| import OAI.Probability.FreeIsing.Brownian.ArrayLaw |
| import OAI.Probability.FreeIsing.Brownian.Characterization |
| import OAI.Probability.FreeIsing.Conditioning.GaussianVector |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
|
|
| namespace Problem367.GaussianArrayCharacterization |
|
|
| theorem isGaussianProcess_array_of_finite_coordinate_processes |
| {ι T Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} |
| {B : ι → T → Ω → ℝ} |
| (h : ∀ I : Finset ι, |
| IsGaussianProcess (fun t ω (i : I) => B i t ω) P) : |
| IsGaussianProcess (fun p : (_ : ι) × T => B p.1 p.2) P := by |
| classical |
| constructor |
| intro S |
| let I : Finset ι := S.image Sigma.fst |
| let J : Finset T := S.image Sigma.snd |
| let L : (J → I → ℝ) →L[ℝ] (S → ℝ) := |
| { toFun x p := x |
| ⟨p.1.2, Finset.mem_image_of_mem Sigma.snd p.2⟩ |
| ⟨p.1.1, Finset.mem_image_of_mem Sigma.fst p.2⟩ |
| map_add' x y := by rfl |
| map_smul' c x := by rfl } |
| exact (h I |>.hasGaussianLaw J).map L |
|
|
| open ConditionalCharacteristic ConditionalGaussianVector |
|
|
| theorem gaussianPi_law_and_indep_of_weighted_characters |
| {ι Ω : Type*} [Fintype ι] {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] |
| {P : Measure Ω} [IsProbabilityMeasure P] (hm : m ≤ mΩ) |
| {X : Ω → ι → ℝ} (hX : Measurable X) (v : ι → ℝ≥0) |
| (hchar : ∀ a : ι → ℝ, |
| P[character (fun ω => ∑ i, a i * X ω i) 1 | m] =ᵐ[P] |
| fun _ => Complex.exp (-(∑ i, ((v i : ℝ) : ℂ) * (a i : ℂ)^2) / 2)) |
| {E : Type*} [MeasurableSpace E] {Y : Ω → E} (hY : Measurable[m] Y) : |
| HasLaw X (Measure.pi (fun i => gaussianReal 0 (v i))) P ∧ IndepFun X Y P := by |
| apply product_gaussian_law_and_indep_of_condExp_linear_character hm hX v _ hY |
| intro a |
| have hc : character (fun ω => ∑ i, a i * X ω i) 1 = |
| fun ω => Complex.exp (((∑ i, a i * X ω i : ℝ) : ℂ) * Complex.I) := by |
| funext ω |
| simp [character] |
| have h := hchar a |
| rw [hc] at h |
| simpa only [Complex.ofReal_sum, Complex.ofReal_mul, Complex.ofReal_pow] using h |
|
|
| theorem isGaussianProcess_vector_of_conditional_weighted_characters |
| {ι Ω : Type*} [Fintype ι] [DecidableEq ι] [mΩ : MeasurableSpace Ω] |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| {X : ℝ≥0 → Ω → ι → ℝ} (F : Filtration ℝ≥0 mΩ) (hX : Adapted F X) |
| (hzero : ∀ᵐ ω ∂P, X 0 ω = 0) |
| (hchar : ∀ s t : ℝ≥0, s ≤ t → ∀ a : ι → ℝ, |
| P[character (fun ω => ∑ i, a i * (X t ω i - X s ω i)) 1 | F s] =ᵐ[P] |
| fun _ => Complex.exp (-(∑ i, (((t - s : ℝ≥0) : ℝ) : ℂ) * (a i : ℂ)^2) / 2)) : |
| IsGaussianProcess X P := by |
| have hm (s t : ℝ≥0) : Measurable (fun ω => X t ω - X s ω) := |
| hX.measurable.sub hX.measurable |
| have hlaw (s t : ℝ≥0) (hst : s ≤ t) : |
| HasLaw (fun ω => X t ω - X s ω) |
