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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 | import OAI.Probability.FreeIsing.Brownian.ArbitraryHorizon
import OAI.Probability.FreeIsing.Ising.CanonicalClippedWeight
import OAI.Probability.FreeIsing.Ising.CanonicalDriftRegularity
import OAI.Probability.FreeIsing.Ising.CanonicalIntegralRepresentation
import OAI.Probability.FreeIsing.Ising.CanonicalResponseIntegrals
import OAI.Probability.FreeIsing.Martingales.StoppedIdentityExhaustion
import OAI.Probability.FreeIsing.Martingales.StoppedRowNormalization
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation BrownianGrid
open DominatedMartingaleRightContinuation
theorem canonical_stopped_field_eq_of_integral_limits
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {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)
{I J : TreeVertex d → ℝ≥0 → Paths d → ℝ} {Z : ℝ≥0 → Paths d → ℝ}
(hIp : ∀ a, IsStronglyProgressive (filtration d) (I a))
(hIM : ∀ a, Martingale (I a) (filtration d) (observationLaw μ))
(hJM : ∀ a, Martingale (J a) (filtration d) (observationLaw μ))
(hZM : Martingale Z (filtration d) (observationLaw μ))
(hI2 : ∀ a t, MemLp (I a t) 2 (observationLaw μ))
(hJ2 : ∀ a t, MemLp (J a t) 2 (observationLaw μ))
(hZ2 : ∀ t, MemLp (Z t) 2 (observationLaw μ))
(hcI : ∀ a, ∀ᵐ x ∂observationLaw μ, Continuous (fun t => I a t x))
(hcJ : ∀ a, ∀ᵐ x ∂observationLaw μ, Continuous (fun t => J a t x))
(hcZ : ∀ᵐ x ∂observationLaw μ, Continuous (fun t => Z t x))
(n : ℕ) (T : ℝ≥0)
(hsumI : HasSum (fun a => (hI2 a T).toLp (I a T))
((posterior_memLp beta μ (ConnectedHull.root d) T 2).toLp
(posterior beta (ConnectedHull.root d) T)))
(hsumJ : HasSum (fun a => (hJ2 a T).toLp (J a T)) ((hZ2 T).toLp (Z T)))
(hIlim : ∀ a, Tendsto (fun k => ∫ x,
(uniformSum (fun t x => innovationMap beta x a t)
(continuousRootCovariance hb a) T k x - I a T x) ^ 2 ∂observationLaw μ)
atTop (𝓝 0))
(hJlim : ∀ a, Tendsto (fun k => ∫ x,
(uniformSum (fun t x => innovationMap beta x a t)
(continuousRootResponse hb a) T k x - J a T x) ^ 2 ∂observationLaw μ)
atTop (𝓝 0)) :
let τ := canonicalPosteriorExit beta (ConnectedHull.root d) n
stoppedProcess (continuousField beta (ConnectedHull.root d)) τ T =ᵐ[observationLaw μ]
(fun x => stoppedProcess Z τ T x +
stoppedProcess (nnrealTimePrimitive (canonicalResponseDrift beta (ConnectedHull.root d))) τ T x) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let M := continuousPosterior beta (ConnectedHull.root d)
let τ := canonicalPosteriorExit beta (ConnectedHull.root d) n
let K := canonicalClippedWeight (d := d) beta n
have hM := continuousPosterior_martingale hd hμ hb hcrit (ConnectedHull.root d)
have hcM := (continuousPosterior_modification hd hμ hb hcrit (ConnectedHull.root d)).1
have hM2 (t : ℝ≥0) : MemLp (M t) 2 (observationLaw μ) :=
MemLp.ae_eq ((continuousPosterior_modification hd hμ hb hcrit (ConnectedHull.root d)).2 t).symm
(posterior_memLp beta μ (ConnectedHull.root d) t 2)
have hsumM : HasSum (fun a => (hI2 a T).toLp (I a T)) ((hM2 T).toLp (M T)) := by
