| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.ExponentialGrowth |
| import OAI.Probability.SKBarriers.Calculus.MarginalDerivatives |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped Topology Interval |
| noncomputable section |
| open MeasureTheory Set |
| open scoped Interval |
| noncomputable section |
| open MeasureTheory Set Filter ProbabilityTheory |
| open scoped Topology |
| namespace SK.Analytic |
| section SecondMarginal |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| def secondLeftRestrict : |
| ((E × ℝ) →L[ℝ] (E × ℝ) →L[ℝ] ℝ) →L[ℝ] E →L[ℝ] E →L[ℝ] ℝ := |
| leftRestrict ∘L |
| ((ContinuousLinearMap.compL ℝ (E × ℝ) ((E × ℝ) →L[ℝ] ℝ) (E →L[ℝ] ℝ)) leftRestrict) |
|
|
| theorem secondLeftRestrict_apply_bound (B : (E × ℝ) →L[ℝ] (E × ℝ) →L[ℝ] ℝ) : |
| ‖secondLeftRestrict B‖ ≤ ‖B‖ := by |
| apply (leftRestrict_apply_bound _).trans |
| exact (ContinuousLinearMap.opNorm_comp_le _ _).trans |
| ((mul_le_mul_of_nonneg_right leftRestrict_norm (norm_nonneg B)).trans_eq (one_mul _)) |
|
|
| theorem secondLeftRestrict_growth {f : E × ℝ → ((E × ℝ) →L[ℝ] (E × ℝ) →L[ℝ] ℝ)} |
| (hg : HasExpGrowth f) : HasExpGrowth (fun x => secondLeftRestrict (f x)) := by |
| exact hg.congr_bound zero_le_one (fun x => |
| (secondLeftRestrict_apply_bound _).trans_eq (one_mul _).symm) |
|
|
| theorem fderiv_fderiv_gaussian_integral (f : E × ℝ → ℝ) (hf : ContDiff ℝ 2 f) |
| (hg : HasExpGrowth f) (hg₁ : HasExpGrowth (fderiv ℝ f)) |
| (hg₂ : HasExpGrowth (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_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_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_growth hg₂) hd x).fderiv |
|
|
| theorem norm_fderiv_fderiv_gaussian_integral_le (f : E × ℝ → ℝ) (hf : ContDiff ℝ 2 f) |
| (hg : HasExpGrowth f) (hg₁ : HasExpGrowth (fderiv ℝ f)) |
| (hg₂ : HasExpGrowth (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_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.norm (fun z => ContinuousLinearMap.opNorm_nonneg _)).integrable_gaussian_section hc.norm x |
| · exact ae_of_all _ (fun y => secondLeftRestrict_apply_bound _) |
|
|
| end SecondMarginal |
| end SK.Analytic |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
