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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 91 92 93 94 95 96 97 98 | import Mathlib
namespace OAI
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology
namespace DirectionalTransience
abbrev Lattice (d : ℕ) := Fin d → ℤ
abbrev Direction (d : ℕ) := Fin d × Bool
abbrev Vector (d : ℕ) := Fin d → ℝ
abbrev Row (d : ℕ) := {p : Direction d → ℝ≥0 // ∑ e, p e = 1}
abbrev Environment (d : ℕ) := Lattice d → Row d
abbrev State (d : ℕ) := Environment d × Lattice d
abbrev Path (d : ℕ) := ℕ → Lattice d
def step {d : ℕ} (e : Direction d) : Lattice d :=
fun i => if i = e.1 then (if e.2 then 1 else -1) else 0
noncomputable def environmentLaw {d : ℕ} (ν : Measure (Row d)) :
Measure (Environment d) := Measure.infinitePi (fun _ => ν)
def UniformElliptic {d : ℕ} (ν : Measure (Row d)) : Prop :=
∃ κ : ℝ≥0, 0 < κ ∧ ∀ᵐ p ∂ν, ∀ e, κ ≤ p.1 e
lemma measurable_current_row {d : ℕ} :
Measurable (fun z : State d => z.1 z.2) := by
apply measurable_from_prod_countable_left
intro x
exact measurable_pi_apply x
noncomputable def transition {d : ℕ} : Kernel (State d) (State d) where
toFun z := ∑ e : Direction d,
((z.1 z.2).1 e : ℝ≥0∞) • Measure.dirac (z.1, z.2 + step e)
measurable' := by
apply Measure.measurable_of_measurable_coe
intro s hs
simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul]
apply Finset.measurable_fun_sum
intro e _
apply Measurable.mul
· exact (measurable_pi_apply e |>.comp
(measurable_subtype_coe.comp measurable_current_row)).coe_nnreal_ennreal
· exact (Measure.measurable_coe hs).comp
(Measure.measurable_dirac.comp (measurable_fst.prodMk
(measurable_snd.add_const (step e))))
instance transition_markov {d : ℕ} : IsMarkovKernel (transition (d := d)) where
isProbabilityMeasure z := by
constructor
change (∑ e : Direction d,
((z.1 z.2).1 e : ℝ≥0∞) • Measure.dirac (z.1, z.2 + step e)) Set.univ = 1
simp only [Measure.finsetSum_apply, Measure.smul_apply, Measure.dirac_apply_of_mem
(Set.mem_univ _), smul_eq_mul, mul_one]
rw [← ENNReal.ofNNReal_finsetSum, (z.1 z.2).2]
rfl
noncomputable def historyTransition {d : ℕ} (n : ℕ) :
Kernel ((i : Finset.Iic n) → State d) (State d) where
toFun z := transition (z ⟨n, Finset.mem_Iic.mpr le_rfl⟩)
measurable' := transition.measurable.comp (measurable_pi_apply _)
instance historyTransition_markov {d : ℕ} (n : ℕ) :
IsMarkovKernel (historyTransition (d := d) n) where
isProbabilityMeasure _ := transition_markov.isProbabilityMeasure _
noncomputable def initialLaw {d : ℕ} (ν : Measure (Row d)) : Measure (State d) :=
(environmentLaw ν).map (fun ω => (ω, (0 : Lattice d)))
noncomputable def annealedLaw {d : ℕ} (ν : Measure (Row d)) : Measure (Path d) :=
(@Kernel.trajMeasure (fun _ => State d) _ (initialLaw ν) (historyTransition (d := d)) _).map
(fun z n => (z n).2)
def dot {d : ℕ} (v ℓ : Vector d) : ℝ := ∑ i, v i * ℓ i
def realPosition {d : ℕ} (x : Lattice d) : Vector d := fun i => (x i : ℝ)
def DirectionallyTransient {d : ℕ} (ν : Measure (Row d)) (ℓ : Vector d) : Prop :=
∀ᵐ X ∂annealedLaw ν,
Tendsto (fun n => dot (realPosition (X n)) ℓ) atTop atTop
def HasVelocity {d : ℕ} (ν : Measure (Row d)) (v : Vector d) : Prop :=
∀ᵐ X ∂annealedLaw ν,
Tendsto (fun n : ℕ => (fun i => (X n i : ℝ) / (n : ℝ))) atTop (𝓝 v)
open scoped ENNReal NNReal Classical Topology BigOperators
def MainStatement : Prop :=
∀ (d : ℕ), 2 ≤ d → ∀ (ν : Measure (Row d)), IsProbabilityMeasure ν →
UniformElliptic ν → ∀ (ℓ : Vector d), dot ℓ ℓ = 1 →
DirectionallyTransient ν ℓ →
∃ v : Vector d, 0 < dot v ℓ ∧ HasVelocity ν v
end DirectionalTransience
end OAI
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