File size: 4,573 Bytes
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 | import OAI.Probability.FreeIsing.Ising.CanonicalArtanh
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal Topology
namespace Problem367.IsingCopyCanonicalStops
open IsingObservationInnovation IsingObservationFactor IsingCopyEnergy
variable {d : ℕ} {beta : ℝ}
lemma projection_measurable_past (j : Bool) (t : ℝ≥0) :
Measurable[ContinuousCopyBrownian.joinedFiltration t, filtration d t]
(projection (d := d) j) := by
let : MeasurableSpace (Joint d) := ContinuousCopyBrownian.joinedFiltration t
rw [filtration_eq_comap_past, measurable_comap_iff]
apply Measurable.of_eval
intro v
apply Measurable.of_eval
intro s
cases j
· exact ((IsingConditionalCopyBounds.progressive_rightObservation v).stronglyAdapted s).mono
(ContinuousCopyBrownian.joinedFiltration.mono s.2) |>.measurable
· exact ((IsingConditionalCopyBounds.progressive_leftObservation v).stronglyAdapted s).mono
(ContinuousCopyBrownian.joinedFiltration.mono s.2) |>.measurable
lemma measurable_rightCont_of_past
{Ω Ξ : Type*} [mΩ : MeasurableSpace Ω] [mΞ : MeasurableSpace Ξ]
{F : Filtration ℝ≥0 mΩ} {G : Filtration ℝ≥0 mΞ} {φ : Ω → Ξ}
(hφ : ∀ t, Measurable[F t, G t] φ) (t : ℝ≥0) :
Measurable[F.rightCont t, G.rightCont t] φ := by
intro s hs
change MeasurableSet[G.rightCont t] s at hs
rw [G.rightCont_eq t] at hs
rw [F.rightCont_eq t]
simp only [MeasurableSpace.measurableSet_iInf] at hs ⊢
intro u htu
exact hφ u (hs u htu)
lemma projection_measurable_rightCont (j : Bool) (t : ℝ≥0) :
Measurable[ContinuousCopyBrownian.joinedFiltration.rightCont t,
(filtration d).rightCont t] (projection (d := d) j) :=
measurable_rightCont_of_past (projection_measurable_past j) t
def projectedCanonicalExit (beta : ℝ) (v : TreeVertex d) (j : Bool)
(n : ℕ) (q : Joint d) : WithTop ℝ≥0 :=
canonicalPosteriorExit beta v n (projection j q)
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
lemma projectedCanonicalExit_localizing (hd : 3 ≤ d)
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(v : TreeVertex d) (j : Bool) :
IsLocalizingSequence ContinuousCopyBrownian.joinedFiltration.rightCont
(projectedCanonicalExit beta v j) (copies beta μ) := by
have h := canonicalPosteriorExit_localizing hd hμ hb hcrit v
have hp := (copyProjection_measurePreserving beta μ j).quasiMeasurePreserving
exact {
isStoppingTime := fun n t => projection_measurable_rightCont j t (h.isStoppingTime n t)
mono := hp.ae h.mono
tendsto_top := hp.ae h.tendsto_top }
def combinedCanonicalExit (beta : ℝ) (v : TreeVertex d)
(n : ℕ) (q : Joint d) : WithTop ℝ≥0 :=
min (min (projectedCanonicalExit beta v true n q)
(projectedCanonicalExit beta v false n q)) (commonPosteriorExit beta v n q)
lemma combinedCanonicalExit_le_projected (v : TreeVertex d) (j : Bool)
(n : ℕ) (q : Joint d) :
combinedCanonicalExit beta v n q ≤ projectedCanonicalExit beta v j n q := by
cases j
· exact (min_le_left _ _).trans (min_le_right _ _)
· exact (min_le_left _ _).trans (min_le_left _ _)
lemma combinedCanonicalExit_le_common (v : TreeVertex d) (n : ℕ) (q : Joint d) :
combinedCanonicalExit beta v n q ≤ commonPosteriorExit beta v n q :=
min_le_right _ _
lemma combinedCanonicalExit_localizing (hd : 3 ≤ d)
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(v : TreeVertex d) :
IsLocalizingSequence ContinuousCopyBrownian.joinedFiltration.rightCont
(combinedCanonicalExit beta v) (copies beta μ) :=
((projectedCanonicalExit_localizing hd hμ hb hcrit v true).min
(projectedCanonicalExit_localizing hd hμ hb hcrit v false)).min
(isLocalizingSequence_commonPosteriorExit hd hμ hb hcrit v)
lemma ae_eventually_le_combinedCanonicalExit (hd : 3 ≤ d)
(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) :
∀ᵐ q ∂copies beta μ, ∀ᶠ n in atTop,
(T : WithTop ℝ≥0) ≤ combinedCanonicalExit beta v n q := by
filter_upwards [(combinedCanonicalExit_localizing hd hμ hb hcrit v).tendsto_top] with q hq
exact ((WithTop.tendsto_nhds_top_iff _).mp hq T).mono (fun _ hn => hn.le)
end Problem367.IsingCopyCanonicalStops
end
end OAI
|