have he := (posterior_memLp beta μ (ConnectedHull.root d) T 2).toLp_congr (hM2 T)
((continuousPosterior_modification hd hμ hb hcrit (ConnectedHull.root d)).2 T).symm
exact he ▸ hsumI
have hτ : IsStoppingTime (filtration d).rightCont τ :=
(canonicalPosteriorExit_localizing hd hμ hb hcrit (ConnectedHull.root d)).isStoppingTime n
have hk : 0 ≤ 1 / (1 - PosteriorStoppingTimes.threshold n ^ 2) := by
apply one_div_nonneg.mpr
nlinarith [PosteriorStoppingTimes.threshold_nonneg n,
PosteriorStoppingTimes.threshold_lt_one n]
have hbK : ∀ᵐ x ∂observationLaw μ, ∀ t, |K t x| ≤
1 / (1 - PosteriorStoppingTimes.threshold n ^ 2) :=
Eventually.of_forall (fun x t => canonicalClippedWeight_bound beta n t x)
have hK := (canonicalClippedWeight_progressive (d := d) beta n).stronglyAdapted
have hnorm := StoppedRowNormalization.stopped_row_normalization
(fun a => (innovationMap_isBrownianReal hμ a).toIsPreBrownianReal)
(innovationMap_stronglyAdapted beta)
(fun a s t hst => BrownianArrayFiltration.coordinate_increment_indep_past
(B := fun a t x => innovationMap beta x a t) (F := filtration d)
(innovationMap_increment_indep_filtration hμ s t hst) a)
(fun a => (innovationMap_isBrownianReal hμ a).cont)
(fun a => (continuousRootCovariance_progressive hb a).stronglyAdapted)
(fun a => (continuousRootResponse_progressive hb a).stronglyAdapted) hK
(continuousRootCovariance_continuous_ae hd hμ hb hcrit)
(canonicalClippedWeight_continuous_ae hd hμ hb hcrit n)
(c := fun _ => 1) (r := fun _ => 1) (fun _ => by norm_num) hk
(fun a => Eventually.of_forall (fun x t => continuousRootCovariance_abs_le_one hb a t x))
(fun a => Eventually.of_forall (fun x t => continuousRootResponse_abs_le_one hb a t x)) hbK
hIM hJM hM hZM hI2 hJ2 hM2 hZ2 hcI hcJ hcM hcZ hIp
(continuousPosterior_progressive beta (ConnectedHull.root d)) T hsumM hsumJ hIlim hJlim hτ
(fun a => (canonicalClippedWeight_mul_covariance_before_exit hd hμ hb hcrit n).mono
(fun x hx t ht => hx t ht a))
obtain ⟨L, hmL, hL2, hlimL, hfield⟩ :=
canonical_artanh_decomposition hd hμ hb hcrit (ConnectedHull.root d) n T
have hlimL' : Tendsto (fun k => ∫ x,
(uniformSum (stoppedProcess M τ) K T k x - L x) ^ 2 ∂observationLaw μ)
atTop (𝓝 0) := by
apply hlimL.congr
intro k
apply integral_congr_ae
filter_upwards [canonicalClippedWeight_uniformSum_eq hd hμ hb hcrit n T k] with x hx
rw [hx]
have hMs2 (t : ℝ≥0) := StoppedMartingaleL2.memLp_stoppedProcess_two
(martingale_rightCont_of_ae_continuous_memLp_two hM hM2 hcM) hM2 hcM hτ t
have hZs2 := StoppedMartingaleL2.memLp_stoppedProcess_two
(martingale_rightCont_of_ae_continuous_memLp_two hZM hZ2 hcZ) hZ2 hcZ hτ T
have heq := ae_eq_of_mean_square_limits
(fun k => BrownianContinuousAssociativity.uniformSum_memLp hMs2 hK hbK T k)
hL2 hZs2 hlimL' hnorm
filter_upwards [hfield, heq] with x hx he
exact hx.trans (congrArg (fun z => z +
stoppedProcess (nnrealTimePrimitive (canonicalResponseDrift beta (ConnectedHull.root d))) τ T x) he)
theorem canonical_field_eq_normalized_integral
