File size: 2,535 Bytes
b37e936 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | import OAI.Geometry.RCD.Transport.ExponentialTiltVariation
namespace OAI
/-! # Bounded exponential integrals for compact-support kernel scaling
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.
-/
open Set Filter MeasureTheory
open scoped NNReal
noncomputable section
namespace RegularPointCoordinates
variable {Y : Type*} [MeasurableSpace Y] (μ : Measure Y)
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
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]
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'
end RegularPointCoordinates
end
end OAI
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