| import OAI.Probability.FreeIsing.Estimates.DiscrepancyDensityModification |
| import OAI.Probability.FreeIsing.Ising.CanonicalPosteriorStops |
| import OAI.Probability.FreeIsing.Ising.ContinuousRootResponse |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
|
|
| namespace Problem367.IsingContinuousCopyDensity |
| open IsingObservationInnovation IsingObservationFactor IsingCopyEnergy |
|
|
| variable {d : ℕ} {beta : ℝ} |
|
|
| def response (hb : 0 ≤ beta) (v : TreeVertex d) (j : Bool) |
| (t : ℝ≥0) (q : Joint d) : ℝ := |
| continuousRootResponse hb v t (projection j q) |
|
|
| theorem response_progressive (hb : 0 ≤ beta) (v : TreeVertex d) (j : Bool) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration (response hb v j) := by |
| intro T |
| let : MeasurableSpace (Joint d) := ContinuousCopyBrownian.joinedFiltration T |
| exact ((continuous_siteRootResponse hb v).measurable.comp |
| (Measurable.of_eval (fun i => |
| (copyContinuousField_progressive beta i j T).measurable))).stronglyMeasurable |
|
|
| theorem response_eq_raw_ae |
| (hd : 3 ≤ d) {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) (j : Bool) (t : ℝ≥0) : |
| response hb v j t =ᵐ[copies beta μ] |
| fun q => responseRow beta (ConnectedHull.root d) v ((t : ℝ), projection j q) := by |
| have h := (copyProjection_measurePreserving beta μ j).quasiMeasurePreserving.ae |
| (continuousRootResponse_eq_response_ae hd hμ hb hcrit v t) |
| filter_upwards [h] with q hq |
| simpa only [response, responseRow, Real.toNNReal_coe] using hq |
|
|
| private theorem rawObservation_progressive (j : Bool) (v : TreeVertex d) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration |
| (fun t (q : Joint d) => projection j q v t) := by |
| cases j |
| · exact IsingConditionalCopyBounds.progressive_rightObservation v |
| · exact IsingConditionalCopyBounds.progressive_leftObservation v |
|
|
| private theorem rawPosterior_progressive (v : TreeVertex d) (j : Bool) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration |
| (fun t (q : Joint d) => posterior beta v t (projection j q)) := |
| isStronglyProgressive_infiniteGibbsSpinMean _ beta v _ (rawObservation_progressive j) |
| |
| private theorem rawResponse_progressive (v : TreeVertex d) (j : Bool) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration |
| (fun t (q : Joint d) => |
| responseRow beta (ConnectedHull.root d) v ((t : ℝ), projection j q)) := by |
| intro T |
| let : MeasurableSpace (Joint d) := ContinuousCopyBrownian.joinedFiltration T |
| simpa only [Function.comp_def, responseRow, Real.toNNReal_coe] using |
| ((measurable_infiniteGibbsResponse beta (ConnectedHull.root d) v).comp |
| (Measurable.of_eval (fun i => |
| (rawObservation_progressive j i T).measurable))).stronglyMeasurable |
|
|
| def continuousDensity (hb : 0 ≤ beta) (t : ℝ≥0) : Joint d → ℝ := |
| DiscrepancyEnergy.density (fun v => response hb v true t) |
| (fun v => response hb v false t) (continuousDifference beta (ConnectedHull.root d) t) |
| (copyContinuousPosterior beta (ConnectedHull.root d) true t) |
| (copyContinuousPosterior beta (ConnectedHull.root d) false t) |
|
|
| theorem continuousDensity_progressive (hb : 0 ≤ beta) : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration |
| (continuousDensity (d := d) hb) := |
