energyLogSobolev_tilted

Plain-English statement

Independently compiled at proof commit 9ba98ec42d9fa6abd78c045315cdd8d795e37c3c; current five packet-bound audits ASTIS-RT-20261020-BoundedEntropy-1 through -5 accept only this finite-integral scope. Normalization, positive partition function and both density comparisons are derived. Geometric reference LSI and Sobolev extension remain separate.

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.

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Lean statement

theorem energyLogSobolev_tilted {μ : Measure X} [IsProbabilityMeasure μ]
    {Γ : (X → ℝ) → X → ℝ} {admissible : (X → ℝ) → Prop} {C l u : ℝ} {B : X → ℝ}
    (hC : 0 ≤ C) (hB : AEMeasurable B μ)
    (hl : ∀ᵐ x ∂μ, l ≤ B x) (hu : ∀ᵐ x ∂μ, B x ≤ u)
    (hLS : EnergyLogSobolev μ Γ admissible C) :
    EnergyLogSobolev (μ.tilted B) Γ admissible (C * Real.exp (u - l)) := by
  have hExp : Integrable (fun x => Real.exp (B x)) μ :=
    (integrable_const (Real.exp u)).mono'
      (Real.measurable_exp.comp_aemeasurable hB).aestronglyMeasurable (by
        filter_upwards [hu] with x hx
        simpa only [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)] using Real.exp_le_exp.mpr hx)
  have := isProbabilityMeasure_tilted hExp
  let Z := ∫ x, Real.exp (B x) ∂μ
  have hZ : 0 < Z := integral_exp_pos hExp
  have hupper : μ.tilted B ≤ ENNReal.ofReal (Real.exp u / Z) • μ := by
    rw [Measure.tilted, ← withDensity_const]
    apply withDensity_mono
    filter_upwards [hu] with x hx
    exact ENNReal.ofReal_le_ofReal (div_le_div_of_nonneg_right (Real.exp_le_exp.mpr hx) hZ.le)
  have hlower : μ ≤ ENNReal.ofReal (Z / Real.exp l) • μ.tilted B := by
    calc
      μ = μ.withDensity (fun _ => 1) := withDensity_one.symm
      _ ≤ μ.withDensity (fun x => ENNReal.ofReal (Z / Real.exp l) *
          ENNReal.ofReal (Real.exp (B x) / Z)) := by
        apply withDensity_mono
        filter_upwards [hl] with x hx
        rw [← ENNReal.ofReal_mul (by positivity)]
        have heq : (Z / Real.exp l) * (Real.exp (B x) / Z) =
            Real.exp (B x) / Real.exp l := by field_simp
        rw [heq, ← ENNReal.ofReal_one]
        exact ENNReal.ofReal_le_ofReal
          ((one_le_div (Real.exp_pos _)).2 (Real.exp_le_exp.mpr hx))
      _ = ENNReal.ofReal (Z / Real.exp l) • μ.tilted B :=
        withDensity_smul' _ _ ENNReal.ofReal_ne_top
  have hconst : C * (Z / Real.exp l) * (Real.exp u / Z) = C * Real.exp (u - l) := by
    rw [Real.exp_sub]
    field_simp [ne_of_gt hZ]
  rw [← hconst]
  exact energyLogSobolev_of_measure_comparison hC (by positivity) (by positivity)
    hlower hupper hLS

end
end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BoundedPerturbation

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/BoundedPerturbation.lean:160published source at ed709a7f320d

Proof architecture

A bounded measurable exponential tilt transfers the fixed-energy finite-integral LSI from C to C*exp(u-l).

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Measurability is represented explicitly or must be supplied by a dependency.
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