| import Mathlib |
| import OAI.Probability.Ballisticity.Geometry.HeightPolygon |
|
|
| 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 |
| namespace DirectionalTransience |
|
|
| lemma norm_heightPolygon_le (F : ℕ → ℝ) (r n T : ℝ) (hq : 0 ≤ T*n) |
| {E : ℝ} (hE : 0 ≤ E) (hb : ∀ j ≤ ⌊T*n⌋₊+1, |F j/r| ≤ E) : |
| ‖heightPolygon F r n T‖ ≤ E := by |
| apply (ContinuousMap.norm_le _ hE).mpr |
| intro t |
| rw [Real.norm_eq_abs,heightPolygon_formula F r n T hq t] |
| let u := T*n*(t:ℝ) |
| let j := ⌊u⌋₊ |
| have hu : 0 ≤ u := mul_nonneg hq t.2.1 |
| have hv0 : 0 ≤ u-(j:ℝ) := sub_nonneg.mpr (Nat.floor_le hu) |
| have hv1 : u-(j:ℝ) ≤ 1 := by have := Nat.lt_floor_add_one u; change u < (j:ℝ)+1 at this; linarith |
| have hj : j ≤ ⌊T*n⌋₊ := Nat.floor_mono (mul_le_of_le_one_right hq t.2.2) |
| have he : (F j+(u-j)*(F (j+1)-F j))/r = (1-(u-j))*(F j/r)+(u-j)*(F (j+1)/r) := by ring |
| change |(F j+(u-j)*(F (j+1)-F j))/r| ≤ E |
| rw [he] |
| calc |
| _ ≤ |(1-(u-j))*(F j/r)|+|(u-j)*(F (j+1)/r)| := abs_add_le _ _ |
| _ = (1-(u-j))*|F j/r|+(u-j)*|F (j+1)/r| := by |
| rw [abs_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr hv1),abs_of_nonneg hv0] |
| _ ≤ (1-(u-j))*E+(u-j)*E := by |
| gcongr |
| · exact hb j (by omega) |
| · exact hb (j+1) (by omega) |
| _ = E := by ring |
|
|
| lemma heightPolygon_tendsto_zero_of_grid (F : ℕ → ℕ → ℝ) (r n : ℕ → ℝ) |
| (hn : Tendsto n atTop atTop) (T : ℝ) (hT : 0 ≤ T) |
| (hb : ∀ ε > 0, ∀ᶠ i in atTop, ∀ j : ℕ, (j:ℝ) ≤ (T+1)*n i → |F i j/r i| < ε) : |
| Tendsto (fun i => heightPolygon (F i) (r i) (n i) T) atTop (𝓝 0) := by |
| apply Metric.tendsto_nhds.mpr |
| intro ε hε |
| filter_upwards [hb (ε/2) (half_pos hε),hn.eventually (eventually_ge_atTop (1:ℝ))] with i hi hni |
| rw [dist_zero_right] |
| apply lt_of_le_of_lt (norm_heightPolygon_le _ _ _ _ (mul_nonneg hT (by positivity)) (half_pos hε).le _) (half_lt_self hε) |
| intro j hj |
| have hjr : (j:ℝ) ≤ (⌊T*n i⌋₊:ℝ)+1 := by exact_mod_cast hj |
| have hfl := Nat.floor_le (mul_nonneg hT (show 0 ≤ n i by positivity)) |
| exact (hi j (by nlinarith)).le |
|
|
| lemma heightPolygon_change_center (F b c : ℕ → ℝ) (r n T : ℝ) : |
| heightPolygon (fun h => F h-c h) r n T-heightPolygon (fun h => F h-b h) r n T = |
| heightPolygon (fun h => b h-c h) r n T := by |
| rw [← heightPolygon_sub] |
| congr 1 |
| funext h |
| ring |
|
|
| theorem height_path_change_center {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (F : ℕ → Ω → ℝ) (hF : ∀ h, Measurable (F h)) |
| (b : ℕ → ℝ) (c : ℕ → ℕ → ℝ) (r n : ℕ → ℝ) (hn : Tendsto n atTop atTop) |
| (T : ℝ) (hT : 0 ≤ T) |
| (hb : ∀ ε > 0, ∀ᶠ i in atTop, ∀ j : ℕ, (j:ℝ) ≤ (T+1)*n i → |(b j-c i j)/r i| < ε) |
| (W : ProbabilityMeasure C(unitInterval,ℝ)) |
| (hlim : TendstoInDistribution (fun i x => heightPolygon (fun h => F h x-b h) (r i) (n i) T) |
| atTop id (fun _ => μ) W) : |
| TendstoInDistribution (fun i x => heightPolygon (fun h => F h x-c i h) (r i) (n i) T) |
| atTop id (fun _ => μ) W := by |
| apply tendstoInDistribution_of_tendstoInMeasure_sub _ id hlim |
| · have hh := heightPolygon_tendsto_zero_of_grid (fun i j => b j-c i j) r n hn T hT hb |
| apply tendstoInMeasure_of_tendsto_ae |
| · intro i |
| have he : (fun x => heightPolygon (fun h => F h x-c i h) (r i) (n i) T) - |
| (fun x => heightPolygon (fun h => F h x-b h) (r i) (n i) T) = |
| (fun _ => heightPolygon (fun h => b h-c i h) (r i) (n i) T) := by |
| funext x |
| exact heightPolygon_change_center _ _ _ _ _ _ |
| change AEStronglyMeasurable ((fun x => heightPolygon (fun h => F h x-c i h) (r i) (n i) T) - |
| (fun x => heightPolygon (fun h => F h x-b h) (r i) (n i) T)) μ |
| rw [he] |
| exact aestronglyMeasurable_const |
| · exact Eventually.of_forall (fun x => by simpa only [Pi.sub_apply,heightPolygon_change_center,Pi.zero_apply] using hh) |
| · intro i |
| exact (measurable_heightPolygon _ (fun h => (hF h).sub_const _) _ _ _).aemeasurable |
|
|
| end DirectionalTransience |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
|
|
| end |
| end |
|
|
| end OAI |
|
|
