| import OAI.Probability.FreeIsing.Ising.CanonicalArtanh |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped NNReal |
|
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| namespace Problem367.IsingObservationFactor |
| open IsingObservationInnovation IsingCopyEnergy |
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|
| theorem norm_canonicalResponseDrift_le {d : ℕ} (beta : ℝ) |
| (v : TreeVertex d) (t : ℝ≥0) (x : Paths d) : |
| ‖canonicalResponseDrift beta v t x‖ ≤ |
| infiniteGibbsResponseRow beta v (fun i => x i t) := by |
| have hnonneg : 0 ≤ infiniteGibbsResponseRow beta v (fun i => x i t) := |
| tsum_nonneg (fun _ => sq_nonneg _) |
| rw [canonicalResponseDrift, norm_mul, Real.norm_of_nonneg hnonneg] |
| exact mul_le_of_le_one_left hnonneg (posterior_norm_le_one beta v t x) |
|
|
| theorem measurable_canonicalResponseDrift_real {d : ℕ} (beta : ℝ) |
| (v : TreeVertex d) : |
| Measurable (fun p : ℝ × Paths d => canonicalResponseDrift beta v p.1.toNNReal p.2) := by |
| have hc : Measurable (fun p : ℝ≥0 × Paths d => canonicalResponseDrift beta v p.1 p.2) := |
| (stronglyMeasurable_joint_of_isStronglyProgressive |
| (canonicalResponseDrift_progressive beta v)).measurable |
| have hn : Measurable (fun p : ℝ × Paths d => (p.1.toNNReal, p.2)) := |
| (measurable_real_toNNReal.comp measurable_fst).prodMk measurable_snd |
| exact Measurable.comp |
| (g := fun p : ℝ≥0 × Paths d => canonicalResponseDrift beta v p.1 p.2) |
| (f := fun p : ℝ × Paths d => (p.1.toNNReal, p.2)) hc hn |
|
|
| theorem canonicalResponseDrift_integrable |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (v : TreeVertex d) (T : ℝ≥0) : |
| Integrable (fun p : ℝ × Paths d => canonicalResponseDrift beta v p.1.toNNReal p.2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) := by |
| have hi : Integrable (fun p : ℝ × Paths d => |
| infiniteGibbsResponseRow beta v (fun i => p.2 i p.1.toNNReal)) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) := by |
| simpa only [infiniteGibbsResponseRow, responseRow] using |
| (observation_row_energy_integrable hμ hb v T.coe_nonneg).2.1 |
| exact hi.mono' (measurable_canonicalResponseDrift_real beta v).aestronglyMeasurable |
| (Eventually.of_forall (fun p => norm_canonicalResponseDrift_le beta v p.1.toNNReal p.2)) |
| |
| theorem canonicalResponseDrift_primitive_continuous |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (v : TreeVertex d) : |
| ∀ᵐ x ∂observationLaw μ, |
| Continuous (fun t => nnrealTimePrimitive (canonicalResponseDrift beta v) t x) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| apply ae_continuous_nnrealTimePrimitive_of_joint |
| intro n |
| simpa only [NNReal.coe_natCast] using |
| canonicalResponseDrift_integrable hμ hb v (n : ℝ≥0) |
| |
| theorem canonicalResponseDrift_primitive_integrable |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (v : TreeVertex d) (T : ℝ≥0) : |
| Integrable (nnrealTimePrimitive (canonicalResponseDrift beta v) T) |
| (observationLaw μ) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| exact nnrealTimePrimitive_integrable (canonicalResponseDrift_integrable hμ hb v T) |
| |
| theorem copy_canonicalResponseDrift_primitive_continuous |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (v : TreeVertex d) (j : Bool) : |
| ∀ᵐ q ∂copies beta μ, Continuous (fun t => |
| nnrealTimePrimitive (canonicalResponseDrift beta v) t (projection j q)) := by |
| exact (copyProjection_measurePreserving beta μ j).quasiMeasurePreserving.ae |
| (canonicalResponseDrift_primitive_continuous hμ hb v) |
| |
| end Problem367.IsingObservationFactor |
| |
| end |
| |
| end OAI |
| |
