| import Mathlib |
| import OAI.Probability.SKBarriers.SpinGlass.Basic |
| import OAI.Probability.SKBarriers.Gaussian.GaussianDomination |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
|
|
| def GaussianLocallyDominated {E F : Type} [Norm E] [Norm F] |
| (f : E × ℝ → F) : Prop := |
| ∀ R : ℝ, ∃ b : ℝ → ℝ, Integrable b (gaussianReal 0 1) ∧ |
| ∀ x y, ‖x‖ ≤ R → ‖f (x,y)‖ ≤ b y |
|
|
| theorem GaussianLocallyDominated.of_norm_le {E F G : Type} [Norm E] [Norm F] [Norm G] |
| {f : E × ℝ → F} |
| (hg : GaussianLocallyDominated f) {g : E × ℝ → G} {K : ℝ} (hK : 0 ≤ K) |
| (h : ∀ z, ‖g z‖ ≤ K*‖f z‖) : GaussianLocallyDominated g := by |
| intro R |
| obtain ⟨b,hb,hh⟩ := hg R |
| exact ⟨fun y => K*b y,hb.const_mul _,fun x y hx => |
| (h (x,y)).trans (mul_le_mul_of_nonneg_left (hh x y hx) hK)⟩ |
|
|
| section LocalGaussian |
| variable {E F G : Type} [NormedAddCommGroup E] [NormedAddCommGroup F] |
| [NormedAddCommGroup G] |
|
|
| theorem HasExpGrowth.locallyDominated {f : E × ℝ → F} (hf : HasExpGrowth f) : |
| GaussianLocallyDominated f := hf.gaussian_section_bound |
|
|
| theorem GaussianLocallyDominated.integrable_section {f : E × ℝ → F} |
| (hg : GaussianLocallyDominated f) (hf : Continuous f) (x : E) : |
| Integrable (fun y => f (x,y)) (gaussianReal 0 1) := by |
| obtain ⟨b,hb,hbound⟩ := hg ‖x‖ |
| exact hb.mono' (hf.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| (ae_of_all _ (fun y => hbound x y le_rfl)) |
|
|
| variable [NormedSpace ℝ F] [NormedSpace ℝ G] |
|
|
| theorem GaussianLocallyDominated.clm {f : E × ℝ → F} |
| (hg : GaussianLocallyDominated f) (L : F →L[ℝ] G) : |
| GaussianLocallyDominated (fun z => L (f z)) := |
| hg.of_norm_le (norm_nonneg L) (fun z => L.le_opNorm (f z)) |
|
|
| theorem GaussianLocallyDominated.continuous_integral {f : E × ℝ → F} |
| (hg : GaussianLocallyDominated f) (hf : Continuous f) : |
| Continuous (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) := by |
| apply continuous_iff_continuousAt.mpr |
| intro x₀ |
| obtain ⟨b,hb,hbound⟩ := hg (‖x₀‖+1) |
| apply continuousAt_of_dominated (bound := b) |
| · exact Eventually.of_forall (fun x => (hf.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable) |
| · filter_upwards [Metric.ball_mem_nhds x₀ (by norm_num : (0:ℝ)<1)] with x hx |
| exact ae_of_all _ (fun y => hbound x y (by |
| have hnorm : ‖x‖ ≤ ‖x-x₀‖+‖x₀‖ := by |
| simpa only [sub_add_cancel] using norm_add_le (x-x₀) x₀ |
| rw [← dist_eq_norm] at hnorm |
| have hdist : dist x x₀ < 1 := hx |
| linarith)) |
| · exact hb |
| · exact ae_of_all _ (fun y => (hf.comp (continuous_id.prodMk continuous_const)).continuousAt) |
|
|
| variable [NormedSpace ℝ E] |
|
|
| theorem hasFDerivAt_local_gaussian_integral (f : E × ℝ → F) |
| (f₁ : E × ℝ → E →L[ℝ] F) (hf : Continuous f) (hf₁ : Continuous f₁) |
| (hg : GaussianLocallyDominated f) (hg₁ : GaussianLocallyDominated f₁) |
| (hd : ∀ x y, HasFDerivAt (fun z => f (z,y)) (f₁ (x,y)) x) (x₀ : E) : |
| HasFDerivAt (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) |
| (∫ y, f₁ (x₀,y) ∂gaussianReal 0 1) x₀ := by |
| obtain ⟨b,hb,hbound⟩ := hg₁ (‖x₀‖+1) |
