| import Mathlib |
| import OAI.Probability.Ballisticity.Model |
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| namespace OAI |
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| section |
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| section |
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| open MeasureTheory ProbabilityTheory Filter Function |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
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| noncomputable def canonicalLogExcess {Ω : Type*} (A : ℝ) (p : ℕ → Ω → ℝ) (ω : Ω) : ℝ := |
| (⨆ n : ℕ, ENNReal.ofReal (-Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2))).toReal |
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| lemma measurable_canonicalLogExcess {Ω : Type*} [MeasurableSpace Ω] |
| (A : ℝ) (p : ℕ → Ω → ℝ) (hp : ∀ n, Measurable (p n)) : |
| Measurable (canonicalLogExcess A p) := by |
| unfold canonicalLogExcess |
| apply Measurable.ennreal_toReal |
| apply Measurable.iSup |
| intro n |
| exact ((hp n).log.neg.sub_const _).ennreal_ofReal |
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| lemma canonicalLogExcess_nonneg {Ω : Type*} (A : ℝ) (p : ℕ → Ω → ℝ) (ω : Ω) : |
| 0 ≤ canonicalLogExcess A p ω := ENNReal.toReal_nonneg |
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| lemma canonicalLogExcess_bounds {Ω : Type*} {A D W : ℝ} (hA : 0 ≤ A) (hD : 0 ≤ D) |
| (hW : 0 ≤ W) (p : ℕ → Ω → ℝ) (ω : Ω) |
| (hp : ∀ n, -Real.log (p n ω) ≤ A*Real.log ((n:ℝ)+2)+D*(W+1)) : |
| canonicalLogExcess A p ω ≤ D*(W+1) ∧ |
| (∀ n, -Real.log (p n ω) ≤ 2*A*Real.log ((n:ℝ)+2)+canonicalLogExcess A p ω) := by |
| let S := ⨆ n : ℕ, ENNReal.ofReal (-Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2)) |
| have hS : S ≤ ENNReal.ofReal (D*(W+1)) := by |
| apply iSup_le |
| intro n |
| apply ENNReal.ofReal_le_ofReal |
| have hn := hp n |
| have hl : 0 ≤ Real.log ((n:ℝ)+2) := Real.log_nonneg (by have := Nat.cast_nonneg (α:=ℝ) n; linarith) |
| have hm : 0 ≤ A*Real.log ((n:ℝ)+2) := mul_nonneg hA hl |
| linarith |
| have hSt : S ≠ ⊤ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hS |
| constructor |
| · have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top hS |
| simpa only [canonicalLogExcess,S,ENNReal.toReal_ofReal (show 0 ≤ D*(W+1) by positivity)] using hh |
| · intro n |
| have hn : ENNReal.ofReal (-Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2)) ≤ S := le_iSup (fun k : ℕ => ENNReal.ofReal (-Real.log (p k ω)-2*A*Real.log ((k:ℝ)+2))) n |
| have hr := (ENNReal.ofReal_le_iff_le_toReal hSt).mp hn |
| change -Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2) ≤ canonicalLogExcess A p ω at hr |
| linarith |
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| noncomputable def finiteLogExcess {Ω : Type*} (A : ℝ) (p : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) : ℝ := |
| (⨆ n : Fin (N+1), ENNReal.ofReal (-Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2))).toReal |
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| lemma canonicalLogExcess_eq_finite {Ω : Type*} {A D W M : ℝ} (hA : 0 ≤ A) (hD : 0 ≤ D) |
| (hWM : W ≤ M) (p : ℕ → Ω → ℝ) (ω : Ω) |
| (hp : ∀ n, -Real.log (p n ω) ≤ A*Real.log ((n:ℝ)+2)+D*(W+1)) |
| (N : ℕ) (hN : D*(M+1) ≤ A*Real.log ((N:ℝ)+2)) : |
| canonicalLogExcess A p ω = finiteLogExcess A p N ω := by |
| unfold canonicalLogExcess finiteLogExcess |
| congr 1 |
| apply le_antisymm |
| · apply iSup_le |
| intro n |
| by_cases hn : n ≤ N |
| · exact le_iSup (fun k : Fin (N+1) => ENNReal.ofReal (-Real.log (p k ω)-2*A*Real.log ((k:ℝ)+2))) ⟨n,by omega⟩ |
| · have hlog : Real.log ((N:ℝ)+2) ≤ Real.log ((n:ℝ)+2) := |
| Real.log_le_log (by positivity) (by exact_mod_cast (show N+2 ≤ n+2 by omega)) |
| have hAl := mul_le_mul_of_nonneg_left hlog hA |
| have hDW := mul_le_mul_of_nonneg_left (add_le_add_right hWM 1) hD |
| have hp' := hp n |
| have hzero : -Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2) ≤ 0 := by linarith |
| rw [ENNReal.ofReal_eq_zero.mpr hzero] |
| exact zero_le |
| · apply iSup_le |
| intro n |
| exact le_iSup (fun k : ℕ => ENNReal.ofReal (-Real.log (p k ω)-2*A*Real.log ((k:ℝ)+2))) n.val |
| end DirectionalTransience |
| |
| end |
| |
| end |
| |
| end OAI |
| |
