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9b6e014 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 | import Mathlib
import OAI.Probability.Ballisticity.Walk.Kernel
namespace OAI
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal Topology
namespace TailDecorrelation
lemma sampled_kernel_sections {D E J W : Type*}
[MeasurableSpace D] [MeasurableSpace E] [MeasurableSpace J] [MeasurableSpace W]
[Countable J] [MeasurableSingletonClass J]
(δ : Measure D) (σ : Measure E) (P : Measure W)
[IsProbabilityMeasure δ] [IsProbabilityMeasure σ] [IsProbabilityMeasure P]
(κ : Kernel D J) (s : D → E → J)
(hs : Measurable (Function.uncurry s)) (hmap : ∀ d, σ.map (s d) = κ d)
{f : D × W → ℝ} (hf : Measurable f) {Z : J → W → ℝ}
(hZ : ∀ a, Measurable (Z a))
(hi : Integrable (fun x : (D × E) × W => f (x.1.1,x.2) * Z (s x.1.1 x.1.2) x.2)
((δ.prod σ).prod P)) :
∀ᵐ x ∂δ.prod P, f x ≠ 0 → Integrable (fun a => Z a x.2) (κ x.1) := by
have hz : Measurable (Function.uncurry Z) := measurable_from_prod_countable_right hZ
let G : (D × E) × W → ℝ := fun x => f (x.1.1,x.2) * Z (s x.1.1 x.1.2) x.2
have hG : Measurable G :=
(hf.comp ((measurable_fst.comp measurable_fst).prodMk measurable_snd)).mul
(hz.comp ((hs.comp measurable_fst).prodMk measurable_snd))
have hp := swap_seed_preserving δ σ P
have hj : Integrable (fun x : (D × W) × E => f x.1 * Z (s x.1.1 x.2) x.1.2)
((δ.prod P).prod σ) := (hp.integrable_comp hG.aestronglyMeasurable).mpr hi
filter_upwards [hj.prod_right_ae] with x hx
intro hfx
have hzx : Integrable (fun u => Z (s x.1 u) x.2) σ := by
have hh := hx.const_mul (f x)⁻¹
simpa only [←mul_assoc, inv_mul_cancel₀ hfx, one_mul] using hh
rw [←hmap]
exact (integrable_map_measure (measurable_of_countable _).aestronglyMeasurable
(hs.comp measurable_prodMk_left).aemeasurable).mpr hzx
lemma kernel_density_sections {D I A J : Type*} [MeasurableSpace D] [MeasurableSpace A] [Encodable I]
[MeasurableSpace J] [Countable J] [MeasurableSingletonClass J]
[Nonempty J] [StandardBorelSpace J]
(δ : Measure D) (ν : Measure A) [IsProbabilityMeasure δ] [IsProbabilityMeasure ν]
{f : D × (I → A) → ℝ} {Z : J → (I → A) → ℝ}
{c Q : ℝ} (hc : 0 < c)
(hf : Measurable f) (hf0 : ∀ x, 0 ≤ f x)
(hfi : Integrable f (δ.prod (Measure.infinitePi (fun _ : I => ν))))
(hH : Integrable (fun x => entropyPlus (f x)) (δ.prod (Measure.infinitePi (fun _ : I => ν))))
(hZ : ∀ a, Measurable (Z a)) (hZ0 : ∀ a ω, 0 ≤ Z a ω)
(he : ∀ a, Integrable (fun ω => Real.exp (c * Z a ω)) (Measure.infinitePi (fun _ : I => ν)))
(heQ : ∀ a, (∫ ω, Real.exp (c * Z a ω) ∂Measure.infinitePi (fun _ : I => ν)) ≤ Q)
(κ : Kernel D J) [IsMarkovKernel κ] :
∀ᵐ x ∂δ.prod (Measure.infinitePi (fun _ : I => ν)),
f x ≠ 0 → Integrable (fun a => Z a x.2) (κ x.1) := by
obtain ⟨s,hs,hmap⟩ := Kernel.exists_measurable_map_eq_unitInterval κ
let σ : Measure unitInterval := volume
let g : (D × unitInterval) × (I → A) → ℝ := fun x => f (x.1.1,x.2)
have hg : Measurable g := hf.comp ((measurable_fst.comp measurable_fst).prodMk measurable_snd)
obtain ⟨_,hgH,_⟩ := randomized_density δ σ ν hf hfi hH
have hi := integrable_density_mul_of_entropy hc hg
(measurable_countable_sample hZ hs) (fun x => hf0 _) (fun x => hZ0 _ _) hgH
