File size: 3,967 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 | import OAI.Probability.FreeIsing.Ising.CanonicalArtanh
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation IsingCopyEnergy
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
|