File size: 4,797 Bytes
135052a | 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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import Mathlib
import OAI.Probability.SKBarriers.Calculus.MarginalSecondDerivative
import OAI.Probability.SKBarriers.Gaussian.LocalGaussian
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section LocalMarginalDerivatives
variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
theorem fderiv_local_gaussian_integral (f : E × ℝ → F) (hf : ContDiff ℝ 1 f)
(hg : GaussianLocallyDominated f) (hg₁ : GaussianLocallyDominated (fderiv ℝ f)) (x : E) :
fderiv ℝ (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) x =
∫ y, leftRestrict (fderiv ℝ f (x,y)) ∂gaussianReal 0 1 := by
exact (hasFDerivAt_local_gaussian_integral f _ hf.continuous
(leftRestrict.continuous.comp ((hf.fderiv_right (m := 0) (by norm_num)).continuous))
hg (hg₁.clm leftRestrict)
(leftRestrict_hasFDerivAt f (hf.differentiable (by norm_num))) x).fderiv
theorem norm_fderiv_local_gaussian_integral_le (f : E × ℝ → F) (hf : ContDiff ℝ 1 f)
(hg : GaussianLocallyDominated f) (hg₁ : GaussianLocallyDominated (fderiv ℝ f)) (x : E) :
‖fderiv ℝ (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) x‖ ≤
∫ y, ‖fderiv ℝ f (x,y)‖ ∂gaussianReal 0 1 := by
rw [fderiv_local_gaussian_integral f hf hg hg₁ x]
apply (norm_integral_le_integral_norm _).trans
apply integral_mono
· exact ((hg₁.clm leftRestrict).integrable_section
(leftRestrict.continuous.comp ((hf.fderiv_right (m := 0) (by norm_num)).continuous)) x).norm
· exact (hg₁.integrable_section ((hf.fderiv_right (m := 0) (by norm_num)).continuous) x).norm
· exact fun y => leftRestrict_apply_bound _
end LocalMarginalDerivatives
section LocalSecondMarginal
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
theorem secondLeftRestrict_local_growth {f : E × ℝ → ((E × ℝ) →L[ℝ] (E × ℝ) →L[ℝ] ℝ)}
(hg : GaussianLocallyDominated f) : GaussianLocallyDominated (fun x => secondLeftRestrict (f x)) := by
exact hg.of_norm_le zero_le_one (fun point =>
(secondLeftRestrict_apply_bound (f point)).trans_eq (one_mul _).symm)
theorem fderiv_fderiv_local_gaussian_integral (f : E × ℝ → ℝ) (hf : ContDiff ℝ 2 f)
(hg : GaussianLocallyDominated f) (hg₁ : GaussianLocallyDominated (fderiv ℝ f))
(hg₂ : GaussianLocallyDominated (fderiv ℝ (fderiv ℝ f))) (x : E) :
fderiv ℝ (fderiv ℝ (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1)) x =
∫ y, secondLeftRestrict (fderiv ℝ (fderiv ℝ f) (x,y)) ∂gaussianReal 0 1 := by
have he : fderiv ℝ (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) =
fun x => ∫ y, leftRestrict (fderiv ℝ f (x,y)) ∂gaussianReal 0 1 := by
funext x
exact fderiv_local_gaussian_integral f (hf.of_le (by norm_num)) hg hg₁ x
rw [he]
let T : ((E × ℝ) →L[ℝ] ℝ) →L[ℝ] E →L[ℝ] ℝ := leftRestrict
have hd : ∀ a b, HasFDerivAt (fun z => T (fderiv ℝ f (z,b)))
(secondLeftRestrict (fderiv ℝ (fderiv ℝ f) (a,b))) a := by
intro a b
have h₁ := T.hasFDerivAt.comp (a,b)
((hf.fderiv_right (m := 1) (by norm_num)).differentiable (by norm_num) (a,b)).hasFDerivAt
have hxy : HasFDerivAt (fun z : E => (z,b)) (ContinuousLinearMap.inl ℝ E ℝ) a :=
(hasFDerivAt_id a).prodMk (hasFDerivAt_const b a)
convert! h₁.comp a hxy using 1
apply (hasFDerivAt_local_gaussian_integral (fun z => T (fderiv ℝ f z)) _
(T.continuous.comp (hf.fderiv_right (m := 1) (by norm_num)).continuous)
(secondLeftRestrict.continuous.comp (hf.fderiv_right (m := 1) (by norm_num)
|>.fderiv_right (m := 0) (by norm_num)).continuous)
(hg₁.clm T) (secondLeftRestrict_local_growth hg₂) hd x).fderiv
theorem norm_fderiv_fderiv_local_gaussian_integral_le (f : E × ℝ → ℝ) (hf : ContDiff ℝ 2 f)
(hg : GaussianLocallyDominated f) (hg₁ : GaussianLocallyDominated (fderiv ℝ f))
(hg₂ : GaussianLocallyDominated (fderiv ℝ (fderiv ℝ f))) (x : E) :
‖fderiv ℝ (fderiv ℝ (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1)) x‖ ≤
∫ y, ‖fderiv ℝ (fderiv ℝ f) (x,y)‖ ∂gaussianReal 0 1 := by
rw [fderiv_fderiv_local_gaussian_integral f hf hg hg₁ hg₂ x]
apply (norm_integral_le_integral_norm _).trans
have hc := (hf.fderiv_right (m := 1) (by norm_num) |>.fderiv_right (m := 0) (by norm_num)).continuous
apply integral_mono_of_nonneg (ae_of_all _ (fun y =>
ContinuousLinearMap.opNorm_nonneg (secondLeftRestrict (fderiv ℝ (fderiv ℝ f) (x,y)))))
· exact (hg₂.integrable_section hc x).norm
· exact ae_of_all _ (fun y => secondLeftRestrict_apply_bound _)
end LocalSecondMarginal
end SK.Analytic
end
end
end
end OAI
|