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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 99 100 101 102 103 104 105 106 107 108 109 110 | import OAI.Probability.FreeIsing.Brownian.SpatialPrimitive
import OAI.Probability.FreeIsing.Ising.ContinuousResponseEnergy
import OAI.Probability.FreeIsing.Ising.ObservationIndependence
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.IsingObservationInnovation
theorem innovationMap_increments_indep
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (s t : ℝ≥0)
{v w : TreeVertex d} (hvw : v ≠ w) :
IndepFun (fun x : Paths d => innovationMap beta x v t - innovationMap beta x v s)
(fun x => innovationMap beta x w t - innovationMap beta x w s)
(observationLaw μ) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have hp : MeasurePreserving (innovationMap beta) (observationLaw μ)
(Measure.infinitePi (fun _ : TreeVertex d => WienerSampler.wienerMeasure)) :=
⟨measurable_innovationMap beta, map_innovationMap_observationLaw hμ⟩
apply indepFun_comp_measurePreserving hp
((measurable_evaluation v t).sub (measurable_evaluation v s))
((measurable_evaluation w t).sub (measurable_evaluation w s))
exact (iIndepFun_infinitePi (P := fun _ : TreeVertex d => WienerSampler.wienerMeasure)
(fun _ => (continuous_eval_const t).measurable.sub
(continuous_eval_const s).measurable)).indepFun hvw
end Problem367.IsingObservationInnovation
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation BrownianArrayFiltration
theorem exists_canonical_root_response_integrals
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hd : 3 ≤ d) (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) :
∃ (I : TreeVertex d → ℝ≥0 → Paths d → ℝ) (Y : ℝ≥0 → Paths d → ℝ)
(hI2 : ∀ a t, MemLp (I a t) 2 (observationLaw μ))
(hY2 : ∀ t, MemLp (Y t) 2 (observationLaw μ)),
(∀ a, IsStronglyPredictable (filtration d) (I a) ∧
Martingale (I a) (filtration d) (observationLaw μ) ∧
(∀ᵐ x ∂observationLaw μ, Continuous (fun t => I a t x))) ∧
(∀ a (N : ℕ), 0 < N → ∀ t ≤ (N : ℝ≥0),
Tendsto (fun n => ∫ x,
(BrownianIntegralProcess.stoppedUniformSum
(fun s x => innovationMap beta x a s) (continuousRootResponse hb a)
N n t x - I a t x) ^ 2 ∂observationLaw μ) atTop (𝓝 0)) ∧
IsStronglyPredictable (filtration d) Y ∧
Martingale Y (filtration d) (observationLaw μ) ∧
(∀ᵐ x ∂observationLaw μ, Continuous (fun t => Y t x)) ∧
Y 0 =ᵐ[observationLaw μ] 0 ∧
(∀ t, (∫ x, Y t x ^ 2 ∂observationLaw μ) =
∑' a, ∫ s in (0 : ℝ)..(t : ℝ), ∫ x,
continuousRootResponse hb a s.toNNReal x ^ 2 ∂observationLaw μ) ∧
(∀ t, HasSum (fun a => (hI2 a t).toLp (I a t)) ((hY2 t).toLp (Y t))) ∧
(∀ T : ℝ≥0, Tendsto (fun q : Finset (TreeVertex d) =>
∫ x, MartingaleStrongMaximal.maximalSquare
(MartingaleSpatialLimit.finiteSum I q - Y) T x ∂observationLaw μ)
atTop (𝓝 0)) ∧
Martingale (fun t x => Y t x ^ 2 -
nnrealTimePrimitive (spatialSquareDensity (continuousRootResponse hb)) t x)
(filtration d) (observationLaw μ) ∧
(∀ t, Integrable
(nnrealTimePrimitive (spatialSquareDensity (continuousRootResponse hb)) t)
(observationLaw μ)) ∧
(∀ᵐ x ∂observationLaw μ, Continuous (fun t =>
nnrealTimePrimitive (spatialSquareDensity (continuousRootResponse hb)) t x)) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have hc (a : TreeVertex d) := continuousRootResponse_continuous_ae hd hμ hb hcrit a
have henergy (N : ℕ) : Summable (fun a =>
∫ s in (0 : ℝ)..(N : ℝ), ∫ x,
continuousRootResponse hb a s.toNNReal x ^ 2 ∂observationLaw μ) := by
simpa only [NNReal.coe_natCast] using
(continuousRootResponse_hasSum_timeEnergy hd hμ hb hcrit (N : ℝ≥0)).summable
obtain ⟨I, Y, hI2, hY2, hI, hIlim, hYp, hYM, hYc, hY0,
hYiso, hYsum, hYmax, hQ, hAint, hAsum⟩ :=
exists_global_continuous_spatial_integral_compensatedSquare
(B := fun a s x => innovationMap beta x a s)
(F := filtration d) (H := continuousRootResponse hb)
(fun a => (innovationMap_isBrownianReal hμ a).toIsPreBrownianReal)
(innovationMap_stronglyAdapted beta)
(innovationMap_increment_indep_filtration hμ)
(fun s t _ _ _ hab => innovationMap_increments_indep hμ s t hab)
(fun a => (innovationMap_isBrownianReal hμ a).cont)
(fun a => (continuousRootResponse_progressive hb a).stronglyAdapted) hc
(C := fun _ => 1) (fun _ => by norm_num)
(fun a => Eventually.of_forall (fun x t => continuousRootResponse_abs_le_one hb a t x))
henergy
have hs (N : ℕ) := hAsum (N : ℝ≥0)
have heq := ae_spatialCompensator_eq_primitive hc hs
have hpQ := compensatedSquare_spatial_primitive
(fun a => continuousRootResponse_progressive hb a) hc hs hYM.stronglyAdapted hQ
refine ⟨I, Y, hI2, hY2, hI, hIlim, hYp, hYM, hYc, hY0,
hYiso, hYsum, hYmax, hpQ, ?_, ?_⟩
· intro t
apply (hAint t).congr
filter_upwards [heq] with x hx
exact hx.1 t
· filter_upwards [heq] with x hx
exact hx.2.1
end Problem367.IsingObservationFactor
end
end OAI
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