AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer

- Positive integer horizon used by the global localization ladder.

def integerHorizon (n : ℕ) : ℝ≥0 := (n + 1 : ℕ)

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:35published source at 7bcd37294df1

No declaration docstring.

@[simp] theorem integerHorizon_pos (n : ℕ) : 0 < integerHorizon n := by
  simp [integerHorizon]

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:37published source at 7bcd37294df1

No declaration docstring.

@[simp] theorem integerHorizon_succ (n : ℕ) :
    integerHorizon (n + 1) = integerHorizon n + 1 := by
  simp [integerHorizon]

/-- Integer horizons are monotone. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:40published source at 7bcd37294df1

- Integer horizons are monotone.

theorem integerHorizon_mono {n m : ℕ} (hnm : n ≤ m) :
    integerHorizon n ≤ integerHorizon m := by
  change ((n + 1 : ℕ) : ℝ≥0) ≤ ((m + 1 : ℕ) : ℝ≥0)
  exact_mod_cast Nat.add_le_add_right hnm 1

/-- The countable exceptional set where local square integrability fails on at
least one positive integer horizon. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:45published source at 7bcd37294df1

- The countable exceptional set where local square integrability fails on at least one positive integer horizon.

def globalBadSet
    (eta : GlobalLocalProgressiveL2Integrand filtration mu) : Set Omega :=
  ⋃ n : ℕ, badEnergySet (eta.onHorizon (integerHorizon n))

/-- The global exceptional set is null. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:52published source at 7bcd37294df1

- The global exceptional set is null.

theorem measure_globalBadSet_zero
    (eta : GlobalLocalProgressiveL2Integrand filtration mu) :
    mu (globalBadSet eta) = 0 := by
  apply measure_iUnion_null
  intro n
  exact measure_badEnergySet_zero (eta.onHorizon (integerHorizon n))

/-- Completeness puts the shared exceptional set in every filtration sigma
algebra. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:57published source at 7bcd37294df1

- Completeness puts the shared exceptional set in every filtration sigma algebra.

theorem measurableSet_globalBadSet
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu) (t : ℝ≥0) :
    MeasurableSet[filtration t] (globalBadSet eta) :=
  hUsual.completeAt t (globalBadSet eta) (measure_globalBadSet_zero eta)

/-- A globally good path is good on every positive integer horizon. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:66published source at 7bcd37294df1

- A globally good path is good on every positive integer horizon.

theorem not_bad_on_integerHorizon
    (eta : GlobalLocalProgressiveL2Integrand filtration mu)
    {omega : Omega} (homega : omega ∉ globalBadSet eta) (n : ℕ) :
    omega ∉ badEnergySet (eta.onHorizon (integerHorizon n)) := by
  intro hbad
  apply homega
  exact Set.mem_iUnion.mpr ⟨n, hbad⟩

/-- On a globally good path, completed accumulated energy before the smaller
horizon is independent of which larger integer horizon is used. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:73published source at 7bcd37294df1

- On a globally good path, completed accumulated energy before the smaller horizon is independent of which larger integer horizon is used.

theorem completedEnergy_eq_of_le_horizons
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu)
    {n m : ℕ} (hnm : n ≤ m) {t : ℝ≥0}
    (ht : t ≤ integerHorizon n) {omega : Omega}
    (homega : omega ∉ globalBadSet eta) :
    completedEnergy hUsual (eta.onHorizon (integerHorizon n)) t omega =
      completedEnergy hUsual (eta.onHorizon (integerHorizon m)) t omega := by
  have hbadn := not_bad_on_integerHorizon eta homega n
  have hbadm := not_bad_on_integerHorizon eta homega m
  simp only [CompletedEnergy.completedEnergy, if_neg hbadn, if_neg hbadm]
  rw [accumulatedEnergyReal_eq_prefixIntegral,
    accumulatedEnergyReal_eq_prefixIntegral]
  simpa only [GlobalLocalProgressiveL2Integrand.onHorizon_process] using
    PrefixIntegral.prefixIntegral_eq_of_le_horizons
      (fun s => eta.process s omega ^ 2) ht (integerHorizon_mono hnm)

/-- Global canonical localizer: zero on the shared null set; otherwise use the
usual finite-horizon energy hitting time at matching level and horizon. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:83published source at 7bcd37294df1

- Global canonical localizer: zero on the shared null set; otherwise use the usual finite-horizon energy hitting time at matching level and horizon.

noncomputable def globalLocalizingTime
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu)
    (n : ℕ) (omega : Omega) : ℝ≥0 := by
  classical
  exact if omega ∈ globalBadSet eta then 0 else
    canonicalEnergyLocalizer hUsual
      (eta.onHorizon (integerHorizon n)) (n + 1 : ℝ) omega

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:102published source at 7bcd37294df1

No declaration docstring.

@[simp] theorem globalLocalizingTime_of_bad
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu) (n : ℕ)
    {omega : Omega} (homega : omega ∈ globalBadSet eta) :
    globalLocalizingTime hUsual eta n omega = 0 := by
  simp [globalLocalizingTime, homega]

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:111published source at 7bcd37294df1

No declaration docstring.

