| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.ConfinementTemplates |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Classical Topology BigOperators |
| namespace DirectionalTransience |
|
|
| def StripExcursion {d : ℕ} (ℓ : Vector d) (H R : ℝ) : Set (Path d) := |
| {X | ∃ n, X ∈ BoundedHeightPrefix ℓ H n ∧ ∃ u : Direction d, R < signedCoordinate u (X n)} |
|
|
| lemma measurableSet_stripExcursion {d : ℕ} (ℓ : Vector d) (H R : ℝ) : |
| MeasurableSet (StripExcursion ℓ H R) := by |
| simp only [StripExcursion, Set.ofPred_exists, Set.ofPred_and] |
| apply MeasurableSet.iUnion |
| intro n |
| refine (measurableSet_boundedHeightPrefix ℓ H n).inter (MeasurableSet.iUnion fun u => ?_) |
| exact measurableSet_lt measurable_const ((measurable_of_countable (signedCoordinate u)).comp (measurable_pi_apply n)) |
|
|
| lemma stripExcursion_subset_ratio {d : ℕ} (ℓ : Vector d) (X : Path d) |
| (h0 : X 0 = 0) (hNN : ∀ n, ∃ e : Direction d, X (n+1) = X n+step e) |
| (b a D : ℝ) (hb : 0 ≤ b) (ha : 1 ≤ a) (hD : 0 ≤ D) (N : ℕ) : |
| X ∈ StripExcursion ℓ N (a*(2*b+2+D)*(N+1)^2) → |
| X ∈ ⋃ u : Direction d, RatioHit ℓ u b a N (1+N*(2*b+1+D)) := by |
| rintro ⟨n,hn,u,hu⟩ |
| refine Set.mem_iUnion.mpr ⟨u,Set.mem_iUnion.mpr ⟨n,⟨⟨h0,fun j _ => hNN j⟩,hn⟩,?_⟩⟩ |
| have hm := runningHeight_le ℓ X n N (Nat.cast_nonneg _) (fun j hj => (hn j hj).2) |
| have hm0 := runningHeight_nonneg ℓ X n |
| have ht : 0 ≤ 1+(N:ℝ)*(2*b+1+D) := by positivity |
| have hfactor : 1+(N:ℝ)*(2*b+1+D) ≤ (2*b+2+D)*(N+1) := by nlinarith |
| have hbnd : (1+(N:ℝ)*(2*b+1+D))*(a*(1+runningHeight ℓ X n)) ≤ |
| a*(2*b+2+D)*(N+1)^2 := by |
| calc |
| _ ≤ (1+(N:ℝ)*(2*b+1+D))*(a*(1+N)) := |
| mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left (by linarith) (by linarith)) ht |
| _ ≤ ((2*b+2+D)*(N+1))*(a*(1+N)) := |
| mul_le_mul_of_nonneg_right hfactor (by positivity) |
| _ = _ := by ring |
| change _ < tiltedCoordinate ℓ u b a (X n)/(a*(1+runningHeight ℓ X n)) |
| apply (lt_div_iff₀ (mul_pos (by linarith) (by linarith))).mpr |
| have hz := mul_nonneg (mul_nonneg hb (show 0 ≤ a by linarith)) (hn n le_rfl).1 |
| dsimp only [tiltedCoordinate] |
| linarith |
|
|
| theorem spatial_confinement {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (hue : UniformElliptic ν) (e : Direction d) |
| (htrans : DirectionallyTransient ν (realPosition (step e))) : |
| ∃ C ε : ℝ, 0 < C ∧ 0 < ε ∧ ε ≤ 1 ∧ ∀ N : ℕ, |
| (annealedLaw ν).real (StripExcursion (realPosition (step e)) N (C*(N+1)^2)) ≤ |
| (2*d : ℕ)*(1-ε)^N := by |
| obtain ⟨b,a,D,ε,hb,ha,hD,hε,hε1,hA⟩ := strip_templates ν hue e htrans |
| refine ⟨a*(2*b+2+D),ε,mul_pos (by linarith) (by linarith),hε,hε1,?_⟩ |
| intro N |
| obtain ⟨A,hAm,hAp,hAc⟩ := hA N |
| apply le_trans ?_ (ratio_hit_union_bound ν (realPosition (step e)) |
| (unit_direction_coordinate_bound _ (signed_direction_unit e)) b a D N ε hb ha hD hε.le hε1 A hAm hAp hAc N) |
| apply ENNReal.toReal_mono (measure_ne_top _ _) |
| apply measure_mono_ae |
| filter_upwards [annealed_initial ν,annealed_nearest_neighbor ν] with X h0 hNN |
| exact stripExcursion_subset_ratio _ X h0 hNN b a D hb ha hD N |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end OAI |
|
|
