| import Mathlib |
| import OAI.Probability.Ballisticity.Coupling.QuenchedPrefixDeterminedFuture |
| import OAI.Probability.Ballisticity.Estimates.RecordOrPrefixAdd |
|
|
| namespace OAI |
|
|
| section |
| section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| lemma tubePrefix_mono_radius {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) |
| (θ z z' : ℝ) (hzz : z ≤ z') (s : ℕ) : TubePrefix ℓ f x θ z s ⊆ TubePrefix ℓ f x θ z' s := by |
| rintro X hX |
| obtain ⟨n,hn⟩ := Set.mem_iUnion.mp hX |
| apply Set.mem_iUnion.mpr |
| refine ⟨n,hn.1,?_⟩ |
| intro j hj |
| obtain ⟨a,ha,hr,hg⟩ := hn.2 j hj |
| exact ⟨a,ha,hr,hg.trans hzz⟩ |
| |
| lemma tubePrefixMass_mono_radius {d : ℕ} (ℓ : Vector d) (f : Direction d) |
| (θ z z' : ℝ) (hzz : z ≤ z') (s : ℕ) (π : Measure (Lattice d)) (ω : Environment d) : |
| tubePrefixMass ℓ f θ z s π ω ≤ tubePrefixMass ℓ f θ z' s π ω := |
| lintegral_mono fun x => measure_mono (tubePrefix_mono_radius ℓ f x θ z z' hzz s) |
| |
| lemma tubePrefixMass_comp_le {d : ℕ} (e f : Direction d) |
| (θ z₁ z₂ : ℝ) (hz₂ : 0 ≤ z₂) (s k : ℕ) (π : Measure (Lattice d)) (ω : Environment d) : |
| tubePrefixMass (realPosition (step e)) f θ z₂ k |
| (tubeEndpointMixture (realPosition (step e)) f θ z₁ s π ω) ω ≤ |
| tubePrefixMass (realPosition (step e)) f θ (z₁+z₂) (s+k) π ω := by |
| let ℓ := realPosition (step e) |
| rw [tubePrefixMass,tubeEndpointMixture_future ℓ f θ z₁ s π ω |
| (fun y => TubePrefix ℓ f y θ z₂ k) (fun y => measurableSet_tubePrefix ℓ f y θ z₂ k)] |
| apply lintegral_mono; intro x |
| apply measure_mono_ae |
| filter_upwards [quenched_initial_ae (ω,x),quenched_nearest_neighbor (ω,x)] with X h0 hnn |
| intro hX |
| obtain ⟨n,hn⟩ := Set.mem_iUnion.mp hX |
| obtain ⟨m,hm⟩ := Set.mem_iUnion.mp hn.2 |
| exact Set.mem_iUnion.mpr ⟨n+m,tubePrefixAt_concatenate e f x θ z₁ z₂ hz₂ s n k m X h0 hnn hn.1 hm⟩ |
| |
| lemma directBridge_lintegral {d : ℕ} (e : Direction d) (π : Measure (Lattice d)) |
| (ω : Environment d) (G : Lattice d → ℝ≥0∞) : |
| (∫⁻ y, G y ∂directBridge e π ω) = ∫⁻ x, ((ω x).1 e : ℝ≥0∞)*G (x+step e) ∂π := by |
| rw [directBridge,lintegral_sum_measure,lintegral_countable'] |
| apply tsum_congr; intro x |
| rw [lintegral_smul_measure,lintegral_dirac',smul_eq_mul] |
| · ring |
| · exact measurable_of_countable G |
| |
| lemma single_step_tubePrefixAt {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (x : Lattice d) (θ : ℝ) (X : Path d) (h0 : X 0=x) (h1 : X 1=x+step e) : |
| X ∈ TubePrefixAt (realPosition (step e)) f x θ |θ| 1 1 := by |
| let ℓ := realPosition (step e) |
| have hheight : dot (realPosition (X 1)) ℓ = dot (realPosition x) ℓ+1 := by |
| rw [h1,signedHeight_projection,signedHeight_add_step_self,Int.cast_add,Int.cast_one,signedHeight_projection] |
