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f61e547 | 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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 | import OAI.Probability.FreeIsing.Energy.TimeRowEnergyHasSum
import OAI.Probability.FreeIsing.Ising.ContinuousRootResponse
import OAI.Probability.FreeIsing.Ising.CopyEnergyIntegrability
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal Topology
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation
variable {d : ℕ} {beta : ℝ}
private theorem measurable_real_of_progressive
{Ω : Type*} [mΩ : MeasurableSpace Ω] {F : Filtration ℝ≥0 mΩ}
{M : ℝ≥0 → Ω → ℝ} (hM : IsStronglyProgressive F M) :
Measurable (fun p : ℝ × Ω => M p.1.toNNReal p.2) := by
have hj : Measurable (fun p : ℝ≥0 × Ω => M p.1 p.2) :=
(stronglyMeasurable_joint_of_isStronglyProgressive hM).measurable
have ht : Measurable (fun p : ℝ × Ω => (p.1.toNNReal, p.2)) :=
(measurable_real_toNNReal.comp measurable_fst).prodMk measurable_snd
exact hj.comp ht
private theorem measurable_response_real (hb : 0 ≤ beta) (v : TreeVertex d) :
Measurable (fun p : ℝ × Paths d => continuousRootResponse hb v p.1.toNNReal p.2) :=
measurable_real_of_progressive (continuousRootResponse_progressive hb v)
private theorem measurable_covariance_real (hb : 0 ≤ beta) (v : TreeVertex d) :
Measurable (fun p : ℝ × Paths d => continuousRootCovariance hb v p.1.toNNReal p.2) :=
measurable_real_of_progressive (continuousRootCovariance_progressive hb v)
theorem continuousRootResponse_row_eq_ae
(hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (T : ℝ) :
∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ),
∀ v, continuousRootResponse hb v p.1.toNNReal p.2 =
IsingCopyEnergy.responseRow beta (ConnectedHull.root d) v p := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
apply ae_all_iff.mpr
intro v
apply (Measure.ae_prod_iff_ae_ae (measurableSet_eq_fun
(measurable_response_real hb v)
(IsingCopyEnergy.measurable_responseRow beta (ConnectedHull.root d) v))).mpr
exact Eventually.of_forall fun t =>
continuousRootResponse_eq_response_ae hd hμ hb hcrit v t.toNNReal
theorem continuousRootResponse_row_energy
(hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
{T : ℝ} (hT : 0 ≤ T) :
(∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ),
Summable (fun v => continuousRootResponse hb v p.1.toNNReal p.2 ^ 2)) ∧
Integrable (fun p : ℝ × Paths d => ∑' v,
continuousRootResponse hb v p.1.toNNReal p.2 ^ 2)
((volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) ∧
(∫ p : ℝ × Paths d, ∑' v, continuousRootResponse hb v p.1.toNNReal p.2 ^ 2
∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) ≤
2 * ((d : ℝ) * beta + T + Real.sqrt T) := by
have hraw := IsingCopyEnergy.observation_row_energy_integrable hμ hb
(ConnectedHull.root d) hT
have he := continuousRootResponse_row_eq_ae hd hμ hb hcrit T
have hsum : (fun p : ℝ × Paths d => ∑' v,
continuousRootResponse hb v p.1.toNNReal p.2 ^ 2) =ᵐ[
(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)]
(fun p => ∑' v, IsingCopyEnergy.responseRow beta (ConnectedHull.root d) v p ^ 2) := by
filter_upwards [he] with p hp
exact tsum_congr (fun v => congrArg (fun r : ℝ => r ^ 2) (hp v))
refine ⟨?_, hraw.2.1.congr hsum.symm, (integral_congr_ae hsum).trans_le hraw.2.2⟩
filter_upwards [he, hraw.1] with p hp hs
simpa only [hp] using hs
theorem continuousRootResponse_hasSum_timeEnergy
(hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (T : ℝ≥0) :
HasSum (fun v => ∫ t in (0 : ℝ)..(T : ℝ),
∫ x, continuousRootResponse hb v t.toNNReal x ^ 2 ∂observationLaw μ)
