| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.UpperPairPatch |
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| namespace OAI |
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| section |
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| section |
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| open MeasureTheory ProbabilityTheory Filter Function |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
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| 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'] |
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| 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⟩ |
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| 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 |
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| end |
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| section |
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| open MeasureTheory ProbabilityTheory Filter Function |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
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| 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 |
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| 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 |
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| end |
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| end |
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| end OAI |
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