(sampled_exponential_integrable (δ.prod σ) ν hZ he heQ hs).1
exact sampled_kernel_sections δ σ (Measure.infinitePi (fun _ : I => ν))
κ s hs hmap hf hZ hi
end TailDecorrelation
end
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal Topology
namespace DirectionalTransience
lemma cut_neighborhoods_escape {d : ℕ} (e : Direction d) (a : ℝ)
(k : Fin d) (L U : ℕ → ℤ)
(hL : Tendsto L atTop atBot) (hU : Tendsto U atTop atTop)
(N : ℕ) (S : Set (Lattice d)) (hS : S.Finite) :
∀ᶠ r in atTop, ∀ p : Lattice d × Lattice d,
p.1 k ≤ L r → U r ≤ p.2 k → Disjoint S (upperPairNeighborhood e a p N) := by
have hz : ∀ z ∈ S, ∀ᶠ r in atTop, ∀ p : Lattice d × Lattice d,
p.1 k ≤ L r → U r ≤ p.2 k → z ∉ LatticeBall p.1 N ∪ LatticeBall p.2 N := by
intro z _
filter_upwards [(tendsto_atBot.mp hL) (z k - (N : ℤ) - 1),
(tendsto_atTop.mp hU) (z k + (N : ℤ) + 1)] with r hrx hry
intro p hp hq hz
rcases hz with hz | hz
· have hh := (abs_le.mp (hz k)).2
omega
· have hh := (abs_le.mp (hz k)).1
omega
filter_upwards [hS.eventually_all.mpr hz] with r hr
intro p hp hq
exact Set.disjoint_left.mpr fun z hz hz' => hr z hz p hp hq hz'.1
theorem actual_kernel_tail_decorrelation {d : ℕ}
(ν : Measure (Row d)) [IsProbabilityMeasure ν] (hue : UniformElliptic ν)
(e f : Direction d) (hef : e.1 ≠ f.1)
(htrans : DirectionallyTransient ν (realPosition (step e))) :
∃ A : ℝ, 0 < A ∧
(∀ᵐ ω ∂environmentLaw ν, ∀ x : Lattice d × Lattice d,
dot (realPosition x.1) (realPosition (step e)) =
dot (realPosition x.2) (realPosition (step e)) →
∀ t : ℕ, 0 < outwardKernelMass e f (t+1) x ω ∧
-Real.log (outwardKernelMass e f (t+1) x ω) ≤
2*A*Real.log ((t:ℝ)+2)+outwardCanonicalExcess e f A x ω) ∧
let J := {p : Lattice d × Lattice d // p ∈ PairAtHeight (realPosition (step e)) 0}
let Z : J → Environment d → ℝ := fun p => outwardCanonicalExcess e f A p.1
BddAbove (Set.range (fun p => ∫ ω, Z p ω ∂environmentLaw ν)) ∧
∀ (D : Type) [MeasurableSpace D] (δ : Measure D) [IsProbabilityMeasure δ]
(g : D × Environment d → ℝ),
Measurable g → (∀ x, 0 ≤ g x) → Integrable g (δ.prod (environmentLaw ν)) →
(∫ x, g x ∂δ.prod (environmentLaw ν)) = 1 →
Integrable (fun x => TailDecorrelation.entropyPlus (g x)) (δ.prod (environmentLaw ν)) →
∀ (κ : ℕ → Kernel D J) [∀ r, IsMarkovKernel (κ r)] (L U : ℕ → D → ℤ),
(∀ᵐ z ∂δ, Tendsto (fun r => L r z) atTop atBot ∧ Tendsto (fun r => U r z) atTop atTop) →
(∀ r, ∀ᵐ z ∂δ, ∀ᵐ p ∂κ r z, p.1.1 f.1 ≤ L r z ∧ U r z ≤ p.1.2 f.1) →
(∀ r, ∀ᵐ z ∂δ.prod (environmentLaw ν), g z ≠ 0 →
Integrable (fun p => Z p z.2) (κ r z.1)) ∧
(∀ r, Integrable (fun z => g z * (∫ p, Z p z.2 ∂κ r z.1)) (δ.prod (environmentLaw ν))) ∧
(∀ ε : ℝ, 0 < ε → ∀ᶠ r in atTop,
(∫ z, g z * (∫ p, Z p z.2 ∂κ r z.1) ∂δ.prod (environmentLaw ν)) ≤
(⨆ p, ∫ ω, Z p ω ∂environmentLaw ν) + ε) ∧
Filter.limsup (fun r => ∫ z, g z * (∫ p, Z p z.2 ∂κ r z.1)
∂δ.prod (environmentLaw ν)) atTop ≤ ⨆ p, ∫ ω, Z p ω ∂environmentLaw ν := by
classical
obtain ⟨A,c,B,hA,hc,hB,hbound,hloc⟩ := actual_outward_order ν hue e f hef htrans
let J := {p : Lattice d × Lattice d // p ∈ PairAtHeight (realPosition (step e)) 0}
let Z : J → Environment d → ℝ := fun p => outwardCanonicalExcess e f A p.1
have hZm : ∀ p : J, Measurable (Z p) := fun p =>
