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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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | 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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