SatisfiesLogSobolev · Samplinglib

Plain-English statement

A generator log-Sobolev inequality controls density entropy by entropy production.

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

KL(rho mu || mu) <= (C/2) E(rho,log rho) for every admissible density rho.

Intuition

Entropy production rules out densities that retain large information while dissipating too slowly.

Conditions

mu is a probability measure
rho is nonnegative with unit mass
entropy and generator energy are integrable
C is positive

Why these conditions cannot be dropped

unit mass makes rho a probability density
finite integrals give mathematical meaning to both sides

Proof route

this is the exact source definition
KL decay and the Langevin Fisher-information form are separate routes

Lean interface notes

Real.log is totalized and rho*log(rho) implements the zero-density convention
all density conditions are bundled in LogSobolevAdmissible

Lean learning studio · mathematics → formal proof

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Lean statement

def SatisfiesLogSobolev
    (mu : Measure E) (generator : (E → ℝ) → E → ℝ)
    (C : ℝ) : Prop :=
  IsProbabilityMeasure mu ∧ 0 < C ∧
    ∀ rho : E → ℝ, LogSobolevAdmissible mu generator rho →
      densityEntropy mu rho ≤
        (C / 2) * dirichletForm mu generator rho (fun x => Real.log (rho x))

end Generator
end FunctionalInequalities
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Generator.lean:67published source at 7bcd37294df1