differentiable one_ne_zero _)
              (hf.differentiable (by norm_num) _)
          have hwrapped :=
            _root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt
              (ι := Fin (n + 1)) (hF.differentiable one_ne_zero x).hasFDerivAt
          change
            Real.exp (-V (e.symm x)) *
                (Laplacian.laplacian f (e.symm x) -
                  inner ℝ (gradient V (e.symm x)) (gradient f (e.symm x))) =
              _root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence
                (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
                  (WithLp.toLp 2 (F (WithLp.ofLp y)) :
                    EuclideanSpace ℝ (Fin (n + 1))))
                (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))
          calc
            _ = ∑ i, fderiv ℝ F x (Pi.single i (1 : ℝ)) i := by
              simpa [e] using htrace.symm
            _ = _ := hwrapped.symm
      _ = 0 := by simpa [F, e] using hdiv
  rw [← (PiLp.volume_preserving_toLp (Fin (n + 1))).integral_comp
    (MeasurableEquiv.toLp 2 _).measurableEmbedding]
  simpa [e] using hraw

/-- Whole-space integrability of the Gibbs-weighted coordinate derivative
field from finiteness of the unnormalized Gibbs mass and a uniform operator
norm bound on the test-function derivative.

This theorem proves only source-field integrability.  It does not prove a
cutoff main-term limit, weighted integration by parts, stationarity, or an
invariant Gibbs law. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1164published source at 7bcd37294df1