isInvariantOn_of_integral_generator_eq_zero

Plain-English statement

If a nonnegative-time semigroup preserves its generator domain, its integrated pairing has the declared right derivative, and the generator has zero mean throughout that domain, then the measure is invariant on that domain.

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 d/dt integral P_t f dmu = integral A(P_t f) dmu = 0 and P_0 f = f, then integral P_t f dmu = integral f dmu for t >= 0.

Intuition

The expected value along the semigroup orbit has zero right derivative on every finite time interval, so the mean-value theorem makes it constant.

Conditions

Identity and semigroup laws hold for nonnegative times.
The semigroup orbit remains in the declared generator domain.
The integral pairing is continuous and has the stated right derivative.
The integrated generator vanishes on the entire domain.

Why these conditions cannot be dropped

Core-level mean-zero is insufficient when the semigroup orbit leaves that core.
A formal generator formula does not justify differentiating an integral pairing.
Right derivatives respect the one-sided time domain of a Markov semigroup.

Proof route

Use domain preservation to apply generator mean-zero at every orbit point.
Rewrite every supplied right derivative of the pairing as zero.
Apply Mathlib's constant_of_has_deriv_right_zero on [0,t].
Use the time-zero identity to obtain invariance.

Lean interface notes

Time is represented by real numbers, while all semigroup laws are guarded by nonnegativity.
The theorem proves invariance only on the explicit domain/test class.
Concrete Langevin semigroup construction and the C_c^2 domain-extension step remain open.

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Lean statement

theorem isInvariantOn_of_integral_generator_eq_zero
    {E : Type*} [MeasurableSpace E]
    {P : ℝ → (E → ℝ) → E → ℝ}
    {generator : (E → ℝ) → E → ℝ}
    {domain : Set (E → ℝ)} {μ : Measure E}
    (hsemigroup : IntegratedSemigroupGeneratorContract P generator domain μ)
    (hgenerator_zero : ∀ f ∈ domain, ∫ x, generator f x ∂μ = 0) :
    IsInvariantOn P μ domain := by
  intro t ht f hf
  have hconstant := constant_of_has_deriv_right_zero
    (hsemigroup.pairing_continuousOn t ht f hf) (fun s hs => by
      have hs0 : 0 ≤ s := hs.1
      simpa [hgenerator_zero (P s f)
        (hsemigroup.orbit_mem_domain s hs0 f hf)] using
        hsemigroup.pairing_hasDerivWithinAt s hs0 f hf)
  have htmem : t ∈ Set.Icc (0 : ℝ) t := ⟨ht, le_rfl⟩
  simpa [hsemigroup.map_zero f] using hconstant t htmem

/-- Move a supplied sample-space generator derivative to a named law path.

In SDE applications, `hderiv` is usually the Ito-generator derivative for a
test function composed with a process, while `hDrift` and `hDiffusion` identify
the sample drift and diffusion-generator terms with law-level weak-test
integrals.  The lemma proves only the reusable rewrite; it does not construct
the process, conditional drift, or Ito theorem.
-/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakGenerator.lean:58published source at 7bcd37294df1