langevinCarreDuChamp_eq_inner

Plain-English statement

For Langevin diffusion, the generator product defect is exactly the inner product of the two gradients.

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

Gamma_L(f,g)(x) = inner(gradient f(x), gradient g(x)); hence Gamma_L(f,f)(x) = norm(gradient f(x)) squared.

Intuition

The Laplacian contributes two cross derivatives, while every potential-drift product term cancels. The remaining local energy is the gradient pairing.

Conditions

a finite-dimensional Euclidean state space
both observables are globally C2
L is the displayed Laplacian-minus-gradient-drift operator

Why these conditions cannot be dropped

C2 regularity justifies the second-order product rule
finite dimension supports the orthonormal-basis trace defining the Laplacian
the displayed operator is distinct from its later closed semigroup-generator realization

Proof route

differentiate the first product derivative to obtain the diagonal second derivative formula
sum over a standard orthonormal basis to prove the Laplacian product rule
rewrite the gradient of a product
expand the three Langevin operator terms and cancel the drift

Lean interface notes

iteratedFDeriv_two_apply exposes the second Frechet derivative
OrthonormalBasis.sum_inner_mul_inner identifies the cross-term sum
the proof does not assume the desired Gamma identity or a generator domain

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Lean statement

theorem langevinCarreDuChamp_eq_inner
    {n : ℕ}
    (V f g : EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
    (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g)
    (x : EuclideanSpace ℝ (Fin (n + 1))) :
    (2 : ℝ)⁻¹ *
        (LangevinGenerator.operator V (f * g) x -
          f x * LangevinGenerator.operator V g x -
          g x * LangevinGenerator.operator V f x) =
      inner ℝ (gradient f x) (gradient g x) := by
  have hf1 : Differentiable ℝ f := hf.differentiable (by norm_num)
  have hg1 : Differentiable ℝ g := hg.differentiable (by norm_num)
  simp only [LangevinGenerator.operator]
  rw [laplacian_mul f g hf hg x, gradient_mul f g hf1 hg1 x]
  simp only [inner_add_right, real_inner_smul_right]
  ring

/-- Diagonal form of Chewi Example 1.2.17. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinCarreDuChamp.lean:114published source at 7bcd37294df1