| import Mathlib |
| import OAI.Probability.Ballisticity.Stationary.ArrayWindowMarks |
| import OAI.Probability.Ballisticity.Stationary.ArrayPermutation |
| import OAI.Probability.Ballisticity.Walk.StationaryGrowth |
| import OAI.Probability.Ballisticity.Stationary.ArrayMarkTests |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Classical Topology BigOperators |
| namespace DirectionalTransience |
|
|
| noncomputable def arrayRealCost {d : ℕ} (e : Direction d) (Y : ActualEpisodeArray e) : ℝ := (Y.1 0).2.2.toReal |
| lemma arrayRealCost_measurable {d : ℕ} (e : Direction d) : Measurable (arrayRealCost e) := by |
| exact (show Measurable (fun Y : ActualEpisodeArray e => (Y.1 0).2.2) from by fun_prop).ennreal_toReal |
| lemma arrayRealCost_nonneg {d : ℕ} (e : Direction d) (Y : ActualEpisodeArray e) : 0≤arrayRealCost e Y := ENNReal.toReal_nonneg |
|
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| lemma arrayWindowCost_toReal {d : ℕ} (e : Direction d) (Y : ActualEpisodeArray e) |
| (hc : ∀ j, (Y.1 j).2.2≠⊤) (m : ℕ) : |
| (arrayWindowCost e 0 m Y).toReal=∑ j∈Finset.range m, arrayRealCost e (StationaryCompact.shift^[j] Y) := by |
| rw [arrayWindowCost,ENNReal.toReal_sum (fun (j : ℕ) _ => hc (0+(j:ℤ)))] |
| apply Finset.sum_congr rfl |
| intro j _ |
| simp only [arrayRealCost,arrayMark_iterate] |
|
|
| lemma expNeg_toReal_of_ne_top (C : ℝ≥0∞) (hc : C≠⊤) : (expNeg C).toReal=Real.exp (-C.toReal) := by |
| rw [←ENNReal.ofReal_toReal hc] |
| unfold expNeg |
| rw [EReal.coe_ennreal_ofReal,max_eq_left ENNReal.toReal_nonneg,←EReal.coe_neg,EReal.exp_coe, |
| ENNReal.toReal_ofReal (Real.exp_nonneg _)] |
| rw [ENNReal.toReal_ofReal ENNReal.toReal_nonneg] |
|
|
| lemma array_cost_decay_of_growth {d : ℕ} (e : Direction d) (Y : ActualEpisodeArray e) |
| (hc : ∀ j, (Y.1 j).2.2≠⊤) (ζ : ℝ) (hζ : 0<ζ) (ns : ℕ → ℕ) (hns : StrictMono ns) |
| (hg : ∀ᶠ k in atTop, ζ*(ns k:ℝ)<∑ j∈Finset.range (ns k), arrayRealCost e (StationaryCompact.shift^[j] Y)) : |
| ∀ ε : ℝ, 0<ε → ∃ m, (expNeg (arrayWindowCost e 0 m Y)).toReal<ε := by |
| have ht : Tendsto (fun k => ζ*(ns k:ℝ)) atTop atTop := |
| (tendsto_natCast_atTop_atTop.comp hns.tendsto_atTop).const_mul_atTop hζ |
| have he := Real.tendsto_exp_neg_atTop_nhds_zero.comp ht |
| intro ε hε |
| obtain ⟨k,hk,hk'⟩ := (hg.and (he.eventually (gt_mem_nhds hε))).exists |
| refine ⟨ns k,?_⟩ |
| have hf : arrayWindowCost e 0 (ns k) Y≠⊤ := by |
| unfold arrayWindowCost |
| exact ENNReal.sum_ne_top.mpr fun j _ => hc (0+j) |
| rw [expNeg_toReal_of_ne_top _ hf,arrayWindowCost_toReal e Y hc] |
| exact (Real.exp_lt_exp.mpr (neg_lt_neg hk)).trans hk' |
|
|
| lemma array_growth_symmetry {d : ℕ} (e : Direction d) (μ : Measure (ActualEpisodeArray e)) |
| (g : ActualEpisodeArray e → ℝ) (ns : ℕ → ℕ) |
| (hlim : ∀ᵐ Y ∂μ, Tendsto (fun k => orbitAverage StationaryCompact.shift |
| (fun Y => min (arrayRealCost e Y) 1) (ns k) Y) atTop (𝓝 (g Y))) |
| (hperm : ∀ σ, MeasurePreserving (arrayPerm e σ) μ μ) (σ : Equiv.Perm ℕ) : |
| ∀ᵐ Y ∂μ, g (arrayPerm e σ Y)=g Y := by |
| exact growth_limit_respects_symmetry μ ns hlim (hperm σ) (fun Y => rfl) (fun Y => rfl) |
|
|
| end DirectionalTransience |
|
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| end |
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| end OAI |
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|