| (Measure.pi (fun _ : ι => gaussianReal 0 (t - s))) P := |
| (gaussianPi_law_and_indep_of_weighted_characters (F.le s) (hm s t) |
| (fun _ : ι => t - s) (hchar s t hst) |
| (measurable_const : Measurable[F s] (fun _ : Ω => ()))).1 |
| have hi : HasIndepIncrements X P := by |
| apply BrownianCharacterization.hasIndepIncrements_of_past_events_general F hX |
| intro s t hst A hA S hS |
| let Y : Ω → ℝ := A.indicator (fun _ => 1) |
| have hY : Measurable[F s] Y := measurable_const.indicator hA |
| have hind := (gaussianPi_law_and_indep_of_weighted_characters |
| (F.le s) (hm s t) (fun _ : ι => t - s) (hchar s t hst) hY).2 |
| have hAe : Y ⁻¹' {1} = A := by |
| ext ω |
| by_cases hω : ω ∈ A <;> simp [Y, hω] |
| have hh := (indepFun_iff_measure_inter_preimage_eq_mul.mp hind) |
| S {1} hS (measurableSet_singleton 1) |
| simpa only [hAe, Set.inter_comm, mul_comm] using hh |
| apply hi.isGaussianProcess _ hzero |
| change ∀ t : ℝ≥0, HasGaussianLaw (X t) P |
| intro t |
| have hg : HasGaussianLaw (id : (ι → ℝ) → (ι → ℝ)) |
| (Measure.pi (fun _ : ι => gaussianReal 0 (t - 0))) := |
| (iIndepFun_pi (X := fun _ : ι => id) (fun _ => aemeasurable_id)).hasGaussianLaw |
| (fun i => (measurePreserving_eval (fun _ : ι => gaussianReal 0 (t - 0)) i).hasLaw.hasGaussianLaw) |
| have : IsGaussian (Measure.pi (fun _ : ι => gaussianReal 0 (t - 0))) := by |
| simpa only [Measure.map_id] using hg.isGaussian_map |
| apply (hlaw 0 t bot_le).hasGaussianLaw.congr |
| filter_upwards [hzero] with ω hω |
| simp [hω] |
|
|
| theorem isGaussianProcess_array_of_conditional_finite_characters |
| {ι Ω : Type*} [mΩ : MeasurableSpace Ω] |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| {B : ι → ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ) |
| (hB : ∀ i, Adapted F (B i)) |
| (hzero : ∀ i, ∀ᵐ ω ∂P, B i 0 ω = 0) |
| (hchar : ∀ (I : Finset ι) (s t : ℝ≥0), s ≤ t → ∀ a : I → ℝ, |
| P[character (fun ω => ∑ i : I, a i * (B i t ω - B i s ω)) 1 | F s] =ᵐ[P] |
| fun _ => Complex.exp (-(∑ i : I, (((t - s : ℝ≥0) : ℝ) : ℂ) * (a i : ℂ)^2) / 2)) : |
| IsGaussianProcess (fun p : (_ : ι) × ℝ≥0 => B p.1 p.2) P := by |
| classical |
| apply isGaussianProcess_array_of_finite_coordinate_processes |
| intro I |
| apply isGaussianProcess_vector_of_conditional_weighted_characters F |
| · intro t |
| let : MeasurableSpace Ω := F t |
| exact Measurable.of_eval (fun index : I => hB index t) |
| · have hz : ∀ᵐ ω ∂P, ∀ i : I, B i 0 ω = 0 := ae_all_iff.mpr (fun i => hzero i) |
| filter_upwards [hz] with ω hω |
| exact funext hω |
| · exact hchar I |
|
|
| theorem conditional_character_coordinate |
| {ι Ω : Type*} [mΩ : MeasurableSpace Ω] |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| {B : ι → ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ) |
