import Mathlib import OAI.Probability.Ballisticity.Walk.QuenchedRegularPathAe import OAI.Probability.Ballisticity.Crossings.CoordinateHitExact 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 namespace DirectionalTransience noncomputable def stoppedTubeEndpoint {d : ℕ} (ℓ : Vector d) (f : Direction d) (θ z : ℝ) (s : ℕ) (π : Measure (Lattice d)) (ω : Environment d) : Measure (Lattice d) := if s=0 then π else tubeEndpointMixture ℓ f θ z s π ω instance stoppedTubeEndpoint_finite {d : ℕ} (ℓ : Vector d) (f : Direction d) (θ z : ℝ) (s : ℕ) (π : Measure (Lattice d)) [IsFiniteMeasure π] (ω : Environment d) : IsFiniteMeasure (stoppedTubeEndpoint ℓ f θ z s π ω) := by unfold stoppedTubeEndpoint split_ifs <;> infer_instance lemma stoppedTubeEndpoint_rows {d : ℕ} (ℓ : Vector d) (f : Direction d) (a θ z : ℝ) (s : ℕ) (π : Measure (Lattice d)) (hπ : ∀ᵐ x ∂π, dot (realPosition x) ℓ = a) : @Measurable _ _ (rowSigma {y | dot (realPosition y) ℓ < a+s}) _ (stoppedTubeEndpoint ℓ f θ z s π) := by change @Measurable _ _ (rowSigma {y | dot (realPosition y) ℓ < a+s}) _ (fun ω => stoppedTubeEndpoint ℓ f θ z s π ω) by_cases hs : s=0 · simp only [stoppedTubeEndpoint,hs,ite_eq_left] apply @measurable_const · simp only [stoppedTubeEndpoint,hs] exact tubeEndpointMixture_lower_rows ℓ f a θ z (Nat.pos_of_ne_zero hs) π hπ lemma stoppedTubeEndpoint_supported {d : ℕ} (e f : Direction d) (a : ℤ) (θ z : ℝ) (s : ℕ) (π : Measure (Lattice d)) (hπ : ∀ᵐ x ∂π, signedHeight e x = a) (ω : Environment d) : ∀ᵐ y ∂stoppedTubeEndpoint (realPosition (step e)) f θ z s π ω, signedHeight e y = a+s := by by_cases hs : s=0 · simpa only [stoppedTubeEndpoint,hs,ite_eq_left,Nat.cast_zero,add_zero] using hπ · simp only [stoppedTubeEndpoint,hs] exact tubeEndpointMixture_supported e f a θ z s π hπ ω lemma signedHeight_add_nsmul_step {d : ℕ} (e : Direction d) (x : Lattice d) (s : ℕ) : signedHeight e (x+s • step e) = signedHeight e x+s := by induction s with | zero => simp | succ s ih => rw [succ_nsmul,← add_assoc,signedHeight_add_step_self,ih] simp only [Nat.cast_add,Nat.cast_one,add_assoc] noncomputable def bridgedRestart {d : ℕ} (e f : Direction d) (x₀ : Lattice d) (θ z : ℝ) (π : Measure (Lattice d)) (s : ℕ) (ω : Environment d) : Measure (Lattice d) := if s=0 then π else probabilityNormalize (directBridge e (stoppedTubeEndpoint (realPosition (step e)) f θ z (s-1) π ω) ω) (x₀+s • step e) lemma bridgedRestart_probability {d : ℕ} (e f : Direction d) (x₀ : Lattice d) (θ z : ℝ) (π : Measure (Lattice d)) [IsProbabilityMeasure π] (s : ℕ) (ω : Environment d) : IsProbabilityMeasure (bridgedRestart e f x₀ θ z π s ω) := by unfold bridgedRestart split_ifs · infer_instance · exact probabilityNormalize_probability _ _ lemma bridgedRestart_supported {d : ℕ} (e f : Direction d) (x₀ : Lattice d) (a : ℤ) (hx₀ : signedHeight e x₀ = a) (θ z : ℝ) (π : Measure (Lattice d)) (hπ : ∀ᵐ x ∂π, signedHeight e x = a) (s : ℕ) (ω : Environment d) : ∀ᵐ y ∂bridgedRestart e f x₀ θ z π s ω, signedHeight e y = a+s := by by_cases hs : s=0 · simpa only [bridgedRestart,hs,ite_eq_left,Nat.cast_zero,add_zero] using hπ · rw [bridgedRestart,ite_eq_right hs] apply probabilityNormalize_supported · rw [signedHeight_add_nsmul_step,hx₀] · have hh := directBridge_supported e (a+(s-1:ℕ)) _ ω (stoppedTubeEndpoint_supported e f a θ z (s-1) π hπ ω) have he : a+((s-1:ℕ):ℤ)+1=a+s := by omega rwa [he] at hh lemma bridgedRestart_rows {d : ℕ} (e f : Direction d) (x₀ : Lattice