| 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 |
| |