| (hchar : ∀ (I : Finset ι) (s t : ℝ≥0), s ≤ t → ∀ a : I → ℝ, |
| P[character (fun ω => ∑ i : I, a i * (B i t ω - B i s ω)) 1 | F s] =ᵐ[P] |
| fun _ => Complex.exp (-(∑ i : I, (((t - s : ℝ≥0) : ℝ) : ℂ) * (a i : ℂ)^2) / 2)) |
| (i : ι) (s t : ℝ≥0) (hst : s ≤ t) (u : ℝ) : |
| P[character (fun ω => B i t ω - B i s ω) u | F s] =ᵐ[P] |
| fun _ => charFun (gaussianReal 0 (t - s)) u := by |
| classical |
| have h := hchar {i} s t hst (fun _ => u) |
| have hc : character (fun ω => u * (B i t ω - B i s ω)) 1 = |
| character (fun ω => B i t ω - B i s ω) u := by |
| funext ω |
| simp [character, Complex.ofReal_mul] |
| simpa [hc, charFun_gaussianReal, neg_div] using h |
|
|
| theorem preBrownian_coordinate_of_conditional_finite_characters |
| {ι Ω : Type*} [mΩ : MeasurableSpace Ω] |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| {B : ι → ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ) |
| (hB : ∀ i, Adapted F (B i)) |
| (hzero : ∀ i, ∀ᵐ ω ∂P, B i 0 ω = 0) |
| (hchar : ∀ (I : Finset ι) (s t : ℝ≥0), s ≤ t → ∀ a : I → ℝ, |
| P[character (fun ω => ∑ i : I, a i * (B i t ω - B i s ω)) 1 | F s] =ᵐ[P] |
| fun _ => Complex.exp (-(∑ i : I, (((t - s : ℝ≥0) : ℝ) : ℂ) * (a i : ℂ)^2) / 2)) |
| (i : ι) : IsPreBrownianReal (B i) P := by |
| classical |
| exact BrownianCharacterization.isPreBrownianReal_of_conditional_charFun |
| F (hB i).stronglyAdapted (hzero i) (conditional_character_coordinate F hchar i) |
|
|
| theorem law_finite_coordinates_of_conditional_finite_characters |
| {ι Ω : Type*} [mΩ : MeasurableSpace Ω] |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| {B : ι → ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ) |
| (hB : ∀ i, Adapted F (B i)) |
| (hzero : ∀ i, ∀ᵐ ω ∂P, B i 0 ω = 0) |
| (hchar : ∀ (I : Finset ι) (s t : ℝ≥0), s ≤ t → ∀ a : I → ℝ, |
| P[character (fun ω => ∑ i : I, a i * (B i t ω - B i s ω)) 1 | F s] =ᵐ[P] |
| fun _ => Complex.exp (-(∑ i : I, (((t - s : ℝ≥0) : ℝ) : ℂ) * (a i : ℂ)^2) / 2)) |
| (I : Finset ι) (t : ℝ≥0) : |
| HasLaw (fun ω (i : I) => B i t ω) |
| (Measure.pi (fun _ : I => gaussianReal 0 t)) P := by |
| classical |
| have hm : Measurable (fun ω (i : I) => B i t ω - B i 0 ω) := |
| Measurable.of_eval (fun index => (hB index).measurable.sub (hB index).measurable) |
| have hh := (gaussianPi_law_and_indep_of_weighted_characters |
| (F.le 0) hm (fun _ : I => t - 0) (hchar I 0 t bot_le) |
| (measurable_const : Measurable[F 0] (fun _ : Ω => ()))).1 |
| simp only [tsub_zero] at hh |
| apply hh.congr |
| have hz : ∀ᵐ ω ∂P, ∀ i : I, B i 0 ω = 0 := ae_all_iff.mpr (fun i => hzero i) |
| filter_upwards [hz] with ω hω |
| funext i |
| simp [hω i] |
|
|
| theorem covariance_coordinate_ne_of_gaussianPi_law |
