| import OAI.Geometry.RCD.Transport.ExponentialTiltVariation |
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| namespace OAI |
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| /-! # Bounded exponential integrals for compact-support kernel scaling |
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| Only a bounded measurable exponent and an integrable weight are needed. |
| The logarithmic derivative is used for the positive heat-kernel scaling |
| in the entropic approximation of bounded-support transport plans. |
| -/ |
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|
| open Set Filter MeasureTheory |
| open scoped NNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {Y : Type*} [MeasurableSpace Y] (μ : Measure Y) |
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| theorem bounded_exp_mul_integrable {r g : Y → ℝ} (hr : Integrable r μ) |
| (hg : Measurable g) {M : ℝ≥0} (hgb : ∀ x, |g x| ≤ M) (t : ℝ) : |
| Integrable (fun x => Real.exp (t*g x)*r x) μ := by |
| have hb : ∀ᵐ x ∂μ, ‖Real.exp (t*g x)‖ ≤ Real.exp (|t| * M) := by |
| apply Eventually.of_forall |
| intro x |
| rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| apply Real.exp_le_exp.mpr |
| calc |
| t*g x ≤ |t*g x| := le_abs_self _ |
| _ = |t| * |g x| := abs_mul _ _ |
| _ ≤ |t| * M := mul_le_mul_of_nonneg_left (hgb x) (abs_nonneg t) |
| exact hr.bdd_mul (Real.measurable_exp.comp (measurable_const.mul hg)).aestronglyMeasurable hb |
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| theorem exponential_tilt_integral_hasDerivAt {r g : Y → ℝ} (hr : Integrable r μ) |
| (hg : Measurable g) {M : ℝ≥0} (hgb : ∀ x, |g x| ≤ M) (t : ℝ) : |
| HasDerivAt (fun s : ℝ => ∫ x, Real.exp (s*g x)*r x ∂μ) |
| (∫ x, g x*(Real.exp (t*g x)*r x) ∂μ) t := by |
| let w := fun x => Real.exp (t*g x)*r x |
| have hw : Integrable w μ := bounded_exp_mul_integrable μ hr hg hgb t |
| have hz := exponential_tilt_integral_hasDerivAt_zero μ hw hg hgb |
| have hshift : HasDerivAt (fun s : ℝ => s-t) 1 t := by |
| simpa using (hasDerivAt_id t).sub_const t |
| have hh := hz.comp_of_eq t hshift (by simp) |
| convert hh using 1 |
| · funext s |
| apply integral_congr_ae |
| apply Eventually.of_forall |
| intro x |
| dsimp only [Function.comp_def,w] |
| rw [← mul_assoc,← Real.exp_add] |
| congr 2 |
| ring |
| · simp only [w,mul_one] |
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| theorem log_exponential_tilt_hasDerivAt {r g : Y → ℝ} (hr : Integrable r μ) |
| (hg : Measurable g) {M : ℝ≥0} (hgb : ∀ x, |g x| ≤ M) (t : ℝ) |
| (hpos : 0 < ∫ x, Real.exp (t*g x)*r x ∂μ) : |
| HasDerivAt (fun s : ℝ => Real.log (∫ x, Real.exp (s*g x)*r x ∂μ)) |
| ((∫ x, g x*(Real.exp (t*g x)*r x) ∂μ)/(∫ x, Real.exp (t*g x)*r x ∂μ)) t := |
| (exponential_tilt_integral_hasDerivAt μ hr hg hgb t).log hpos.ne' |
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| end RegularPointCoordinates |
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| end |
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| end OAI |
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