weakGeneratorFromSampleDerivative

Plain-English statement

A derivative identity for expectations on the sample space can be rewritten as a weak generator identity for the corresponding time-indexed laws.

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 rho_s is the pushforward law of X_s, the sample expectation has the stated derivative, and drift/diffusion expectations equal law integrals, then the law integral has the corresponding derivative.

Intuition

The theorem is a transport and rewrite layer. Ito's formula or another stochastic argument must first provide the sample-space derivative.

Conditions

rho_s is exactly the pushforward of X_s for every s.
X_s and the test function have the stated almost-everywhere measurability.
The sample derivative and drift/diffusion integral identifications are supplied.

Why these conditions cannot be dropped

Integrals cannot be transported through Measure.map without measurability.
The theorem does not infer a drift or diffusion field from process notation.

Proof route

Apply the generic law-integral derivative transport theorem.
Obtain the derivative with sample-space drift and diffusion terms.
Rewrite those terms by the supplied law-level identities.

Lean interface notes

HasDerivAt gives a one-dimensional time derivative.
AEMeasurable and AEStronglyMeasurable are measure-relative conditions.
simpa only performs the final named-integral rewrite.

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Lean statement

theorem weakGeneratorFromSampleDerivative {Ω E : Type*}
    [MeasurableSpace Ω] [MeasurableSpace E]
    {P : Measure Ω} {X : ℝ → Ω → E} {ρ : ℝ → Measure E}
    {φ driftTerm diffusionTerm : E → ℝ}
    {sampleDrift sampleDiffusion : Ω → ℝ} {s0 σ : ℝ}
    (hρ : ∀ s, ρ s = Measure.map (X s) P)
    (hX : ∀ s, AEMeasurable (X s) P)
    (hφ : ∀ s, AEStronglyMeasurable φ (ρ s))
    (hderiv :
      HasDerivAt (fun s => ∫ ω, φ (X s ω) ∂P)
        ((∫ ω, sampleDrift ω ∂P) +
          (σ ^ 2 / 2) * (∫ ω, sampleDiffusion ω ∂P)) s0)
    (hDrift :
      (∫ ω, sampleDrift ω ∂P) = ∫ x, driftTerm x ∂ρ s0)
    (hDiffusion :
      (∫ ω, sampleDiffusion ω ∂P) = ∫ x, diffusionTerm x ∂ρ s0) :
    HasDerivAt (fun s => ∫ x, φ x ∂ρ s)
      ((∫ x, driftTerm x ∂ρ s0) +
        (σ ^ 2 / 2) * (∫ x, diffusionTerm x ∂ρ s0)) s0 := by
  have hbase :
      HasDerivAt (fun s => ∫ x, φ x ∂ρ s)
        ((∫ ω, sampleDrift ω ∂P) +
          (σ ^ 2 / 2) * (∫ ω, sampleDiffusion ω ∂P)) s0 :=
    AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndSample
      (P := P) (X := X) (ρ := ρ) (φ := φ)
      hρ hX hφ hderiv
  simpa [hDrift, hDiffusion] using hbase

end WeakGenerator
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakGenerator.lean:84published source at 7bcd37294df1