| {ι Ω : Type*} [Fintype ι] [MeasurableSpace Ω] |
| {P : Measure Ω} {X : Ω → ι → ℝ} (v : ι → ℝ≥0) |
| (hX : HasLaw X (Measure.pi (fun i => gaussianReal 0 (v i))) P) |
| (i j : ι) (hij : i ≠ j) : |
| cov[fun ω => X ω i, fun ω => X ω j; P] = 0 := by |
| rw [hX.covariance_fun_comp (f := fun x : ι → ℝ => x i) (g := fun x : ι → ℝ => x j) |
| (measurable_pi_apply i).aemeasurable (measurable_pi_apply j).aemeasurable] |
| exact (iIndepFun_pi (X := fun _ : ι => id) (fun _ => aemeasurable_id)).indepFun hij |
| |>.covariance_eq_zero |
| ((measurePreserving_eval (fun i => gaussianReal 0 (v i)) i).hasLaw.hasGaussianLaw.memLp_two) |
| ((measurePreserving_eval (fun i => gaussianReal 0 (v i)) j).hasLaw.hasGaussianLaw.memLp_two) |
|
|
| theorem crossCovariance_of_conditional_finite_characters |
| {ι Ω : Type*} [mΩ : MeasurableSpace Ω] |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| {B : ι → ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ) |
| (hB : ∀ i, Adapted F (B i)) |
| (hzero : ∀ i, ∀ᵐ ω ∂P, B i 0 ω = 0) |
| (hchar : ∀ (I : Finset ι) (s t : ℝ≥0), s ≤ t → ∀ a : I → ℝ, |
| P[character (fun ω => ∑ i : I, a i * (B i t ω - B i s ω)) 1 | F s] =ᵐ[P] |
| fun _ => Complex.exp (-(∑ i : I, (((t - s : ℝ≥0) : ℝ) : ℂ) * (a i : ℂ)^2) / 2)) : |
| ∀ i j, i ≠ j → ∀ s t, cov[B i s, B j t; P] = 0 := by |
| classical |
| have hb (i : ι) := preBrownian_coordinate_of_conditional_finite_characters F hB hzero hchar i |
| have hL2 (i : ι) (t : ℝ≥0) : MemLp (B i t) 2 P := |
| (hb i).isGaussianProcess.hasGaussianLaw_eval t |>.memLp_two |
| have hsame (i j : ι) (hij : i ≠ j) (t : ℝ≥0) : cov[B i t, B j t; P] = 0 := by |
| let I : Finset ι := {i, j} |
| let ii : I := ⟨i, by simp [I]⟩ |
| let jj : I := ⟨j, by simp [I]⟩ |
| have hne : ii ≠ jj := fun h => hij (congrArg Subtype.val h) |
| exact covariance_coordinate_ne_of_gaussianPi_law (fun _ : I => t) |
| (law_finite_coordinates_of_conditional_finite_characters F hB hzero hchar I t) |
| ii jj hne |
| have hordered (i j : ι) (hij : i ≠ j) (s t : ℝ≥0) (hst : s ≤ t) : |
| cov[B i s, B j t; P] = 0 := by |
| have hi : IndepFun (fun ω => B j t ω - B j s ω) (B i s) P := |
| indepFun_of_condExp_character (F.le s) |
| ((hB j).measurable.sub (hB j).measurable) (gaussianReal 0 (t - s)) |
| (conditional_character_coordinate F hchar j s t hst) (hB i s) |
| change IndepFun (B j t - B j s) (B i s) P at hi |
| have heq : B j t = (B j t - B j s) + B j s := by |
| funext ω |
| simp |
| rw [heq, covariance_add_right (hL2 i s) ((hL2 j t).sub (hL2 j s)) (hL2 j s), |
| hi.symm.covariance_eq_zero (hL2 i s) ((hL2 j t).sub (hL2 j s)), hsame i j hij s, |
| add_zero] |
| intro i j hij s t |
| rcases le_total s t with hst | hts |
| · exact hordered i j hij s t hst |
| · rw [covariance_comm] |