| DiscrepancyEnergy.isStronglyProgressive_density (ι := TreeVertex d) (Ω := Joint d) |
| (fun v => response_progressive hb v true) (fun v => response_progressive hb v false) |
| (continuousDifference_progressive beta (ConnectedHull.root d)) |
| (copyContinuousPosterior_progressive beta (ConnectedHull.root d) true) |
| (copyContinuousPosterior_progressive beta (ConnectedHull.root d) false) |
|
|
| theorem actualDensity_progressive : |
| IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration |
| (fun t : ℝ≥0 => IsingCopyEnergy.density beta (ConnectedHull.root d) (t : ℝ)) := by |
| unfold IsingCopyEnergy.density |
| simp only [Real.toNNReal_coe] |
| apply DiscrepancyEnergy.isStronglyProgressive_density |
| · intro v |
| exact rawResponse_progressive (beta := beta) v true |
| · intro v |
| exact rawResponse_progressive (beta := beta) v false |
| · exact (isStronglyProgressive_artanh _ |
| (rawPosterior_progressive (beta := beta) (ConnectedHull.root d) true)).sub |
| (isStronglyProgressive_artanh _ |
| (rawPosterior_progressive (beta := beta) (ConnectedHull.root d) false)) |
| · exact rawPosterior_progressive (beta := beta) (ConnectedHull.root d) true |
| · exact rawPosterior_progressive (beta := beta) (ConnectedHull.root d) false |
|
|
| theorem continuousDensity_eq_actual_ae |
| (hd : 3 ≤ d) {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) (t : ℝ≥0) : |
| continuousDensity hb t =ᵐ[copies beta μ] |
| IsingCopyEnergy.density beta (ConnectedHull.root d) (t : ℝ) := by |
| unfold continuousDensity IsingCopyEnergy.density |
| simp only [Real.toNNReal_coe] |
| apply DiscrepancyEnergy.density_congr_ae |
| · intro v |
| exact response_eq_raw_ae hd hμ hb hcrit v true t |
| · intro v |
| exact response_eq_raw_ae hd hμ hb hcrit v false t |
| · exact (continuousDifference_modification hd hμ hb hcrit (ConnectedHull.root d)).2 t |
| · exact (copyContinuousPosterior_modification hd hμ hb hcrit |
| (ConnectedHull.root d) true).2 t |
| · exact (copyContinuousPosterior_modification hd hμ hb hcrit |
| (ConnectedHull.root d) false).2 t |
|
|
| theorem continuousDensity_eq_actual_prod_ae |
| (hd : 3 ≤ d) {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) (T : ℝ≥0) : |
| (fun p : ℝ × Joint d => continuousDensity hb p.1.toNNReal p.2) =ᵐ[ |
| (volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod (copies beta μ)] |
| (fun p => IsingCopyEnergy.density beta (ConnectedHull.root d) p.1 p.2) := by |
| have hmap : Measurable (fun p : ℝ × Joint d => (p.1.toNNReal, p.2)) := |
| (measurable_real_toNNReal.comp measurable_fst).prodMk measurable_snd |
| have hl := (stronglyMeasurable_joint_of_isStronglyProgressive |
| (continuousDensity_progressive hb)).comp_measurable hmap |
| have hr := (stronglyMeasurable_joint_of_isStronglyProgressive |
| (actualDensity_progressive (d := d) (beta := beta))).comp_measurable hmap |
| have he : (fun p : ℝ × Joint d => continuousDensity hb p.1.toNNReal p.2) =ᵐ[ |
| (volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod (copies beta μ)] |
| (fun p => IsingCopyEnergy.density beta (ConnectedHull.root d) |
| (p.1.toNNReal : ℝ) p.2) := by |
| apply (Measure.ae_prod_iff_ae_ae (measurableSet_eq_fun hl.measurable hr.measurable)).2 |
| exact Filter.Eventually.of_forall fun t => |
| continuousDensity_eq_actual_ae hd hμ hb hcrit t.toNNReal |
| simpa only [IsingCopyEnergy.density, responseRow, Real.toNNReal_coe] using he |
