| import OAI.Probability.FirstPassage.CompanionModel |
| import OAI.Probability.FirstPassage.LatticeLimitExtension |
| import OAI.Probability.FirstPassage.TimeConstantObligation |
| import OAI.Probability.LatticePassage.TimeConstantNorm |
|
|
| namespace OAI |
|
|
| namespace PlanarFPP |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology |
|
|
| noncomputable def actualSourceTimeConstant (v : Plane) : ℝ := |
| CompanionFPP.timeSeminorm (v 0,v 1) |
|
|
| theorem actualSourceTimeConstant_norm : TimeConstantNorm actualSourceTimeConstant where |
| nonneg v := apply_nonneg _ _ |
| zero_iff v := by |
| constructor |
| · intro hv |
| have he := CompanionFPP.actualTimeNorm.definite (v 0,v 1) hv |
| ext i |
| fin_cases i |
| · exact congrArg Prod.fst he |
| · exact congrArg Prod.snd he |
| · rintro rfl |
| exact map_zero _ |
| triangle v w := CompanionFPP.timeSeminorm.add_le' (v 0,v 1) (w 0,w 1) |
| homogeneous a v := by |
| change CompanionFPP.timeSeminorm (a • (v 0,v 1)) = _ |
| rw [map_smul_eq_mul,Real.norm_eq_abs] |
| rfl |
|
|
| theorem actualSourceTimeConstant_bounds (v : Plane) : |
| (Real.log 2/20)*‖v‖ ≤ actualSourceTimeConstant v ∧ |
| actualSourceTimeConstant v ≤ 2*‖v‖ := by |
| constructor |
| · exact (mul_le_mul_of_nonneg_left (plane_norm_le_coordinates v) (by positivity)).trans |
| (CompanionFPP.timeSeminorm_linear_lower (v 0,v 1)) |
| · apply (CompanionFPP.timeSeminorm_bound (v 0,v 1)).trans |
| have hh : ‖(v 0,v 1)‖ ≤ ‖v‖ := by |
| rw [Prod.norm_def] |
| exact max_le (plane_coordinate_le_norm v 0) (plane_coordinate_le_norm v 1) |
| linarith |
|
|
| variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ} |
|
|
| theorem actualSourceTimeConstant_lattice_limits (h : ExponentialEnvironment P τ) : |
| ∀ᵐ ω ∂P, ∀ z : Vertex, |
| Tendsto (fun n : ℕ => passageTime (fun e => τ e ω) 0 (n • z)/n) atTop |
| (𝓝 (actualSourceTimeConstant (latticeEmbed z))) := by |
| have hm : MeasurePreserving (fun ω => companionWeights (fun e => τ e ω)) P |
| CompanionFPP.environmentLaw := ⟨measurable_companionWeights h.measurable,h.companionLaw⟩ |
| filter_upwards [hm.quasiMeasurePreserving.ae CompanionFPP.actualTimeNorm_hasTimeConstant] with ω hω |
| intro z |
| have hz := hω z |
| simp only [companion_passageTime_eq] at hz |
| have he (n : ℕ) : CompanionFPP.scaleVertex n z = n • z := by |
| ext <;> simp [CompanionFPP.scaleVertex,nsmul_eq_mul] |
| simp only [he] at hz |
| exact hz |
|
|
| theorem actualSourceTimeConstant_all_real_limits (h : ExponentialEnvironment P τ) : |
| ∀ᵐ ω ∂P, ∀ v : Plane, |
| Tendsto (fun t : ℝ => passageTime (fun e => τ e ω) 0 (latticeFloor (t • v))/t) |
| atTop (𝓝 (actualSourceTimeConstant v)) := by |
| have hc : ∀ᵐ ω ∂P, ∀ k : ℕ, ∀ᶠ n : ℕ in atTop, |
| ∀ p ∈ latticeBox n, ∀ q ∈ latticeBox n, |
| passageTime (fun e => τ e ω) p q < |
| 4*(latticeDistance p q : ℝ)+(1/(k+1 : ℝ))*n := by |
| rw [ae_all_iff] |
| intro k |
| exact ae_eventually_passageTime_connections h (by positivity) |
| filter_upwards [h.ae_positive,actualSourceTimeConstant_lattice_limits h,hc] with ω hp hr hc |
| exact real_direction_limits_of_lattice_limits _ (fun e => (hp e).le) |
| actualSourceTimeConstant actualSourceTimeConstant_norm hr hc |
|
|
| theorem actualSourceTimeConstant_isTimeConstant (h : ExponentialEnvironment P τ) : |
| IsTimeConstant P τ actualSourceTimeConstant := by |
| intro v |
| filter_upwards [actualSourceTimeConstant_all_real_limits h] with ω hω |
| exact hω v |
|
|
| theorem actualSourceTimeConstant_uniform_shape (h : ExponentialEnvironment P τ) : |
| ∀ᵐ ω ∂P, UniformShape (fun e => τ e ω) actualSourceTimeConstant := |
| exponential_uniform_shape h actualSourceTimeConstant actualSourceTimeConstant_norm |
| (actualSourceTimeConstant_isTimeConstant h) |
|
|
| theorem timeConstantExistence : TimeConstantExistenceTarget := by |
| intro Ω _ P τ h |
| exact ⟨actualSourceTimeConstant,actualSourceTimeConstant_norm, |
| actualSourceTimeConstant_isTimeConstant h⟩ |
|
|
| theorem actualSourceTimeConstant_unique (h : ExponentialEnvironment P τ) |
| {μ : Plane → ℝ} (hμ : IsTimeConstant P τ μ) : μ=actualSourceTimeConstant := |
| isTimeConstant_unique h hμ (actualSourceTimeConstant_isTimeConstant h) |
|
|
| end PlanarFPP |
|
|
| end OAI |
|
|
