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180bc03 | 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 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.UpperPairPatch
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter Function
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
lemma finiteLogExcess_eq_ciSup {Ω : Type*} (A : ℝ) (p : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) :
finiteLogExcess A p N ω = ⨆ n : Fin (N+1), max (-Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2)) 0 := by
unfold finiteLogExcess
rw [ENNReal.toReal_iSup (fun _ => ENNReal.ofReal_ne_top)]
simp only [ENNReal.toReal_ofReal']
lemma finiteLogExcess_stability {Ω : Type*} (A : ℝ) (p q : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω)
{ε : ℝ} (hε : 0 ≤ ε) (hpq : ∀ n ≤ N, |Real.log (p n ω)-Real.log (q n ω)| ≤ ε) :
|finiteLogExcess A p N ω-finiteLogExcess A q N ω| ≤ ε := by
have one_side (p q : ℕ → Ω → ℝ)
(hpq : ∀ n ≤ N, |Real.log (p n ω)-Real.log (q n ω)| ≤ ε) :
finiteLogExcess A p N ω ≤ finiteLogExcess A q N ω+ε := by
rw [finiteLogExcess_eq_ciSup]
apply ciSup_le
intro n
have hq : max (-Real.log (q n ω)-2*A*Real.log ((n:ℝ)+2)) 0 ≤ finiteLogExcess A q N ω := by
rw [finiteLogExcess_eq_ciSup]
exact le_ciSup (Set.finite_range (fun k : Fin (N+1) => max (-Real.log (q k ω)-2*A*Real.log ((k:ℝ)+2)) 0)).bddAbove n
have hh := (abs_le.mp (hpq n (by omega))).1
apply (max_le_max (by linarith : -Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2) ≤
-Real.log (q n ω)-2*A*Real.log ((n:ℝ)+2)+ε) (show (0:ℝ) ≤ 0+ε by linarith)).trans
simpa only [max_add_add_right] using add_le_add_left hq ε
have h₁ := one_side p q hpq
have h₂ := one_side q p (fun n hn => by simpa only [abs_sub_comm] using hpq n hn)
exact abs_le.mpr ⟨by linarith,by linarith⟩
lemma log_uniform_on_positive_interval {c ε : ℝ} (hc : 0 < c) (hε : 0 < ε) :
∃ δ > 0, ∀ p q : ℝ, c ≤ p → p ≤ 1 → c ≤ q → q ≤ 1 → |p-q| < δ →
|Real.log p-Real.log q| < ε := by
have hcont : ContinuousOn Real.log (Set.Icc c 1) :=
Real.continuousOn_log.mono (fun x hx => ne_of_gt (hc.trans_le hx.1))
obtain ⟨δ,hδ,hd⟩ := Metric.uniformContinuousOn_iff.mp (isCompact_Icc.uniformContinuousOn_of_continuous hcont) ε hε
refine ⟨δ,hδ,?_⟩
intro p q hp hp1 hq hq1 hdif
exact hd p ⟨hp,hp1⟩ q ⟨hq,hq1⟩ hdif
end DirectionalTransience
end
section
open MeasureTheory ProbabilityTheory Filter Function
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
lemma outwardKernelMass_le_one {d : ℕ} (e f : Direction d) (H : ℕ)
(x : Lattice d × Lattice d) (ω : Environment d) : outwardKernelMass e f H x ω ≤ 1 := by
have hh := ENNReal.toReal_mono ENNReal.one_ne_top
((measure_mono (Set.subset_univ {y : Lattice d × Lattice d | signedCoordinate f (x.2-x.1) ≤ signedCoordinate f (y.2-y.1)})).trans (rawPairEndpointLaw_total_le_one (realPosition (step e)) H ω x))
simpa only [outwardKernelMass,ENNReal.toReal_one] using hh
lemma outwardLog_fixed_height_locality {d : ℕ} {κ : ℝ≥0} (hκ : 0 < κ)
(e f : Direction d) {H : ℕ} (hH : 0 < H) {ε : ℝ} (hε : 0 < ε) :
∃ N : ℕ, ∀ (a : ℝ) (x : Lattice d × Lattice d), x ∈ PairAtHeight (realPosition (step e)) a →
∀ (ω : Environment d), (∀ y g, κ ≤ (ω y).1 g) →
|Real.log (outwardKernelMass e f H x ω)-
Real.log (outwardKernelMass e f H x (upperPairPatch e a x N ω))| < ε := by
have hκr : 0 < (κ:ℝ) := by exact_mod_cast hκ
obtain ⟨δ,hδ,hlog⟩ := log_uniform_on_positive_interval (pow_pos hκr (2*H)) hε
obtain ⟨N,hN⟩ := rawPairEndpointLaw_upper_patch_locality hκ e hH
(ENNReal.ofReal_pos.mpr (show 0 < δ/2 by linarith))
refine ⟨N,?_⟩
intro a x hx ω hω
let η := upperPairPatch e a x N ω
have hη := upperPairPatch_elliptic e a x N ω hω
apply hlog _ _ (outwardKernelMass_lower e f H ω x (fun y => hω y e))
(outwardKernelMass_le_one e f H x ω)
(outwardKernelMass_lower e f H η x (fun y => hη y e))
(outwardKernelMass_le_one e f H x η)
let S := {y : Lattice d × Lattice d | signedCoordinate f (x.2-x.1) ≤ signedCoordinate f (y.2-y.1)}
have hfin (ζ : Environment d) : rawPairEndpointLaw (realPosition (step e)) H ζ x S ≠ ⊤ :=
ne_top_of_le_ne_top ENNReal.one_ne_top ((measure_mono (Set.subset_univ _)).trans (rawPairEndpointLaw_total_le_one _ _ _ _))
have hh := hN a x hx ω hω S
have h₁ := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨hfin η,ENNReal.ofReal_ne_top⟩) hh.1
have h₂ := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨hfin ω,ENNReal.ofReal_ne_top⟩) hh.2
rw [ENNReal.toReal_add (hfin η) ENNReal.ofReal_ne_top,ENNReal.toReal_ofReal (by linarith : 0 ≤ δ/2)] at h₁
rw [ENNReal.toReal_add (hfin ω) ENNReal.ofReal_ne_top,ENNReal.toReal_ofReal (by linarith : 0 ≤ δ/2)] at h₂
change outwardKernelMass e f H x ω ≤ outwardKernelMass e f H x η+δ/2 at h₁
change outwardKernelMass e f H x η ≤ outwardKernelMass e f H x ω+δ/2 at h₂
exact abs_lt.mpr ⟨by linarith,by linarith⟩
end DirectionalTransience
end
end
end OAI
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