| 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 | |