d) (a : ℤ) (θ z : ℝ) (π : Measure (Lattice d)) (hπ : ∀ᵐ x ∂π, signedHeight e x = a) (s : ℕ) : @Measurable _ _ (rowSigma {y | dot (realPosition y) (realPosition (step e)) < (a:ℝ)+s}) _ (bridgedRestart e f x₀ θ z π s) := by let ℓ := realPosition (step e) let S : Set (Lattice d) := {y | dot (realPosition y) ℓ < (a:ℝ)+s} let : MeasurableSpace (Environment d) := rowSigma S change Measurable (fun ω => bridgedRestart e f x₀ θ z π s ω) by_cases hs : s=0 · simp only [bridgedRestart,hs,ite_eq_left] exact measurable_const · simp only [bridgedRestart,hs] apply measurable_probabilityNormalize apply directBridge_rows e S · have hp : ∀ᵐ x ∂π, dot (realPosition x) ℓ = (a:ℝ) := hπ.mono fun x hx => by rw [signedHeight_projection,hx] apply (stoppedTubeEndpoint_rows ℓ f a θ z (s-1) π hp).mono _ le_rfl apply rowSigma_mono intro y hy have hc : ((s-1:ℕ):ℝ) ≤ s := by exact_mod_cast Nat.sub_le s 1 change dot (realPosition y) ℓ < (a:ℝ)+s change dot (realPosition y) ℓ < (a:ℝ)+(s-1:ℕ) at hy linarith · intro ω filter_upwards [stoppedTubeEndpoint_supported e f a θ z (s-1) π hπ ω] with y hy change dot (realPosition y) ℓ < (a:ℝ)+s rw [signedHeight_projection,hy,Int.cast_add,Int.cast_natCast] have hc : ((s-1:ℕ):ℝ) < s := by exact_mod_cast Nat.sub_lt (Nat.pos_of_ne_zero hs) (by decide : 0<1) linarith end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical namespace DirectionalTransience theorem canonical_two_tube_threat {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 ∧ δ < 1/4 ∧ ∃ 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 |θ| ≤ w*r ∧ ∀ (a : ℤ) (x₀ : Lattice d), signedHeight e x₀ = a → ∀ π : Measure (Lattice d), IsProbabilityMeasure π → (∀ᵐ x ∂π, signedHeight e x = a) → (environmentLaw ν).real (⋃ s ∈ Finset.range (H+1), {ω | ω ∈ FirstTubeThreat ℓ f a θ (w*r) δ s π ∧ (bridgedRestart e f x₀ θ (w*r) π s ω,ω) ∈ TubeThreat ℓ f ((a:ℝ)+s) θ (w*r) δ H}) ≤ C*t^2 := by obtain ⟨D,hD,t₀,ht₀,hb⟩ := slope_tube_single_threat ν hue e f hef htrans hw hw1 refine ⟨D^2,sq_pos_of_pos hD,t₀,ht₀,?_⟩ intro t ht htt obtain ⟨δ,hδ,hδ1,R,hR,hb⟩ := hb t ht htt refine ⟨δ,hδ,hδ1,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 refine ⟨hθ,?_⟩ intro a x₀ hx₀ π hπ hsupp let := hπ have hsection (b : ℤ) (η : Measure (Lattice d)) (hη : IsProbabilityMeasure η) (hηsupp : ∀ᵐ x ∂η, signedHeight e x = b) : environmentLaw ν {ω | (η,ω) ∈ TubeThreat ℓ f b θ (w*r) δ H} ≤ ENNReal.ofReal (D*t) := by have hsupp' : ∀ᵐ x ∂η, dot (realPosition x) ℓ = (b:ℝ) := hηsupp.mono fun x hx => by rw [signedHeight_projection,hx] have hh := (tubeThreat_probability_le ν ℓ f b θ (w*r) δ H η hsupp').trans (hb η hη) have h := ENNReal.ofReal_le_ofReal hh rwa [Measure.real,ENNReal.ofReal_toReal (measure_ne_top _ _)] at h have hh := two_tube_threat_bound ν ℓ f a θ (w*r) δ H π (bridgedRestart e f x₀ θ (w*r) π) (fun s => bridgedRestart_rows e f x₀ a θ (w*r) π hsupp s) (ENNReal.ofReal (D*t)) (hsection a π hπ hsupp) (by intro s hs η hη have h := hsection (a+s) (bridgedRestart e f x₀ θ (w*r) π s η) (bridgedRestart_probability e f x₀ θ (w*r) π s η) (bridgedRestart_supported e f x₀ a hx₀ θ (w*r) π hsupp s η) simpa only [Int.cast_add,Int.cast_natCast] using h) have hh' := ENNReal.toReal_mono (ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top) hh rw [ENNReal.toReal_mul,ENNReal.toReal_ofReal (mul_nonneg hD.le ht.le)] at hh' calc _ ≤ D*t*(D*t) := hh' _ = D^2*t^2 := by ring end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical end end end OAI