Plain-English statement
Under the entropy-dissipation identity, the log-Sobolev inequality is equivalent to shifted exponential KL decay.
Mathematical statement
KL(s)≤(C/2)FI(s) for every s iff KL(s+t)≤KL(s)exp(-2t/C), assuming KL'=-FI.
Intuition
LSI converts Fisher information into an entropy decay rate; conversely, differentiating shifted KL decay recovers the same Fisher-information lower bound.
Conditions
- C>0
- KL has exact right derivative -FI
- the decay estimate is valid after every starting time
Why these conditions cannot be dropped
- entropy differentiation requires a concrete regular density flow
- the factor C/2 determines rate 2/C
Proof route
- apply shifted Gronwall to LSI coercivity
- apply one-sided Fermat to shifted KL decay
- rescale the recovered inequality by C/2
Lean interface notes
- this is a full scalar iff but not yet a theorem about a particular Langevin density
- density regularity, FI identification, and domain closure remain recorded obligations
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Lean statement
theorem chewi_theorem_1_2_26_scalar_equivalence
{C : ℝ} (hC : 0 < C) (curve : DissipationCurve 1) :
(∀ s : ℝ,
curve.energy s ≤ (C / 2) * curve.dissipation s) ↔
(∀ s t : ℝ, 0 ≤ t →
curve.energy (s + t) ≤
curve.energy s * Real.exp (-(2 / C) * t)) := by
constructor
· intro hLSI s t ht
have h := chewi_theorem_1_2_26_forward_from
hC curve hLSI (show s ≤ s + t by linarith)
simpa using h
· exact chewi_theorem_1_2_26_backward hC curve
end
end SemigroupDecay
end FunctionalInequalities
end TechnicalLemmas
end AutoSamplingTheoryOpen AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:359published source at 7bcd37294df1