| have hzero : dot (realPosition (0:Lattice d)) ℓ=0 := by simp [dot,realPosition] |
| have hstrict : StrictRecord ℓ (fun k => X k-x) 1 := by |
| intro j hj |
| have hj0 : j=0 := by omega |
| simp only [hj0,h0,sub_self,hzero] |
| rw [dot_realPosition_sub,hheight] |
| linarith |
| refine ⟨⟨?_,?_⟩,?_⟩ |
| · simpa only [Upper,Set.mem_ofPred_eq,Nat.cast_one] using le_of_eq hheight.symm |
| · intro j hj |
| have hj0 : j=0 := by omega |
| simp only [Strip,Set.mem_ofPred_eq,Nat.cast_one,hj0,h0] |
| exact ⟨le_rfl,lt_add_one _⟩ |
| · intro j hj |
| rcases show j=0 ∨ j=1 by omega with rfl | rfl |
| · refine ⟨0,by omega,by simp [RecordOrZeroPrefix],?_⟩ |
| simp [h0,signedCoordinate] |
| · refine ⟨1,le_rfl,?_,?_⟩ |
| · rw [RecordOrZeroPrefix,ite_eq_right (by decide : (1:ℕ) ≠ 0)] |
| refine ⟨by decide,hstrict,?_,?_⟩ |
| · rw [show (1:ℕ)=0+1 from rfl,recordCount_succ,recordCount_zero,ite_eq_left hstrict] |
| · intro j hj |
| rcases show j=0 ∨ j=1 by omega with rfl | rfl |
| · change dot (realPosition (0:Lattice d)) ℓ ≤ dot (realPosition (X 0-x)) ℓ |
| rw [h0,sub_self] |
| · change dot (realPosition (0:Lattice d)) ℓ ≤ dot (realPosition (X 1-x)) ℓ |
| rw [dot_realPosition_sub,hheight,hzero]; linarith |
| · have he : X 1-x = step e := by rw [h1]; abel |
| have hz : signedCoordinate f (step e)=0 := by |
| have hh := signedCoordinate_add_distinct_step e f hef (0:Lattice d) |
| simpa [signedCoordinate] using hh |
| simp only [he,hz,Nat.cast_one,one_mul,zero_sub,abs_neg,le_refl] |
| |
| lemma directBridge_tubeMass_le {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (θ z : ℝ) (hz : 0 ≤ z) (k : ℕ) (π : Measure (Lattice d)) (ω : Environment d) : |
| tubePrefixMass (realPosition (step e)) f θ z k (directBridge e π ω) ω ≤ |
| tubePrefixMass (realPosition (step e)) f θ (|θ|+z) (1+k) π ω := by |
| let ℓ := realPosition (step e) |
| rw [tubePrefixMass,directBridge_lintegral] |
| apply lintegral_mono; intro x |
| let v := wordPath x [e] |
| have hm := quenched_prefix_future ω x v (by simp [v]) 1 |
| (measurableSet_tubePrefix ℓ f (x+step e) θ z k) |
| have hv1 : v 1=x+step e := by simp [v,wordPath] |
| rw [hv1,show pathCylinder v 1=wordCylinder x [e] from rfl,quenched_wordCylinder] at hm |
| have hw : ENNReal.ofReal (wordWeight ω x [e])=((ω x).1 e:ℝ≥0∞) := by simp [wordWeight] |
| rw [hw] at hm |
| change ((ω x).1 e:ℝ≥0∞)*(quenchedKernel (ω,x+step e)) (TubePrefix ℓ f (x+step e) θ z k) ≤ _ |
| rw [← hm] |
| apply measure_mono_ae |
| filter_upwards [quenched_nearest_neighbor (ω,x)] with X hnn |
| rintro ⟨hfut,hcy⟩ |
| have h0 : X 0=x := by simpa [v] using hcy 0 (by omega) |
| have h1 : X 1=x+step e := by simpa [wordPath] using hcy 1 le_rfl |
| obtain ⟨m,hm⟩ := Set.mem_iUnion.mp hfut |
| apply Set.mem_iUnion.mpr |
| exact ⟨1+m,tubePrefixAt_concatenate e f x θ |θ| z hz 1 1 k m X h0 hnn |
