| import OAI.Probability.Ballisticity.Main |
| import OAI.Probability.DirectionalWalk.Main |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
|
|
| namespace DirectionalTransience |
|
|
| namespace RowBridge |
|
|
| def toReal {d : ℕ} (p : Row d) : DirectionalZeroOne.Row d := |
| ⟨fun e => (p.1 e : ℝ), by |
| constructor |
| · intro e |
| exact ⟨(p.1 e).coe_nonneg, by exact_mod_cast row_entry_le_one p e⟩ |
| · change ∑ e : Direction d, (p.1 e : ℝ) = 1 |
| rw [← NNReal.coe_sum, p.2] |
| rfl⟩ |
|
|
| lemma measurable_toReal {d : ℕ} : Measurable (toReal (d := d)) := by |
| apply Measurable.subtype_mk |
| exact Measurable.of_eval fun e => |
| ((measurable_pi_apply e).comp measurable_subtype_coe).coe_nnreal_real |
|
|
| noncomputable def rowLaw {d : ℕ} (ν : Measure (Row d)) : |
| Measure (DirectionalZeroOne.Row d) := ν.map toReal |
|
|
| instance rowLaw_probability {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] : |
| IsProbabilityMeasure (rowLaw ν) := |
| (Measure.isProbabilityMeasure_map_iff measurable_toReal.aemeasurable).2 inferInstance |
|
|
| def environment {d : ℕ} (ω : Environment d) : DirectionalZeroOne.Environment d := |
| fun x => toReal (ω x) |
|
|
| lemma measurable_environment {d : ℕ} : Measurable (environment (d := d)) := |
| Measurable.of_eval fun x => measurable_toReal.comp (measurable_pi_apply x) |
|
|
| lemma environmentLaw_map {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] : |
| (environmentLaw ν).map environment = DirectionalZeroOne.environmentLaw (rowLaw ν) := by |
| exact Measure.infinitePi_map_pi (fun _ : Lattice d => ν) (fun _ => measurable_toReal) |
|
|
| lemma strictEllipticity {d : ℕ} (ν : Measure (Row d)) (hue : UniformElliptic ν) : |
| DirectionalZeroOne.StrictEllipticity (rowLaw ν) := by |
| obtain ⟨κ, hκ, hrows⟩ := hue |
| have hm : MeasurableSet {p : DirectionalZeroOne.Row d | ∀ e, 0 < p.1 e} := by |
| simp only [Set.ofPred_forall] |
| exact MeasurableSet.iInter fun e => measurableSet_lt measurable_const |
| ((measurable_pi_apply e).comp (measurable_subtype_coe : |
| Measurable (fun p : DirectionalZeroOne.Row d => p.1))) |
| apply (ae_map_iff measurable_toReal.aemeasurable hm).mpr |
| filter_upwards [hrows] with p hp e |
| change (0 : ℝ) < (p.1 e : ℝ) |
| exact_mod_cast lt_of_lt_of_le hκ (hp e) |
|
|
| lemma entryWeight {d : ℕ} (ω : Environment d) (x y : Lattice d) : |
| DirectionalZeroOne.entryWeight (environment ω) x y = edgeWeight ω x y := by |
| classical |
| simp only [DirectionalZeroOne.entryWeight, edgeWeight, |
| environment, toReal, ENNReal.ofReal_coe_nnreal] |
| apply Finset.sum_congr rfl |
| intro e _ |
| change (if x + step e = y then ((ω x).1 e : ℝ≥0∞) else 0) = _ |
| simp only [eq_comm] |
|
|
| lemma annealedLaw_eq {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] : |
| annealedLaw ν = DirectionalZeroOne.annealed (rowLaw ν) 0 := by |
| apply pathMeasure_ext |
| intro γ n |
| rw [annealed_apply ν (measurableSet_pathCylinder γ n)] |
| change (∫⁻ ω, quenchedKernel (ω, 0) (pathCylinder γ n) ∂environmentLaw ν) = |
| DirectionalZeroOne.annealed (rowLaw ν) 0 (DirectionalZeroOne.pathCylinder n γ) |
| rw [DirectionalZeroOne.annealed_cylinder] |
| by_cases h0 : γ 0 = 0 |
| · rw [ite_eq_left h0.symm, ← environmentLaw_map ν, |
| lintegral_map (DirectionalZeroOne.measurable_pathWeight n γ) measurable_environment] |
| apply lintegral_congr |
| intro ω |
| rw [quenched_pathCylinder ω 0 γ h0 n] |
| simp only [DirectionalZeroOne.pathWeight, entryWeight] |
| · rw [ite_eq_right (Ne.symm h0)] |
| simp only [quenched_pathCylinder_zero _ _ _ h0, lintegral_zero] |
|
|
| end RowBridge |
|
|
| end DirectionalTransience |
|
|
| end OAI |
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|