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {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)
{J : TreeVertex d → ℝ≥0 → Paths d → ℝ} {Z : ℝ≥0 → Paths d → ℝ}
(hJM : ∀ a, Martingale (J a) (filtration d) (observationLaw μ))
(hZM : Martingale Z (filtration d) (observationLaw μ))
(hJ2 : ∀ a t, MemLp (J a t) 2 (observationLaw μ))
(hZ2 : ∀ t, MemLp (Z t) 2 (observationLaw μ))
(hcJ : ∀ a, ∀ᵐ x ∂observationLaw μ, Continuous (fun t => J a t x))
(hcZ : ∀ᵐ x ∂observationLaw μ, Continuous (fun t => Z t x))
(hsumJ : ∀ t, HasSum (fun a => (hJ2 a t).toLp (J a t)) ((hZ2 t).toLp (Z t)))
(hJlim : ∀ a (N : ℕ), 0 < N → ∀ t ≤ (N : ℝ≥0),
Tendsto (fun k => ∫ x,
(BrownianIntegralProcess.stoppedUniformSum
(fun s x => innovationMap beta x a s) (continuousRootResponse hb a)
N k t x - J a t x) ^ 2 ∂observationLaw μ) atTop (𝓝 0)) :
∀ᵐ x ∂observationLaw μ, ∀ t,
continuousField beta (ConnectedHull.root d) t x = Z t x +
nnrealTimePrimitive (canonicalResponseDrift beta (ConnectedHull.root d)) t x := by
obtain ⟨I, hI2, hIp, hIM, hcI, hI0, hImean, hIenergy, hIlim, hsumI⟩ :=
exists_canonical_root_covariance_integral_representation hd hμ hb hcrit
have hind (a : TreeVertex d) (s t : ℝ≥0) (hst : s ≤ t) :=
BrownianArrayFiltration.coordinate_increment_indep_past
(B := fun a t x => innovationMap beta x a t) (F := filtration d)
(innovationMap_increment_indep_filtration hμ s t hst) a
have hIuniform (a : TreeVertex d) (T : ℝ≥0) :=
BrownianArbitraryHorizon.uniformSum_tendsto_of_integer_stopped_limits
(innovationMap_isBrownianReal hμ a).toIsPreBrownianReal
(innovationMap_stronglyAdapted beta a) (hind a)
(innovationMap_isBrownianReal hμ a).cont
(continuousRootCovariance_progressive hb a).stronglyAdapted
(continuousRootCovariance_continuous_ae hd hμ hb hcrit a)
(show (0 : ℝ) ≤ 1 by norm_num)
(Eventually.of_forall (fun x t => continuousRootCovariance_abs_le_one hb a t x))
(hIM a) (hI2 a) (hcI a) (hIlim a) T
have hJuniform (a : TreeVertex d) (T : ℝ≥0) :=
BrownianArbitraryHorizon.uniformSum_tendsto_of_integer_stopped_limits
(innovationMap_isBrownianReal hμ a).toIsPreBrownianReal
(innovationMap_stronglyAdapted beta a) (hind a)
(innovationMap_isBrownianReal hμ a).cont
(continuousRootResponse_progressive hb a).stronglyAdapted
(continuousRootResponse_continuous_ae hd hμ hb hcrit a)
(show (0 : ℝ) ≤ 1 by norm_num)
(Eventually.of_forall (fun x t => continuousRootResponse_abs_le_one hb a t x))
(hJM a) (hJ2 a) (hcJ a) (hJlim a) T
have hstop (n : ℕ) (T : ℝ≥0) :=
canonical_stopped_field_eq_of_integral_limits hd hμ hb hcrit
(fun a => (hIp a).isStronglyProgressive) hIM hJM hZM hI2 hJ2 hZ2 hcI hcJ hcZ
n T (hsumI T) (hsumJ T) (fun a => hIuniform a T) (fun a => hJuniform a T)
exact StoppedIdentityExhaustion.ae_all_add_of_localizing_stopped_add
(canonicalPosteriorExit_localizing hd hμ hb hcrit (ConnectedHull.root d))
(continuousField_modification hd hμ hb hcrit (ConnectedHull.root d)).1
hcZ (canonicalResponseDrift_primitive_continuous hμ hb (ConnectedHull.root d)) hstop
end Problem367.IsingObservationFactor
end
end OAI
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