@[simp] theorem globalLocalizingTime_of_good
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu) (n : ℕ)
    {omega : Omega} (homega : omega ∉ globalBadSet eta) :
    globalLocalizingTime hUsual eta n omega =
      canonicalEnergyLocalizer hUsual
        (eta.onHorizon (integerHorizon n)) (n + 1 : ℝ) omega := by
  simp [globalLocalizingTime, homega]

/-- Every global localizer is capped by its matching integer horizon. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:118published source at 7bcd37294df1

- Every global localizer is capped by its matching integer horizon.

theorem globalLocalizingTime_le_horizon
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu)
    (n : ℕ) (omega : Omega) :
    globalLocalizingTime hUsual eta n omega ≤ integerHorizon n := by
  by_cases homega : omega ∈ globalBadSet eta
  · simp [globalLocalizingTime, homega]
  · rw [globalLocalizingTime_of_good hUsual eta n homega]
    exact canonicalEnergyLocalizer_le_terminal hUsual
      (eta.onHorizon (integerHorizon n)) (n + 1 : ℝ) omega

/-- Each global localizer is a Chewi stopping time. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:128published source at 7bcd37294df1

- Each global localizer is a Chewi stopping time.

theorem globalLocalizingTime_isChewiStoppingTime
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu) (n : ℕ) :
    IsChewiStoppingTime filtration
      (fun omega => (globalLocalizingTime hUsual eta n omega : WithTop ℝ≥0)) := by
  intro t
  have hbad := measurableSet_globalBadSet hUsual eta t
  have hcan : MeasurableSet[filtration t]
      {omega |
        canonicalEnergyLocalizer hUsual
          (eta.onHorizon (integerHorizon n)) (n + 1 : ℝ) omega ≤ t} :=
    measurableSet_canonicalEnergyLocalizer_le hUsual
      (eta.onHorizon (integerHorizon n)) (by positivity : (0 : ℝ) ≤ n + 1) t
  have heq :
      {omega |
        (globalLocalizingTime hUsual eta n omega : WithTop ℝ≥0) ≤ t} =
        globalBadSet eta ∪
          ((globalBadSet eta)ᶜ ∩
            {omega |
              canonicalEnergyLocalizer hUsual
                (eta.onHorizon (integerHorizon n)) (n + 1 : ℝ) omega ≤ t}) := by
    ext omega
    by_cases homega : omega ∈ globalBadSet eta
    · constructor
      · intro _
        exact Or.inl homega
      · intro _
        change (globalLocalizingTime hUsual eta n omega : WithTop ℝ≥0) ≤
          (t : WithTop ℝ≥0)
        rw [globalLocalizingTime_of_bad hUsual eta n homega]
        exact bot_le
    · constructor
      · intro hleft
        change (globalLocalizingTime hUsual eta n omega : WithTop ℝ≥0) ≤
          (t : WithTop ℝ≥0) at hleft
        rw [globalLocalizingTime_of_good hUsual eta n homega] at hleft
        have hfinite :
            canonicalEnergyLocalizer hUsual
              (eta.onHorizon (integerHorizon n)) (n + 1 : ℝ) omega ≤ t :=
          WithTop.coe_le_coe.mp hleft
        exact Or.inr ⟨homega, hfinite⟩
      · intro hright
        have hfinite :
            canonicalEnergyLocalizer hUsual
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:140published source at 7bcd37294df1

Excerpt truncated; the exact source link is authoritative.

- The global localizing times are pointwise increasing. The proof uses both increasing energy thresholds and the fact that accumulated energy before an earlier time is independent of the larger ambient horizon.

theorem globalLocalizingTime_mono
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu) :
    Monotone (fun n => globalLocalizingTime hUsual eta n) := by
  intro n m hnm omega
  change globalLocalizingTime hUsual eta n omega ≤
    globalLocalizingTime hUsual eta m omega
  by_cases homega : omega ∈ globalBadSet eta
  · simp [globalLocalizingTime, homega]
  · rw [globalLocalizingTime_of_good hUsual eta n homega,
      globalLocalizingTime_of_good hUsual eta m homega]
    let tm := canonicalEnergyLocalizer hUsual
      (eta.onHorizon (integerHorizon m)) (m + 1 : ℝ) omega
    by_cases hHn : integerHorizon n ≤ tm
    · exact (canonicalEnergyLocalizer_le_terminal hUsual
        (eta.onHorizon (integerHorizon n)) (n + 1 : ℝ) omega).trans hHn
    · have htmHn : tm < integerHorizon n := lt_of_not_ge hHn
      have hHnHm : integerHorizon n ≤ integerHorizon m := integerHorizon_mono hnm
      have hnotHm : ¬ integerHorizon m ≤ tm :=
        not_le_of_gt (htmHn.trans_le hHnHm)
      have hcrossm : (m + 1 : ℝ) ≤
          completedEnergy hUsual (eta.onHorizon (integerHorizon m)) tm omega := by
        have hself := (canonicalEnergyLocalizer_le_iff hUsual
          (eta.onHorizon (integerHorizon m))
          (by positivity : (0 : ℝ) ≤ m + 1) omega tm).1 (le_refl tm)
        exact hself.resolve_left hnotHm
      change canonicalEnergyLocalizer hUsual
          (eta.onHorizon (integerHorizon n)) (n + 1 : ℝ) omega ≤ tm
      apply (canonicalEnergyLocalizer_le_iff hUsual
        (eta.onHorizon (integerHorizon n))
        (by positivity : (0 : ℝ) ≤ n + 1) omega tm).2
      right
      calc
        (n + 1 : ℝ) ≤ (m + 1 : ℝ) := by exact_mod_cast Nat.add_le_add_right hnm 1
        _ ≤ completedEnergy hUsual
            (eta.onHorizon (integerHorizon m)) tm omega := hcrossm
        _ = completedEnergy hUsual
            (eta.onHorizon (integerHorizon n)) tm omega :=
          (completedEnergy_eq_of_le_horizons hUsual eta hnm htmHn.le homega).symm

end GlobalCanonicalLocalizer
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:198published source at 7bcd37294df1