| (single_step_tubePrefixAt e f hef x θ X h0 h1) (by simpa only [h1] using hm)⟩ |
| |
| lemma bridged_continuation_le_tube {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (θ z : ℝ) (hz : 0 ≤ z) (hθ : |θ| ≤ z) (H s : ℕ) (hs0 : 0 < s) (hsH : s ≤ H) |
| (π : Measure (Lattice d)) (ω : Environment d) : |
| tubePrefixMass (realPosition (step e)) f θ z (H-s) |
| (directBridge e (stoppedTubeEndpoint (realPosition (step e)) f θ z (s-1) π ω) ω) ω ≤ |
| tubePrefixMass (realPosition (step e)) f θ (3*z) H π ω := by |
| let ℓ := realPosition (step e) |
| have hb := directBridge_tubeMass_le e f hef θ z hz (H-s) |
| (stoppedTubeEndpoint ℓ f θ z (s-1) π ω) ω |
| by_cases hs : s-1=0 |
| · have hs1 : s=1 := by omega |
| have he : 1+(H-s)=H := by omega |
| simp only [stoppedTubeEndpoint,hs,ite_eq_left,he] at hb |
| simpa only [stoppedTubeEndpoint,hs,ite_eq_left] using hb.trans (tubePrefixMass_mono_radius ℓ f θ (|θ|+z) (3*z) (by linarith) H π ω) |
| · simp only [stoppedTubeEndpoint,hs] at hb |
| have hc := tubePrefixMass_comp_le e f θ z (|θ|+z) (by positivity) (s-1) (1+(H-s)) π ω |
| have he : (s-1)+(1+(H-s))=H := by omega |
| rw [he] at hc |
| simpa only [stoppedTubeEndpoint,hs,ite_false] using (hb.trans hc).trans (tubePrefixMass_mono_radius ℓ f θ (z+(|θ|+z)) (3*z) (by linarith) H π ω) |
| |
| end DirectionalTransience |
| |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
| |
| lemma tubePrefixMass_ne_top {d : ℕ} (ℓ : Vector d) (f : Direction d) |
| (θ z : ℝ) (s : ℕ) (π : Measure (Lattice d)) [IsFiniteMeasure π] (ω : Environment d) : |
| tubePrefixMass ℓ f θ z s π ω ≠ ⊤ := by |
| rw [← tubeEndpointMixture_total] |
| exact measure_ne_top _ _ |
| |
| theorem short_block_slope_survival {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))) |
| {w : ℝ} (hw : 0 < w) (hw1 : w ≤ 20) : |
| ∃ C : ℝ, 0 < C ∧ ∃ t₀ : ℝ, 0 < t₀ ∧ ∀ t : ℝ, 0 < t → t ≤ t₀ → |
| ∃ δ : ℝ, 0 < δ ∧ ∃ R : ℝ, 0 < R ∧ |
| ∀ r : ℝ, R ≤ r → ∀ H : ℕ, 0 < H → |
| (H:ℝ) ≤ t*fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) r → |
| let ℓ := realPosition (step e) |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) |
| let θ := (recordMedian ν ℓ hp f H:ℝ)/H |
| ∀ (a : ℤ) (π : Measure (Lattice d)), IsProbabilityMeasure π → |
| (∀ᵐ x ∂π, signedHeight e x = a) → |
| (environmentLaw ν).real {ω | (tubePrefixMass ℓ f θ (3*(w*r)) H π ω).toReal < δ} ≤ C*t^2 := by |
| obtain ⟨C,hC,t₀,ht₀,hb⟩ := canonical_two_tube_threat ν hue e f hef htrans hw hw1 |
| obtain ⟨κ,hκ,hUE⟩ := environment_uniform_elliptic ν hue |
| refine ⟨C,hC,t₀,ht₀,?_⟩ |
| intro t ht htt |
| obtain ⟨δ₀,hδ₀,hδ₀1,R,hR,hb⟩ := hb t ht htt |
| let δ := min δ₀ ((κ:ℝ)*δ₀^2) |
| have hκreal : (0:ℝ) < κ := by exact_mod_cast hκ |
| have hδ : 0 < δ := lt_min hδ₀ (by positivity) |
