| import OAI.Probability.Ballisticity.RowBridge |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
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| namespace DirectionalTransience |
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| lemma uniformElliptic_directional_zero_one {d : ℕ} (hd : 3 ≤ d) |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] (hue : UniformElliptic ν) |
| (ℓ : Vector d) (hℓ : ℓ ≠ 0) : |
| annealedLaw ν (TransientPaths ℓ) = 0 ∨ annealedLaw ν (TransientPaths ℓ) = 1 := by |
| have h := DirectionalZeroOne.directional_zero_one d hd (RowBridge.rowLaw ν) |
| (RowBridge.strictEllipticity ν hue) ℓ hℓ |
| change DirectionalZeroOne.annealed (RowBridge.rowLaw ν) 0 (TransientPaths ℓ) = 0 ∨ |
| DirectionalZeroOne.annealed (RowBridge.rowLaw ν) 0 (TransientPaths ℓ) = 1 at h |
| rwa [← RowBridge.annealedLaw_eq ν] at h |
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| lemma dot_div_right {d : ℕ} (v ℓ : Vector d) (r : ℝ) : |
| dot v (fun i => ℓ i / r) = dot v ℓ / r := by |
| simp only [dot, mul_div_assoc, Finset.sum_div] |
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| lemma dot_self_pos_of_ne_zero {d : ℕ} (ℓ : Vector d) (hℓ : ℓ ≠ 0) : |
| 0 < dot ℓ ℓ := by |
| obtain ⟨i, hi⟩ := Function.ne_iff.mp hℓ |
| change ℓ i ≠ 0 at hi |
| exact (mul_self_pos.mpr hi).trans_le |
| (Finset.single_le_sum (fun j _ => mul_self_nonneg (ℓ j)) (Finset.mem_univ i)) |
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| lemma transientPaths_div_pos {d : ℕ} (ℓ : Vector d) {r : ℝ} (hr : 0 < r) : |
| TransientPaths (fun i => ℓ i / r) = TransientPaths ℓ := by |
| ext X |
| simp only [TransientPaths, Set.mem_ofPred_eq, dot_div_right] |
| exact tendsto_div_const_atTop_of_pos hr |
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| lemma velocity_of_positive_escape {d : ℕ} (hd : 3 ≤ d) |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] (hue : UniformElliptic ν) |
| (ℓ : Vector d) (hℓ : ℓ ≠ 0) (hpos : 0 < annealedLaw ν (TransientPaths ℓ)) : |
| ∃ v : Vector d, 0 < dot v ℓ ∧ HasVelocity ν v := by |
| have hprob : annealedLaw ν (TransientPaths ℓ) = 1 := |
| (uniformElliptic_directional_zero_one hd ν hue ℓ hℓ).resolve_left hpos.ne' |
| have hs : 0 < dot ℓ ℓ := dot_self_pos_of_ne_zero ℓ hℓ |
| let r : ℝ := Real.sqrt (dot ℓ ℓ) |
| have hr : 0 < r := Real.sqrt_pos.mpr hs |
| let u : Vector d := fun i => ℓ i / r |
| have hu : dot u u = 1 := by |
| have he : dot u u = dot ℓ ℓ / (r * r) := by |
| simp only [u, dot, div_mul_div_comm, Finset.sum_div] |
| rw [he, show r * r = dot ℓ ℓ from Real.mul_self_sqrt hs.le, div_self hs.ne'] |
| have htrans : DirectionallyTransient ν u := by |
| apply (mem_ae_iff_prob_eq_one (measurableSet_transientPaths u)).mpr |
| rw [transientPaths_div_pos ℓ hr] |
| exact hprob |
| obtain ⟨v, hvu, hv⟩ := directional_transience_implies_ballisticity_unit |
| (by omega : 2 ≤ d) ν hue u hu htrans |
| refine ⟨v, ?_, hv⟩ |
| have hp : 0 < dot v ℓ / r := by simpa only [u, dot_div_right] using hvu |
| exact (div_pos_iff_of_pos_right hr).mp hp |
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| lemma escape_probability_one_of_velocity {d : ℕ} (ν : Measure (Row d)) |
| [IsProbabilityMeasure ν] (v u : Vector d) (hv : HasVelocity ν v) |
| (hvu : 0 < dot v u) : annealedLaw ν (TransientPaths u) = 1 := by |
| apply (mem_ae_iff_prob_eq_one (measurableSet_transientPaths u)).mp |
| filter_upwards [hv] with X hX |
| have hproj := velocityPaths_projection v u X hX |
| have hprod := hproj.pos_mul_atTop hvu |
| (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop) |
| apply hprod.congr' |
| filter_upwards [eventually_gt_atTop (0 : ℕ)] with n hn |
| exact div_mul_cancel₀ _ (by exact_mod_cast hn.ne') |
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| lemma unit_direction_ne_zero {d : ℕ} (u : Vector d) (hu : dot u u = 1) : u ≠ 0 := by |
| intro h |
| subst u |
| simp [dot] at hu |
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| end DirectionalTransience |
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| end OAI |
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