| exact hordered j i hij.symm t s hts |
|
|
| theorem map_brownianArray_eq_infinitePi_of_conditional_finite_characters |
| {ι Ω Ω' : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace Ω'] |
| {P : Measure Ω} [IsProbabilityMeasure P] {Q : Measure Ω'} |
| {B : ι → ℝ≥0 → Ω → ℝ} {C : ℝ≥0 → Ω' → ℝ} |
| (F : Filtration ℝ≥0 mΩ) (hB : ∀ i, Adapted F (B i)) |
| (hzero : ∀ i, ∀ᵐ ω ∂P, B i 0 ω = 0) |
| (hchar : ∀ (I : Finset ι) (s t : ℝ≥0), s ≤ t → ∀ a : I → ℝ, |
| P[character (fun ω => ∑ i : I, a i * (B i t ω - B i s ω)) 1 | F s] =ᵐ[P] |
| fun _ => Complex.exp (-(∑ i : I, (((t - s : ℝ≥0) : ℝ) : ℂ) * (a i : ℂ)^2) / 2)) |
| (hC : IsPreBrownianReal C Q) (hmC : ∀ t, Measurable (C t)) : |
| P.map (fun ω i t => B i t ω) = |
| Measure.infinitePi (fun _ : ι => Q.map (fun ω t => C t ω)) := by |
| have hb (i : ι) := preBrownian_coordinate_of_conditional_finite_characters F hB hzero hchar i |
| apply map_brownianArray_eq_infinitePi_reference |
| (isGaussianProcess_array_of_conditional_finite_characters F hB hzero hchar) |
| (fun i _ => (hB i).measurable) (fun i t => (hb i).integral_eval t) |
| _ (crossCovariance_of_conditional_finite_characters F hB hzero hchar) hC hmC |
| intro i s t hst |
| simpa only [min_eq_left hst] using (hb i).covariance_eval s t |
|
|
| theorem map_continuousBrownianArray_eq_infinitePi_of_conditional_finite_characters |
| [MeasurableSpace C(ℝ≥0, ℝ)] [BorelSpace C(ℝ≥0, ℝ)] |
| [StandardBorelSpace C(ℝ≥0, ℝ)] |
| {ι Ω Ω' : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace Ω'] |
| {P : Measure Ω} [IsProbabilityMeasure P] {Q : Measure Ω'} |
| {X : ι → Ω → C(ℝ≥0, ℝ)} {Y : Ω' → C(ℝ≥0, ℝ)} |
| (F : Filtration ℝ≥0 mΩ) (hX : ∀ i, Adapted F (fun t ω => X i ω t)) |
| (hzero : ∀ i, ∀ᵐ ω ∂P, X i ω 0 = 0) |
| (hchar : ∀ (I : Finset ι) (s t : ℝ≥0), s ≤ t → ∀ a : I → ℝ, |
| P[character (fun ω => ∑ i : I, a i * (X i ω t - X i ω s)) 1 | F s] =ᵐ[P] |
| fun _ => Complex.exp (-(∑ i : I, (((t - s : ℝ≥0) : ℝ) : ℂ) * (a i : ℂ)^2) / 2)) |
| (hmY : Measurable Y) (hY : IsPreBrownianReal (fun t ω => Y ω t) Q) : |
| P.map (fun ω i => X i ω) = |
| Measure.infinitePi (fun _ : ι => Q.map Y) := by |
| have hb (i : ι) := preBrownian_coordinate_of_conditional_finite_characters F hX hzero hchar i |
| apply map_continuousBrownianArray_eq_infinitePi_reference |
| (fun i => measurable_continuousPath_of_evaluations (fun _ => (hX i).measurable)) hmY |
| (isGaussianProcess_array_of_conditional_finite_characters F hX hzero hchar) |
| (fun i t => (hb i).integral_eval t) _ |
| (crossCovariance_of_conditional_finite_characters F hX hzero hchar) hY |
| intro i s t hst |
| simpa only [min_eq_left hst] using (hb i).covariance_eval s t |
|
|
| end Problem367.GaussianArrayCharacterization |
|
|
| end |
|
|
| end OAI |
|
|