((hbound 0 p.1 p.2).2.1).mono (rowSigma_le _) le_rfl
have hZ0 : ∀ p : J, ∀ ω, 0 ≤ Z p ω := fun p => (hbound 0 p.1 p.2).1
have he : ∀ p : J, Integrable (fun ω => Real.exp (c * Z p ω)) (environmentLaw ν) :=
fun p => (hbound 0 p.1 p.2).2.2.1
have heB : ∀ p : J, (∫ ω, Real.exp (c * Z p ω) ∂environmentLaw ν) ≤ B :=
fun p => (hbound 0 p.1 p.2).2.2.2.1
have hZi : ∀ p : J, Integrable (Z p) (environmentLaw ν) := fun p =>
(integrable_and_mean_le_of_exp (environmentLaw ν) hc (hZm p) (hZ0 p) (he p) (heB p)).1
have hZB : ∀ p : J, (∫ ω, Z p ω ∂environmentLaw ν) ≤ B / c := fun p =>
(integrable_and_mean_le_of_exp (environmentLaw ν) hc (hZm p) (hZ0 p) (he p) (heB p)).2
have hba : BddAbove (Set.range (fun p : J => ∫ ω, Z p ω ∂environmentLaw ν)) :=
⟨B/c, by rintro _ ⟨p,rfl⟩; exact hZB p⟩
let p0 : J := ⟨(0,0), by simp [PairAtHeight, dot, realPosition]⟩
let : Nonempty J := ⟨p0⟩
have hC : 0 ≤ ⨆ p : J, ∫ ω, Z p ω ∂environmentLaw ν :=
(integral_nonneg fun ω => hZ0 p0 ω).trans (le_ciSup hba p0)
refine ⟨A,hA,outward_order_countable_event ν e f A (fun a x hx => (hbound a x hx).2.2.2.2),hba,?_⟩
intro D mD δ hδ g hgm hg0 hgi hg1 hgH κ hκ L U hcuts htail
have hcap : ∀ K η : ℝ, 0 < K → 0 < η → ∃ N : ℕ,
∃ F : J → Environment d → ℝ,
(∀ p, @Measurable _ _ (TailDecorrelation.rows (upperPairNeighborhood e 0 p.1 N)) _ (F p)) ∧
(∀ p ω, 0 ≤ F p ω ∧ F p ω ≤ K) ∧
(∀ p, (∫ ω, |min K (Z p ω) - F p ω| ∂environmentLaw ν) ≤ η) := by
intro K η hK hη
obtain ⟨N,hN⟩ := outward_cap_locality_L1 ν e f A hloc hK hη
have hh : ∀ p : J, ∃ F : Environment d → ℝ,
@Measurable _ _ (rowSigma (upperPairNeighborhood e 0 p.1 N)) _ F ∧
(∀ ω, 0 ≤ F ω ∧ F ω ≤ K) ∧
(∫ ω, |min (Z p ω) K - F ω| ∂environmentLaw ν) < η :=
fun p => hN 0 p.1 p.2
choose F hFm hFb hFe using hh
refine ⟨N,F,?_,hFb,?_⟩
· intro p
rw [analytic_rows_eq]
exact hFm p
· intro p
simpa only [min_comm] using (hFe p).le
have hesc : ∀ N (S : Set (Lattice d)), S.Finite → ∀ᵐ z ∂δ,
∀ᶠ r in atTop, ∀ᵐ p ∂κ r z, Disjoint S (upperPairNeighborhood e 0 p.1 N) := by
intro N S hS
filter_upwards [hcuts, ae_all_iff.mpr htail] with z hz ht
filter_upwards [cut_neighborhoods_escape e 0 f.1 (fun r => L r z) (fun r => U r z)
hz.1 hz.2 N S hS] with r hr
filter_upwards [ht r] with p hp
exact hr p.1 hp.1 hp.2
have he' : ∀ p : J, Integrable (fun ω => Real.exp ((2*(c/2)) * Z p ω)) (environmentLaw ν) := by
simpa only [mul_div_cancel₀ _ (by norm_num : (2:ℝ) ≠ 0)] using he
have heB' : ∀ p : J, (∫ ω, Real.exp ((2*(c/2)) * Z p ω) ∂environmentLaw ν) ≤ B := by
simpa only [show 2*(c/2) = c by ring] using heB
refine ⟨fun r => TailDecorrelation.kernel_density_sections δ ν hc hgm hg0 hgi hgH
hZm hZ0 he heB (κ r), ?_⟩
refine ⟨fun r => TailDecorrelation.kernel_density_integrable δ ν hc hgm hg0 hgi hgH
hZm hZ0 he heB (κ r), ?_⟩
have hE := TailDecorrelation.kernel_entropy_decorrelation δ ν (by linarith : 0 < c/2) hC
hgm hg0 hgi hg1 hgH hZm hZ0 he' heB' hZi (fun p => le_ciSup hba p)
(fun N p => upperPairNeighborhood e 0 p.1 N) κ hcap hesc
exact ⟨hE, TailDecorrelation.limsup_le_of_nonneg_eventual
(fun r => integral_nonneg fun z => mul_nonneg (hg0 z) (integral_nonneg fun p => hZ0 _ _)) hE⟩
end DirectionalTransience
end
end OAI
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