fundamental_integration_by_parts

Plain-English statement

Stationarity and reversibility identify the negative-generator pairing with integrated carré du champ.

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

integral f(-L)g dmu = integral (-L)f g dmu = integral Gamma(f,g) dmu.

Intuition

Stationarity kills L(fg), while reversibility makes the two cross terms in Gamma equal.

Conditions

integrability of every expanded term
stationarity on f*g
generator symmetry for f and g

Why these conditions cannot be dropped

Bochner integrals are totalized outside their domain
the algebra alone cannot create stationarity or reversibility

Proof route

expand Gamma
integrate subtraction term by term
use stationarity
use symmetry
finish scalar algebra

Lean interface notes

uses Generator.dirichletForm
all source-domain conditions are explicit theorem arguments

Lean learning studio · mathematics → formal proof

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Lean statement

theorem fundamental_integration_by_parts
    [MeasurableSpace X]
    (mu : Measure X) (generator : (X → ℝ) →ₗ[ℝ] (X → ℝ))
    (f g : X → ℝ)
    (hLfg : Integrable (generator (f * g)) mu)
    (hfLg : Integrable (fun x => f x * generator g x) mu)
    (hgLf : Integrable (fun x => g x * generator f x) mu)
    (hstationary : (∫ x, generator (f * g) x ∂mu) = 0)
    (hsymmetric :
      (∫ x, f x * generator g x ∂mu) =
        ∫ x, g x * generator f x ∂mu) :
    FunctionalInequalities.Generator.dirichletForm mu generator f g =
        FunctionalInequalities.Generator.dirichletForm mu generator g f ∧
      FunctionalInequalities.Generator.dirichletForm mu generator f g =
        ∫ x, carreDuChamp generator f g x ∂mu := by
  have hsub : Integrable
      (fun x => generator (f * g) x - f x * generator g x) mu :=
    hLfg.sub hfLg
  have hgammaIntegral :
      (∫ x, carreDuChamp generator f g x ∂mu) =
        (2 : ℝ)⁻¹ *
          ((∫ x, generator (f * g) x ∂mu) -
            (∫ x, f x * generator g x ∂mu) -
            ∫ x, g x * generator f x ∂mu) := by
    change
      (∫ x, (2 : ℝ)⁻¹ *
        (generator (f * g) x - f x * generator g x -
          g x * generator f x) ∂mu) = _
    rw [integral_const_mul,
      integral_sub hsub hgLf, integral_sub hLfg hfLg]
  constructor
  · simp only [FunctionalInequalities.Generator.dirichletForm]
    rw [hsymmetric]
  · simp only [FunctionalInequalities.Generator.dirichletForm]
    rw [hgammaIntegral, hstationary, hsymmetric]
    ring

/-- Chewi Corollary 1.2.15: the negative reversible generator has a
nonnegative quadratic form once Gamma is pointwise nonnegative. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:153published source at 7bcd37294df1