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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 | import Mathlib
import OAI.Probability.SKBarriers.Gaussian.GaussianSecondStep
import OAI.Probability.SKBarriers.Calculus.ParameterComposition
import OAI.Probability.SKBarriers.Calculus.ParameterIntegral
import OAI.Probability.SKBarriers.Calculus.ParameterLinear
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section ParameterSmooth
variable {P E : Type} [NormedAddCommGroup P] [NormedSpace ℝ P]
[NormedAddCommGroup E] [NormedSpace ℝ E]
def ParamRegular (f : P × E → ℝ) : Prop :=
ContDiff ℝ 2 f ∧ ParamLinearGrowth f ∧ ParamExpGrowth (fderiv ℝ f) ∧
ParamExpGrowth (fderiv ℝ (fderiv ℝ f))
theorem ParamRegular.const_mul {f : P × E → ℝ} (hf : ParamRegular f) (c : ℝ) :
ParamRegular (fun z => c*f z) := by
refine ⟨contDiff_const.mul hf.1,hf.2.1.const_mul c,?_,?_⟩
· apply hf.2.2.1.of_norm_le (abs_nonneg c)
intro z
rw [((hf.1.differentiable (by norm_num) z).hasFDerivAt.const_mul c).fderiv]
simp only [norm_smul,Real.norm_eq_abs]
exact le_rfl
· apply hf.2.2.2.of_norm_le (abs_nonneg c)
intro z
rw [fderiv_fderiv_const_mul f hf.1 c]
simpa only [Real.norm_eq_abs] using ContinuousLinearMap.opNorm_smul_le c (fderiv ℝ (fderiv ℝ f) z)
theorem param_gaussian_smooth (f : P × (E × ℝ) → ℝ) (hf : ContDiff ℝ 2 f)
(hg : ParamExpGrowth f) (hg₁ : ParamExpGrowth (fderiv ℝ f))
(hg₂ : ParamExpGrowth (fderiv ℝ (fderiv ℝ f))) :
let Z := fun z : P × E => ∫ y, f (z.1,(z.2,y)) ∂gaussianReal 0 1
ContDiff ℝ 2 Z ∧ ParamExpGrowth (fderiv ℝ Z) ∧
ParamExpGrowth (fderiv ℝ (fderiv ℝ Z)) := by
let L : (P × E) × ℝ →L[ℝ] P × (E × ℝ) :=
(ContinuousLinearEquiv.prodAssoc ℝ P E ℝ).toContinuousLinearMap
let ι : P × (E × ℝ) → (P × E) × ℝ := fun z => ((z.1,z.2.1),z.2.2)
let g := fun z => f (L z)
have hc : ContDiff ℝ 2 g := hf.comp L.contDiff
have hg' : ParamExpGrowth (fun z => g (ι z)) := hg
have h₁ : ParamExpGrowth (fun z => fderiv ℝ g (ι z)) :=
param_fderiv_compCLM_growth f L ι (hf.differentiable (by norm_num)) hg₁
have h₂ : ParamExpGrowth (fun z => fderiv ℝ (fderiv ℝ g) (ι z)) :=
param_second_compCLM_growth f L ι hf hg₂
exact ⟨param_contDiff_gaussian_integral g hc hg' h₁ h₂,
param_fderiv_gaussian_integral_growth g (hc.of_le (by norm_num)) hg' h₁,
param_second_gaussian_integral_growth g hc hg' h₁ h₂⟩
theorem ParamRegular.gaussian_integral {f : P × (E × ℝ) → ℝ} (hf : ParamRegular f) :
ParamRegular (fun z : P × E => ∫ y, f (z.1,(z.2,y)) ∂gaussianReal 0 1) := by
have h := param_gaussian_smooth f hf.1 hf.2.1.expGrowth hf.2.2.1 hf.2.2.2
exact ⟨h.1,hf.2.1.gaussian_integral,h.2⟩
end ParameterSmooth
end SK.Analytic
end
end
end
end OAI
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