Plain-English statement
If carré du champ is nonnegative, then the negative generator has nonnegative quadratic energy.
Mathematical statement
Gamma(f,f)>=0 implies integral f(-L)f dmu>=0 under stationarity and integrability.
Intuition
The integration-by-parts identity turns generator energy into an integral of a pointwise nonnegative field.
Conditions
- stationarity
- integrability
- pointwise Gamma nonnegativity
Why these conditions cannot be dropped
- Gamma positivity is the Markov-semigroup input from Lemma 1.2.13
- stationarity connects local generator algebra to the reference law
Proof route
- specialize fundamental integration by parts to g=f
- apply integral_nonneg_of_ae
Lean interface notes
- does not assume the target quadratic inequality
- Lemma 1.2.13 remains an independent source item
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Lean statement
theorem negativeGenerator_quadratic_nonneg
[MeasurableSpace X]
(mu : Measure X) (generator : (X → ℝ) →ₗ[ℝ] (X → ℝ))
(f : X → ℝ)
(hLf2 : Integrable (generator (f * f)) mu)
(hfLf : Integrable (fun x => f x * generator f x) mu)
(hstationary : (∫ x, generator (f * f) x ∂mu) = 0)
(hgamma : ∀ x, 0 ≤ carreDuChamp generator f f x) :
0 ≤ FunctionalInequalities.Generator.dirichletForm mu generator f f := by
have hibp := fundamental_integration_by_parts mu generator f f
hLf2 hfLf hfLf hstationary rfl
rw [hibp.2]
exact integral_nonneg_of_ae (Filter.Eventually.of_forall hgamma)
end CarreDuChamp
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheoryOpen AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:192published source at 7bcd37294df1