File size: 4,271 Bytes
f61e547 | 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 91 92 93 94 95 96 97 98 99 100 | 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
|