| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.GaussianDomination |
|
|
| 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 TwiceDifferentiable |
| variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] |
|
|
| omit [CompleteSpace F] in |
| theorem contDiff_two_gaussian_integral (f : E × ℝ → F) |
| (f₁ : E × ℝ → E →L[ℝ] F) (f₂ : E × ℝ → E →L[ℝ] E →L[ℝ] F) |
| (hf : Continuous f) (hf₁ : Continuous f₁) (hf₂ : Continuous f₂) |
| (hg : HasExpGrowth f) (hg₁ : HasExpGrowth f₁) (hg₂ : HasExpGrowth f₂) |
| (hd : ∀ x y, HasFDerivAt (fun z => f (z,y)) (f₁ (x,y)) x) |
| (hd₁ : ∀ x y, HasFDerivAt (fun z => f₁ (z,y)) (f₂ (x,y)) x) : |
| ContDiff ℝ 2 (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) := by |
| have hc₁ : ContDiff ℝ 1 (fun x => ∫ y, f₁ (x,y) ∂gaussianReal 0 1) := by |
| apply (contDiff_succ_iff_hasFDerivAt (n := 0)).mpr |
| refine ⟨fun x => ∫ y, f₂ (x,y) ∂gaussianReal 0 1, ?_, ?_⟩ |
| · exact contDiff_zero.mpr (hg₂.continuous_gaussian_integral hf₂) |
| · exact hasFDerivAt_gaussian_integral f₁ f₂ hf₁ hf₂ hg₁ hg₂ hd₁ |
| apply (contDiff_succ_iff_hasFDerivAt (n := 1)).mpr |
| exact ⟨_, hc₁, hasFDerivAt_gaussian_integral f f₁ hf hf₁ hg hg₁ hd⟩ |
|
|
| end TwiceDifferentiable |
|
|
| section GrowthAlgebra |
| variable {E F G : Type} [Norm E] [Norm F] [Norm G] |
|
|
| theorem HasExpGrowth.congr_bound {f : E → F} (hf : HasExpGrowth f) |
| {g : E → G} {K : ℝ} (hK : 0 ≤ K) (hg : ∀ z, ‖g z‖ ≤ K*‖f z‖) : HasExpGrowth g := by |
| obtain ⟨C,M,hC,hM,hf⟩ := hf |
| refine ⟨K*C,M,mul_nonneg hK hC,hM,?_⟩ |
| intro z |
| exact (hg z).trans ((mul_le_mul_of_nonneg_left (hf z) hK).trans_eq (mul_assoc _ _ _).symm) |
|
|
| theorem HasExpGrowth.of_bounded {f : E → F} {C : ℝ} (hC : 0 ≤ C) |
| (hf : ∀ z, ‖f z‖ ≤ C) : HasExpGrowth f := by |
| refine ⟨C,0,hC,le_rfl,?_⟩ |
| simpa only [zero_mul, Real.exp_zero, mul_one] using hf |
|
|
| theorem HasExpGrowth.norm {f : E → F} (hf : HasExpGrowth f) (hn : ∀ x, 0 ≤ ‖f x‖) : |
| HasExpGrowth (fun x => ‖f x‖) := by |
| obtain ⟨C,M,hC,hM,hf⟩ := hf |
| refine ⟨C,M,hC,hM,?_⟩ |
| intro x |
| simpa only [Real.norm_eq_abs, abs_of_nonneg (hn x)] using hf x |
|
|
| end GrowthAlgebra |
| end SK.Analytic |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
