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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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import Mathlib
import OAI.Probability.SKBarriers.Gaussian.GaussianAverage
import OAI.Probability.SKBarriers.Calculus.ParameterAlgebra
import OAI.Probability.SKBarriers.Calculus.ParameterSmooth
import OAI.Probability.SKBarriers.Scalar.LocalStepDerivative
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section ParameterStepDerivative
variable {P E : Type} [NormedAddCommGroup P] [NormedSpace ℝ P]
[NormedAddCommGroup E] [NormedSpace ℝ E]
theorem param_reassoc_fderiv_growth {f : P × (E × ℝ) → ℝ}
(hf : Differentiable ℝ f) (hg : ParamExpGrowth (fderiv ℝ f)) :
ParamExpGrowth (fun z : P × (E × ℝ) =>
fderiv ℝ (fun w : (P × E) × ℝ => f (w.1.1,(w.1.2,w.2))) ((z.1,z.2.1),z.2.2)) := 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)
have h : ParamExpGrowth (fun z => fderiv ℝ (fun w => f (L w)) (ι z)) :=
param_fderiv_compCLM_growth f L ι hf hg
exact h
theorem ParamExpGrowth.integrable_paramStepLaw {F : Type} [NormedAddCommGroup F]
[NormedSpace ℝ F] {f : P × (E × ℝ) → ℝ} (hf : ParamRegular f)
{g : P × (E × ℝ) → F} (hg : ParamExpGrowth g) (hc : Continuous g)
(m : ℝ) (z : P × E) : Integrable (fun y => g (z.1,(z.2,y)))
(gaussianStepLaw m (fun w : (P × E) × ℝ => f (w.1.1,(w.1.2,w.2))) z) := by
have he := hf.2.1.exp_mul m
have hce : Continuous (fun w : (P × E) × ℝ => Real.exp (m*f (w.1.1,(w.1.2,w.2)))) :=
Real.continuous_exp.comp (continuous_const.mul (hf.1.continuous.comp
(continuous_fst.fst.prodMk (continuous_fst.snd.prodMk continuous_snd))))
rw [gaussianStepLaw,integrable_tilted_iff (he.weakLocallyDominated.integrable_section hce z)]
exact (he.smul hg).weakLocallyDominated.integrable_section (hce.smul
(hc.comp (continuous_fst.fst.prodMk (continuous_fst.snd.prodMk continuous_snd)))) z
theorem ParamRegular.fderiv_gaussianStep {f : P × (E × ℝ) → ℝ}
(hf : ParamRegular f) (m : ℝ) (z : P × E) :
fderiv ℝ (SK.Analytic.gaussianStep m (fun w : (P × E) × ℝ => f (w.1.1,(w.1.2,w.2)))) z =
∫ y, leftRestrict (fderiv ℝ (fun w : (P × E) × ℝ => f (w.1.1,(w.1.2,w.2))) (z,y))
∂gaussianStepLaw m (fun w : (P × E) × ℝ => f (w.1.1,(w.1.2,w.2))) z := by
have hg₁ := param_reassoc_fderiv_growth (hf.1.differentiable (by norm_num)) hf.2.2.1
have he := (hf.const_mul m).expSmooth
have he₁ := param_reassoc_fderiv_growth (he.1.differentiable (by norm_num)) he.2.2.1
apply fderiv_local_gaussianStep
· exact (hf.1.comp (ContinuousLinearEquiv.prodAssoc ℝ P E ℝ).contDiff).of_le (by norm_num)
· exact hf.2.1.expGrowth.weakLocallyDominated
· exact hg₁.weakLocallyDominated
· exact he.2.1.weakLocallyDominated
· exact he₁.weakLocallyDominated
theorem ParamRegular.parameterDerivative_gaussianStep {f : P × (E × ℝ) → ℝ}
(hf : ParamRegular f) (m : ℝ) (z : P × E) (v : P) :
fderiv ℝ (SK.Analytic.gaussianStep m (fun w : (P × E) × ℝ => f (w.1.1,(w.1.2,w.2)))) z (v,0) =
gaussianAverage m (fun w : (P × E) × ℝ => f (w.1.1,(w.1.2,w.2)))
(fun w : (P × E) × ℝ => fderiv ℝ f (w.1.1,(w.1.2,w.2)) (v,0)) z := by
let L : (P × E) × ℝ →L[ℝ] P × (E × ℝ) :=
(ContinuousLinearEquiv.prodAssoc ℝ P E ℝ).toContinuousLinearMap
let g := fun w : (P × E) × ℝ => f (L w)
have hcg : ContDiff ℝ 2 g := hf.1.comp L.contDiff
have hg := param_reassoc_fderiv_growth (hf.1.differentiable (by norm_num)) hf.2.2.1
have hi := (hg.clm leftRestrict).integrable_paramStepLaw hf
((leftRestrict.continuous.comp (hcg.fderiv_right (m := 1) (by norm_num)).continuous).comp
(continuous_fst.prodMk continuous_snd.fst |>.prodMk continuous_snd.snd)) m z
rw [hf.fderiv_gaussianStep,ContinuousLinearMap.integral_apply hi]
apply integral_congr_ae
filter_upwards [] with y
have hD := ((hf.1.differentiable (by norm_num) (L (z,y))).hasFDerivAt.comp (z,y) L.hasFDerivAt).fderiv
change leftRestrict (fderiv ℝ g (z,y)) (v,0) = _
change fderiv ℝ g (z,y) = _ at hD
rw [hD]
rfl
end ParameterStepDerivative
end SK.Analytic
end
end
end
end OAI
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