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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 | import OAI.Probability.FreeIsing.Ising.CanonicalArtanh
import OAI.Probability.FreeIsing.Ising.CopyVarianceEquation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal Topology
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation IsingCopyEnergy
theorem continuousField_initial_zero
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) :
continuousField beta v 0 =ᵐ[observationLaw μ] 0 := by
filter_upwards [continuousPosterior_initial_zero hd hμ hb hcrit v] with x hx
have he := artanh_continuousPosterior beta v 0 x
rw [hx] at he
simpa only [Pi.zero_apply, Real.artanh_zero] using he.symm
theorem continuousDifference_initial_zero
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) :
continuousDifference beta v 0 =ᵐ[copies beta μ] 0 := by
have h0 := continuousField_initial_zero hd hμ hb hcrit v
filter_upwards [(copyProjection_measurePreserving beta μ true).quasiMeasurePreserving.ae h0,
(copyProjection_measurePreserving beta μ false).quasiMeasurePreserving.ae h0] with q hl hr
change continuousField beta v 0 q.2.1 - continuousField beta v 0 q.2.2 = 0
change continuousField beta v 0 q.2.1 = 0 at hl
change continuousField beta v 0 q.2.2 = 0 at hr
rw [hl, hr, sub_self]
theorem actual_variance_analysis_of_common_local_square
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(v : TreeVertex d) (T : ℝ≥0)
(hlocal : ∀ n, ∀ t ≤ T,
(∫ q, (stoppedProcess (continuousDifference beta v)
(commonPosteriorExit beta v n) t q) ^ 2 ∂copies beta μ) =
∫ s in (0 : ℝ)..(t : ℝ), ∫ q,
if (s.toNNReal : WithTop ℝ≥0) ≤ commonPosteriorExit beta v n q
then density beta v s q else 0 ∂copies beta μ) :
AbsolutelyContinuousOnInterval (fun t : ℝ => ∫ q,
(field beta v t.toNNReal q.2.1 - field beta v t.toNNReal q.2.2) ^ 2
∂copies beta μ) 0 T ∧
(∀ᵐ t ∂volume.restrict (Icc (0 : ℝ) T),
deriv (fun s : ℝ => ∫ q,
(field beta v s.toNNReal q.2.1 - field beta v s.toNNReal q.2.2) ^ 2
∂copies beta μ) t = ∫ q, density beta v t q ∂copies beta μ) := by
obtain ⟨hc, hver⟩ := continuousDifference_modification hd hμ hb hcrit v
have hprog := DominatedMartingaleRightContinuation.isStronglyProgressive_mono_filtration
ContinuousCopyBrownian.joinedFiltration.le_rightCont
(continuousDifference_progressive beta v)
have hτ := isLocalizingSequence_commonPosteriorExit hd hμ hb hcrit v
apply field_difference_energy_analysis_of_stopped_version hμ hb v hprog hc hver
hτ.isStoppingTime (ae_eventually_le_commonPosteriorExit hd hμ hb hcrit v T)
(density_integrable hμ hb v T)
have hzero : (∫ q, (continuousDifference beta v 0 q) ^ 2 ∂copies beta μ) = 0 := by
calc
_ = ∫ _q, (0 : ℝ) ∂copies beta μ := integral_congr_ae
((continuousDifference_initial_zero hd hμ hb hcrit v).mono (fun q hq => by
change continuousDifference beta v 0 q = 0 at hq
rw [hq, zero_pow (by decide : 2 ≠ 0)]))
_ = 0 := integral_zero _ _
intro n t ht
rw [hzero, zero_add]
exact hlocal n t ht
theorem actual_variance_analysis_of_local_square
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(v : TreeVertex d) (T : ℝ≥0)
{τ : ℕ → Joint d → WithTop ℝ≥0}
(hstopping : ∀ n, IsStoppingTime ContinuousCopyBrownian.joinedFiltration.rightCont (τ n))
(hexit : ∀ᵐ q ∂copies beta μ, ∀ᶠ n in atTop, (T : WithTop ℝ≥0) ≤ τ n q)
(hlocal : ∀ n, ∀ t ≤ T,
(∫ q, (stoppedProcess (continuousDifference beta v)
(τ n) t q) ^ 2 ∂copies beta μ) =
∫ s in (0 : ℝ)..(t : ℝ), ∫ q,
if (s.toNNReal : WithTop ℝ≥0) ≤ τ n q
then density beta v s q else 0 ∂copies beta μ) :
AbsolutelyContinuousOnInterval (fun t : ℝ => ∫ q,
(field beta v t.toNNReal q.2.1 - field beta v t.toNNReal q.2.2) ^ 2
∂copies beta μ) 0 T ∧
(∀ᵐ t ∂volume.restrict (Icc (0 : ℝ) T),
deriv (fun s : ℝ => ∫ q,
(field beta v s.toNNReal q.2.1 - field beta v s.toNNReal q.2.2) ^ 2
∂copies beta μ) t = ∫ q, density beta v t q ∂copies beta μ) := by
obtain ⟨hc, hver⟩ := continuousDifference_modification hd hμ hb hcrit v
have hprog := DominatedMartingaleRightContinuation.isStronglyProgressive_mono_filtration
ContinuousCopyBrownian.joinedFiltration.le_rightCont
(continuousDifference_progressive beta v)
apply field_difference_energy_analysis_of_stopped_version hμ hb v hprog hc hver
hstopping hexit
(density_integrable hμ hb v T)
have hzero : (∫ q, (continuousDifference beta v 0 q) ^ 2 ∂copies beta μ) = 0 := by
calc
_ = ∫ _q, (0 : ℝ) ∂copies beta μ := integral_congr_ae
((continuousDifference_initial_zero hd hμ hb hcrit v).mono (fun q hq => by
change continuousDifference beta v 0 q = 0 at hq
rw [hq, zero_pow (by decide : 2 ≠ 0)]))
_ = 0 := integral_zero _ _
intro n t ht
rw [hzero, zero_add]
exact hlocal n t ht
end Problem367.IsingObservationFactor
end
end OAI
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