|
|
| theorem continuousDensity_integrable |
| (hd : 3 ≤ d) {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) (T : ℝ≥0) : |
| Integrable (fun p : ℝ × Joint d => continuousDensity hb p.1.toNNReal p.2) |
| ((volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod (copies beta μ)) := |
| (IsingCopyEnergy.density_integrable hμ hb (ConnectedHull.root d) T).congr |
| (continuousDensity_eq_actual_prod_ae hd hμ hb hcrit T).symm |
|
|
| theorem ae_all_primitive_eq_actual |
| (hd : 3 ≤ d) {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) : |
| ∀ᵐ q ∂copies beta μ, ∀ t, |
| nnrealTimePrimitive (continuousDensity hb) t q = |
| nnrealTimePrimitive |
| (fun s : ℝ≥0 => IsingCopyEnergy.density beta (ConnectedHull.root d) (s : ℝ)) t q := |
| ae_all_nnrealTimePrimitive_eq_of_modification (continuousDensity_progressive hb) |
| actualDensity_progressive (continuousDensity_eq_actual_ae hd hμ hb hcrit) |
|
|
| theorem ae_all_stopped_primitive_eq_actual |
| (hd : 3 ≤ d) {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) |
| (τ : Joint d → WithTop ℝ≥0) : |
| ∀ᵐ q ∂copies beta μ, ∀ t, |
| stoppedProcess (nnrealTimePrimitive (continuousDensity hb)) τ t q = |
| stoppedProcess (nnrealTimePrimitive |
| (fun s : ℝ≥0 => IsingCopyEnergy.density beta (ConnectedHull.root d) (s : ℝ))) τ t q := by |
| filter_upwards [ae_all_primitive_eq_actual hd hμ hb hcrit] with q hq |
| intro t |
| exact hq _ |
|
|
| theorem integral_cutoff_density_eq_actual |
| (hd : 3 ≤ d) {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) |
| (τ : Joint d → WithTop ℝ≥0) (t : ℝ≥0) : |
| (∫ q, if (t : WithTop ℝ≥0) ≤ τ q then continuousDensity hb t q else 0 |
| ∂copies beta μ) = |
| ∫ q, if (t : WithTop ℝ≥0) ≤ τ q then |
| IsingCopyEnergy.density beta (ConnectedHull.root d) (t : ℝ) q else 0 |
| ∂copies beta μ := by |
| apply integral_congr_ae |
| filter_upwards [continuousDensity_eq_actual_ae hd hμ hb hcrit t] with q hq |
| by_cases ht : (t : WithTop ℝ≥0) ≤ τ q |
| · simp only [ht, ite_true, hq] |
| · simp only [ht, ite_false] |
|
|
| theorem integral_stopDensity_eq_actual |
| (hd : 3 ≤ d) {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) |
| (τ : Joint d → WithTop ℝ≥0) (s : ℝ) : |
| (∫ q, StoppedTimePrimitive.stopDensity (continuousDensity hb) τ s.toNNReal q |
| ∂copies beta μ) = |
| ∫ q, if (s.toNNReal : WithTop ℝ≥0) ≤ τ q then |
| IsingCopyEnergy.density beta (ConnectedHull.root d) s q else 0 |
| ∂copies beta μ := by |
| simpa only [StoppedTimePrimitive.stopDensity, IsingCopyEnergy.density, |
| responseRow, Real.toNNReal_coe] using |
| integral_cutoff_density_eq_actual hd hμ hb hcrit τ s.toNNReal |
|
|
| theorem integral_integral_stopDensity_eq_actual |
| (hd : 3 ≤ d) {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) |
| (τ : Joint d → WithTop ℝ≥0) (T : ℝ≥0) : |
| (∫ s in (0 : ℝ)..(T : ℝ), ∫ q, |
| StoppedTimePrimitive.stopDensity (continuousDensity hb) τ s.toNNReal q |
| ∂copies beta μ) = |
| ∫ s in (0 : ℝ)..(T : ℝ), ∫ q, |
| if (s.toNNReal : WithTop ℝ≥0) ≤ τ q then |
| IsingCopyEnergy.density beta (ConnectedHull.root d) s q else 0 |
| ∂copies beta μ := by |
| apply intervalIntegral.integral_congr |
| intro s _ |
| exact integral_stopDensity_eq_actual hd hμ hb hcrit τ s |
|
|
| end Problem367.IsingContinuousCopyDensity |
|
|
| end |
|
|
| end OAI |
|
|
