| import Mathlib |
| import OAI.Probability.Ballisticity.Coupling.SeedUniform |
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| namespace OAI |
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| section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
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| noncomputable def fluctuationRadius {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) (S : Ω → ℝ) (s : ℝ) : ℝ := |
| if h : ∃ r : ℝ, 0<r ∧ fluctuationScale μ S r=s then h.choose else 1 |
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| lemma fluctuationRadius_spec {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (S : Ω → ℝ) (hS : Measurable S) |
| (hI : Integrable S μ) (hne : 0<μ {x | S x≠0}) {R s : ℝ} (hR : 0<R) |
| (hs : fluctuationScale μ S R<s) : |
| R<fluctuationRadius μ S s ∧ fluctuationScale μ S (fluctuationRadius μ S s)=s := by |
| have hcont := (continuousOn_fluctuationScale μ S hS hne).mono |
| (show Set.Ici R ⊆ Set.Ioi 0 from fun r hr => hR.trans_le hr) |
| obtain ⟨r,hr,he⟩ := intermediate_value_Ici hcont (fluctuationScale_tendsto μ S hS hI hne) hs.le |
| have hex : ∃ r : ℝ, 0<r ∧ fluctuationScale μ S r=s := ⟨r,hR.trans_le hr,he⟩ |
| have hpos : 0<fluctuationRadius μ S s := by simp only [fluctuationRadius,dite_eq_left hex]; exact hex.choose_spec.1 |
| have heq : fluctuationScale μ S (fluctuationRadius μ S s)=s := by simp only [fluctuationRadius,dite_eq_left hex]; exact hex.choose_spec.2 |
| refine ⟨?_,heq⟩ |
| by_contra hn |
| have hh := fluctuationScale_mono μ S hS hne hpos (not_lt.mp hn) |
| rw [heq] at hh |
| exact (not_le_of_gt hs) hh |
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| lemma fluctuationRadius_tendsto {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (S : Ω → ℝ) (hS : Measurable S) |
| (hI : Integrable S μ) (hne : 0<μ {x | S x≠0}) : |
| Tendsto (fluctuationRadius μ S) atTop atTop := by |
| apply tendsto_atTop.mpr |
| intro bound |
| filter_upwards [eventually_gt_atTop (fluctuationScale μ S (max bound 1))] with s hs |
| exact (le_max_left bound 1).trans |
| (fluctuationRadius_spec μ S hS hI hne (lt_of_lt_of_le (by norm_num) (le_max_right _ _)) hs).1.le |
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| lemma fluctuationRadius_eventual_unique {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (S : Ω → ℝ) (hS : Measurable S) |
| (hI : Integrable S μ) (hne : 0<μ {x | S x≠0}) : |
| ∀ᶠ s in atTop, ∀ r : ℝ, 0<r → fluctuationScale μ S r=s → r=fluctuationRadius μ S s := by |
| obtain ⟨R,hR,hstrict⟩ := fluctuationScale_eventually_strictMono μ S hS hne |
| filter_upwards [eventually_gt_atTop (fluctuationScale μ S R)] with s hs r hr he |
| obtain ⟨hRR,hRspec⟩ := fluctuationRadius_spec μ S hS hI hne hR hs |
| have hrR : R≤r := by |
| by_contra hn |
| have hh := fluctuationScale_mono μ S hS hne hr (not_le.mp hn).le |
| rw [he] at hh |
| exact (not_le_of_gt hs) hh |
| exact hstrict.injOn hrR hRR.le (he.trans hRspec.symm) |
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| theorem separated_seeds {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))) |
| (k : ℕ) (hk : 2≤k) (p : ℝ) (hp : 0<p) : |
| ∃ C c₀ g₀ : ℝ, 0<C ∧ 0<c₀ ∧ 0<g₀ ∧ ∃ H₀ : ℕ, ∀ H≥H₀, |
| ∀ π : ProbabilityMeasure (Fin k → Lattice d), |
| 1-p < (environmentLaw ν).real {ω | ENNReal.ofReal g₀≤ |
| rawTupleMixture (realPosition (step e)) H π.toMeasure ω {v | TupleSeparated f |
| (c₀*fluctuationRadius (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) (H/C)) v}} := by |
| obtain ⟨C,c₀,g₀,R,hC,hc,hg,hR,hseed⟩ := seed_mass_at_scale ν hue e f hef htrans k hk p hp |
| let μ := independentConditionedPairLaw ν (realPosition (step e)) |
| let S := commonIncrementProcess (realPosition (step e)) f 0 |
| let : IsProbabilityMeasure μ := independentConditionedPairLaw_probability ν _ |
| (ne_of_gt (noDrop_positive_of_directionallyTransient ν _ htrans)) |
| have hS : Measurable S := measurable_commonIncrementProcess _ _ _ |
| have hI : Integrable S μ := independent_commonWordIncrement_integrable ν hue _ (signed_direction_unit e) htrans |
| (signedHeight e) (signedHeight_projection e) (signedHeight_step_le e) f |
| have hne : 0<μ {x | S x≠0} := independent_commonWordIncrement_nonzero ν hue e f hef htrans |
| have hlim : Tendsto (fun H : ℕ => (H:ℝ)/C) atTop atTop := |
| tendsto_natCast_atTop_atTop.atTop_div_const hC |
| obtain ⟨H₀,hH₀⟩ := eventually_atTop.mp (hlim.eventually (eventually_gt_atTop (fluctuationScale μ S R))) |
| refine ⟨C,c₀,g₀,hC,hc,hg,H₀,fun H hH π => ?_⟩ |
| obtain ⟨hr,hspec⟩ := fluctuationRadius_spec μ S hS hI hne hR (hH₀ H hH) |
| apply hseed _ hr.le H _ π |
| change (H:ℝ)=C*fluctuationScale μ S (fluctuationRadius μ S ((H:ℝ)/C)) |
| rw [hspec] |
| field_simp |
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| end DirectionalTransience |
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| end |
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| end OAI |
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