(∫ p : ℝ × Paths d, ∑' v, continuousRootResponse hb v p.1.toNNReal p.2 ^ 2
∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have h := continuousRootResponse_row_energy hd hμ hb hcrit T.coe_nonneg
exact DiscrepancyEnergy.hasSum_intervalIntegral_sq_row T
(fun v => (measurable_response_real hb v).aestronglyMeasurable) h.1 h.2.1
theorem continuousRootCovariance_sq_le_response_sq
(hb : 0 ≤ beta) (v : TreeVertex d) (t : ℝ≥0) (x : Paths d) :
continuousRootCovariance hb v t x ^ 2 ≤ continuousRootResponse hb v t x ^ 2 := by
have h0 : 0 ≤ 1 - Real.tanh (continuousField beta (ConnectedHull.root d) t x) ^ 2 :=
(sub_pos.mpr (Real.tanh_sq_lt_one _)).le
have h1 : 1 - Real.tanh (continuousField beta (ConnectedHull.root d) t x) ^ 2 ≤ 1 :=
sub_le_self _ (sq_nonneg _)
have hs : (1 - Real.tanh (continuousField beta (ConnectedHull.root d) t x) ^ 2) ^ 2 ≤ 1 := by
nlinarith
rw [continuousRootCovariance, mul_pow]
exact (mul_le_mul_of_nonneg_right hs (sq_nonneg _)).trans_eq (one_mul _)
theorem continuousRootCovariance_row_energy
(hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
{T : ℝ} (hT : 0 ≤ T) :
(∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ),
Summable (fun v => continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2)) ∧
Integrable (fun p : ℝ × Paths d => ∑' v,
continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2)
((volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) ∧
(∫ p : ℝ × Paths d, ∑' v, continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2
∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) ≤
2 * ((d : ℝ) * beta + T + Real.sqrt T) := by
have hR := continuousRootResponse_row_energy hd hμ hb hcrit hT
have hs : ∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ),
Summable (fun v => continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2) := by
filter_upwards [hR.1] with p hp
exact Summable.of_nonneg_of_le (fun v => sq_nonneg _)
(fun v => continuousRootCovariance_sq_le_response_sq hb v _ _) hp
have hle : ∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ),
(∑' v, continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2) ≤
∑' v, continuousRootResponse hb v p.1.toNNReal p.2 ^ 2 := by
filter_upwards [hs, hR.1] with p hcp hrp
exact hcp.tsum_le_tsum
(fun v => continuousRootCovariance_sq_le_response_sq hb v _ _) hrp
have hi : Integrable (fun p : ℝ × Paths d => ∑' v,
continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2)
((volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) := by
apply hR.2.1.mono'
(Measurable.tsum (fun v => (measurable_covariance_real hb v).pow_const 2)).aestronglyMeasurable
filter_upwards [hle] with p hp
rw [Real.norm_eq_abs, abs_of_nonneg (tsum_nonneg
(fun v => sq_nonneg (continuousRootCovariance hb v p.1.toNNReal p.2)))]
exact hp
exact ⟨hs, hi, (integral_mono_ae hi hR.2.1 hle).trans hR.2.2⟩
theorem continuousRootCovariance_hasSum_timeEnergy
(hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (T : ℝ≥0) :
HasSum (fun v => ∫ t in (0 : ℝ)..(T : ℝ),
∫ x, continuousRootCovariance hb v t.toNNReal x ^ 2 ∂observationLaw μ)
(∫ p : ℝ × Paths d, ∑' v, continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2
∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have h := continuousRootCovariance_row_energy hd hμ hb hcrit T.coe_nonneg
exact DiscrepancyEnergy.hasSum_intervalIntegral_sq_row T
(fun v => (measurable_covariance_real hb v).aestronglyMeasurable) h.1 h.2.1
end Problem367.IsingObservationFactor
end
end OAI
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