| import Mathlib |
| import OAI.Probability.SKBarriers.Calculus.MarginalSecondDerivative |
| import OAI.Probability.SKBarriers.Gaussian.LocalGaussian |
|
|
| namespace OAI |
|
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| section |
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| 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] |
|
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| 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 |
|
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| end |
|
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| end OAI |
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|