| apply hasFDerivAt_integral_of_dominated_of_fderiv_le (s := Metric.ball x₀ 1) |
| (Metric.ball_mem_nhds x₀ (by norm_num)) |
| · exact Eventually.of_forall (fun x => (hf.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable) |
| · exact hg.integrable_section hf x₀ |
| · exact (hf₁.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| · exact ae_of_all _ (fun y x hx => hbound x y (by |
| have hnorm : ‖x‖ ≤ ‖x-x₀‖+‖x₀‖ := by |
| simpa only [sub_add_cancel] using norm_add_le (x-x₀) x₀ |
| rw [← dist_eq_norm] at hnorm |
| have hdist : dist x x₀ < 1 := hx |
| linarith)) |
| · exact hb |
| · exact ae_of_all _ (fun y x _ => hd x y) |
|
|
| theorem contDiff_two_local_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 : GaussianLocallyDominated f) (hg₁ : GaussianLocallyDominated f₁) |
| (hg₂ : GaussianLocallyDominated 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_integral hf₂) |
| · exact hasFDerivAt_local_gaussian_integral f₁ f₂ hf₁ hf₂ hg₁ hg₂ hd₁ |
| apply (contDiff_succ_iff_hasFDerivAt (n := 1)).mpr |
| exact ⟨_,hc₁,hasFDerivAt_local_gaussian_integral f f₁ hf hf₁ hg hg₁ hd⟩ |
|
|
| theorem contDiff_two_local_gaussian_integral_fderiv (f : E × ℝ → F) |
| (hf : ContDiff ℝ 2 f) (hg : GaussianLocallyDominated f) |
| (hg₁ : GaussianLocallyDominated (fderiv ℝ f)) |
| (hg₂ : GaussianLocallyDominated (fderiv ℝ (fderiv ℝ f))) : |
| ContDiff ℝ 2 (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) := by |
| let L := ContinuousLinearMap.inl ℝ E ℝ |
| let T := (ContinuousLinearMap.compL ℝ E (E × ℝ) F).flip L |
| let T₁ := (ContinuousLinearMap.compL ℝ E (E × ℝ) (E →L[ℝ] F)).flip L |
| let S := (ContinuousLinearMap.compL ℝ (E × ℝ) ((E × ℝ) →L[ℝ] F) (E →L[ℝ] F)) T |
| let f₁ : E × ℝ → E →L[ℝ] F := fun z => T (fderiv ℝ f z) |
| let f₂ : E × ℝ → E →L[ℝ] E →L[ℝ] F := fun z => T₁ (S (fderiv ℝ (fderiv ℝ f) z)) |
| have hf₁ : ContDiff ℝ 1 f₁ := T.contDiff.comp (hf.fderiv_right (by norm_num)) |
| have hf₂ : Continuous f₂ := T₁.continuous.comp (S.continuous.comp |
| ((hf.fderiv_right (by norm_num : (1 : WithTop ℕ∞)+1 ≤ 2)).fderiv_right (m := 0) (by norm_num)).continuous) |
| have hg₂T : GaussianLocallyDominated f₂ := by |
| apply hg₂.of_norm_le (K := ‖T₁‖*‖S‖) (mul_nonneg (norm_nonneg T₁) (norm_nonneg S)) |
| intro z |
| calc |
| ‖f₂ z‖ ≤ ‖T₁‖*‖S (fderiv ℝ (fderiv ℝ f) z)‖ := T₁.le_opNorm _ |
| _ ≤ ‖T₁‖*(‖S‖*‖fderiv ℝ (fderiv ℝ f) z‖) := |
| mul_le_mul_of_nonneg_left (S.le_opNorm _) (norm_nonneg _) |
| _ = _ := by ring |
| apply contDiff_two_local_gaussian_integral f f₁ f₂ hf.continuous hf₁.continuous hf₂ hg |
| (hg₁.clm T) hg₂T |
| · intro x y |
| exact (hf.differentiable (by norm_num) (x,y)).hasFDerivAt.comp x |
| ((hasFDerivAt_id x).prodMk (hasFDerivAt_const y x)) |
| · intro x y |
| have hd := T.hasFDerivAt.comp (x,y) |
| ((hf.fderiv_right (by norm_num : (1 : WithTop ℕ∞)+1 ≤ 2)).differentiable (by norm_num) (x,y)).hasFDerivAt |
| have hxy : HasFDerivAt (fun z : E => (z,y)) L x := |
| (hasFDerivAt_id x).prodMk (hasFDerivAt_const y x) |
| convert! hd.comp x hxy using 1 |
| end LocalGaussian |
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