| refine ⟨δ,hδ,R,hR,?_⟩ |
| intro r hr H hH hscale |
| obtain ⟨hθ,hb⟩ := hb r hr H hH hscale |
| let ℓ := realPosition (step e) |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) |
| let θ := (recordMedian ν ℓ hp f H:ℝ)/H |
| dsimp only |
| intro a π hπ hsupp |
| let := hπ |
| obtain ⟨x₀,hx₀⟩ := hsupp.exists |
| have hb' := hb a x₀ hx₀ π hπ hsupp |
| have hz : 0 ≤ w*r := mul_nonneg hw.le (hR.trans_le hr).le |
| have hδ1 : δ₀ ≤ 1 := by linarith |
| have hπ' : ∀ᵐ x ∂π, dot (realPosition x) ℓ = (a:ℝ) := |
| hsupp.mono fun x hx => by rw [signedHeight_projection,hx] |
| have hδEN : ENNReal.ofReal δ ≤ (κ:ℝ≥0∞)*ENNReal.ofReal δ₀*ENNReal.ofReal δ₀ := by |
| have he : ENNReal.ofReal ((κ:ℝ)*δ₀^2) = (κ:ℝ≥0∞)*ENNReal.ofReal δ₀*ENNReal.ofReal δ₀ := by |
| rw [pow_two,← mul_assoc,ENNReal.ofReal_mul (mul_nonneg hκreal.le hδ₀.le), |
| ENNReal.ofReal_mul hκreal.le,ENNReal.ofReal_coe_nnreal] |
| rw [← he] |
| exact ENNReal.ofReal_le_ofReal (min_le_right _ _) |
| have hle : environmentLaw ν {ω | (tubePrefixMass ℓ f θ (3*(w*r)) H π ω).toReal < δ} ≤ |
| environmentLaw ν (⋃ s ∈ Finset.range (H+1), |
| {ω | ω ∈ FirstTubeThreat ℓ f a θ (w*r) δ₀ s π ∧ |
| (bridgedRestart e f x₀ θ (w*r) π s ω,ω) ∈ TubeThreat ℓ f ((a:ℝ)+s) θ (w*r) δ₀ H}) := by |
| apply measure_mono_ae |
| filter_upwards [hUE] with ω hω |
| intro hbad |
| have hcontra (h : ENNReal.ofReal δ ≤ tubePrefixMass ℓ f θ (3*(w*r)) H π ω) : False := by |
| have hh := ENNReal.toReal_mono (tubePrefixMass_ne_top ℓ f θ (3*(w*r)) H π ω) h |
| rw [ENNReal.toReal_ofReal hδ.le] at hh |
| exact (not_lt_of_ge hh) hbad |
| by_cases hfirst : (π,ω) ∈ TubeThreat ℓ f a θ (w*r) δ₀ H |
| · have hh : ω ∈ ⋃ s ≤ H, FirstTubeThreat ℓ f a θ (w*r) δ₀ s π := by |
| rw [firstTubeThreat_iUnion] |
| exact hfirst |
| obtain ⟨s,hs⟩ := Set.mem_iUnion.mp hh |
| obtain ⟨hsH,hs⟩ := Set.mem_iUnion.mp hs |
| by_cases hsecond : (bridgedRestart e f x₀ θ (w*r) π s ω,ω) ∈ TubeThreat ℓ f ((a:ℝ)+s) θ (w*r) δ₀ H |
| · exact Set.mem_iUnion.mpr ⟨s,Set.mem_iUnion.mpr ⟨Finset.mem_range.mpr (Nat.lt_succ_of_le hsH),hs,hsecond⟩⟩ |
| · exfalso |
| apply hcontra |
| exact (hδEN.trans (firstThreat_noSecond_retained_mass e f a x₀ hx₀ θ (w*r) δ₀ hz hδ1 H s hsH π hsupp ω |
| κ (fun y => hω y e) hs hsecond)).trans |
| (bridged_continuation_le_tube e f hef θ (w*r) hz hθ H s hs.1 hsH π ω) |
| · exfalso |
| apply hcontra |
| exact ((ENNReal.ofReal_le_ofReal (min_le_left _ _)).trans |
| (noThreat_mass_lower ℓ f a θ (w*r) δ₀ hz hδ1 H π hπ' ω hfirst le_rfl)).trans |
| (tubePrefixMass_mono_radius ℓ f θ (w*r) (3*(w*r)) (by linarith) H π ω) |
| exact (ENNReal.toReal_mono (measure_ne_top _ _) hle).trans hb' |
| |
| end DirectionalTransience |
| |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| |
| end |
| end |
| |
| end OAI |
| |
