File size: 3,181 Bytes
9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | 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
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
end
end OAI
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