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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 | import Mathlib
import OAI.Probability.SKBarriers.Calculus.ParameterGrowth
import OAI.Probability.SKBarriers.Calculus.ParameterWeakGrowth
import OAI.Probability.SKBarriers.Gaussian.LocalMarginal
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section ParameterIntegral
variable {P E : Type} [NormedAddCommGroup P] [NormedSpace ℝ P]
[NormedAddCommGroup E] [NormedSpace ℝ E]
theorem param_contDiff_gaussian_integral (f : (P × E) × ℝ → ℝ) (hf : ContDiff ℝ 2 f)
(hg : ParamExpGrowth (fun z : P × (E × ℝ) => f ((z.1,z.2.1),z.2.2)))
(hg₁ : ParamExpGrowth (fun z : P × (E × ℝ) => fderiv ℝ f ((z.1,z.2.1),z.2.2)))
(hg₂ : ParamExpGrowth (fun z : P × (E × ℝ) => fderiv ℝ (fderiv ℝ f) ((z.1,z.2.1),z.2.2))) :
ContDiff ℝ 2 (fun z => ∫ y, f (z,y) ∂gaussianReal 0 1) := by
exact contDiff_two_local_gaussian_integral_fderiv f hf hg.weakLocallyDominated
hg₁.weakLocallyDominated hg₂.weakLocallyDominated
theorem param_fderiv_gaussian_integral_growth (f : (P × E) × ℝ → ℝ) (hf : ContDiff ℝ 1 f)
(hg : ParamExpGrowth (fun z : P × (E × ℝ) => f ((z.1,z.2.1),z.2.2)))
(hg₁ : ParamExpGrowth (fun z : P × (E × ℝ) => fderiv ℝ f ((z.1,z.2.1),z.2.2))) :
ParamExpGrowth (fderiv ℝ (fun z => ∫ y, f (z,y) ∂gaussianReal 0 1)) := by
have H : ParamExpGrowth (fun z : P × E => ∫ y, ‖fderiv ℝ f (z,y)‖ ∂gaussianReal 0 1) :=
(hg₁.weakNorm (fun _ => ContinuousLinearMap.opNorm_nonneg _)).gaussian_integral
apply H.of_norm_le zero_le_one
intro z
rw [one_mul,Real.norm_eq_abs,abs_of_nonneg (integral_nonneg (fun y => ContinuousLinearMap.opNorm_nonneg (fderiv ℝ f (z,y))))]
exact norm_fderiv_local_gaussian_integral_le f hf hg.weakLocallyDominated hg₁.weakLocallyDominated z
theorem param_second_gaussian_integral_growth (f : (P × E) × ℝ → ℝ) (hf : ContDiff ℝ 2 f)
(hg : ParamExpGrowth (fun z : P × (E × ℝ) => f ((z.1,z.2.1),z.2.2)))
(hg₁ : ParamExpGrowth (fun z : P × (E × ℝ) => fderiv ℝ f ((z.1,z.2.1),z.2.2)))
(hg₂ : ParamExpGrowth (fun z : P × (E × ℝ) => fderiv ℝ (fderiv ℝ f) ((z.1,z.2.1),z.2.2))) :
ParamExpGrowth (fderiv ℝ (fderiv ℝ (fun z => ∫ y, f (z,y) ∂gaussianReal 0 1))) := by
have H : ParamExpGrowth (fun z : P × E => ∫ y, ‖fderiv ℝ (fderiv ℝ f) (z,y)‖ ∂gaussianReal 0 1) :=
(hg₂.weakNorm (fun _ => ContinuousLinearMap.opNorm_nonneg _)).gaussian_integral
apply H.of_norm_le zero_le_one
intro z
rw [one_mul,Real.norm_eq_abs,abs_of_nonneg (integral_nonneg (fun y => ContinuousLinearMap.opNorm_nonneg (fderiv ℝ (fderiv ℝ f) (z,y))))]
exact norm_fderiv_fderiv_local_gaussian_integral_le f hf hg.weakLocallyDominated
hg₁.weakLocallyDominated hg₂.weakLocallyDominated z
end ParameterIntegral
end SK.Analytic
end
end
end
end OAI
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