| 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 |
| |
