Plain-English statement
A generator Poincare inequality controls every observable's variance by its Dirichlet energy.
Mathematical statement
Var_mu(f) <= C integral f(-L)f dmu for every admissible f.
Intuition
The inequality is the inverse spectral-gap bound on the nonconstant part of an observable.
Conditions
- mu is a probability measure
- C is positive
- mean, variance, and generator energy are finite
Why these conditions cannot be dropped
- probability normalization identifies centering
- finite form-domain data prevents totalized integrals from asserting a false comparison
Proof route
- this is the exact source definition
- variance decay is a separate semigroup theorem
Lean interface notes
- dirichletForm is minus integral f*L g
- the gradient specialization is not built into this predicate
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Lean statement
def SatisfiesPoincare
(mu : Measure E) (generator : (E → ℝ) → E → ℝ)
(C : ℝ) : Prop :=
IsProbabilityMeasure mu ∧ 0 < C ∧
∀ f : E → ℝ, PoincareAdmissible mu generator f →
variance mu f ≤ C * dirichletForm mu generator f f
/-- Relative entropy of a density `rho` with respect to its reference
probability measure. Mathlib's totalized `Real.log 0 = 0` gives the standard
zero-density convention in the product `rho * log rho`. -/Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Generator.lean:42published source at 7bcd37294df1