AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizerLimit

- On a globally good path, every deterministic time lies strictly below all sufficiently late canonical localizers. The proof freezes the energy at that time on one integer horizon and then lets both the energy threshold and the ambient horizon grow.

theorem eventually_lt_globalLocalizingTime_of_good
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu)
    {omega : Omega} (homega : omega ∉ globalBadSet eta) (r : ℝ≥0) :
    ∀ᶠ n : ℕ in atTop, r < globalLocalizingTime hUsual eta n omega := by
  obtain ⟨k, hk⟩ := exists_nat_gt r
  have hkH : (k : ℝ≥0) < integerHorizon k := by
    change (k : ℝ≥0) < ((k + 1 : ℕ) : ℝ≥0)
    exact_mod_cast Nat.lt_succ_self k
  have hrHk : r < integerHorizon k := hk.trans hkH
  let E : ℝ :=
    completedEnergy hUsual (eta.onHorizon (integerHorizon k)) r omega
  obtain ⟨N, hN⟩ := exists_nat_gt E
  filter_upwards [eventually_ge_atTop (max k N)] with n hn
  have hkn : k ≤ n := (le_max_left k N).trans hn
  have hNn : N ≤ n := (le_max_right k N).trans hn
  rw [globalLocalizingTime_of_good hUsual eta n homega]
  apply lt_of_not_ge
  intro hle
  have hcases :=
    (canonicalEnergyLocalizer_le_iff hUsual
      (eta.onHorizon (integerHorizon n))
      (by positivity : (0 : ℝ) ≤ n + 1) omega r).1 hle
  rcases hcases with hH | hE
  · have hHkHn : integerHorizon k ≤ integerHorizon n :=
      integerHorizon_mono hkn
    exact (not_le_of_gt (hrHk.trans_le hHkHn)) hH
  · have hEq := completedEnergy_eq_of_le_horizons hUsual eta hkn hrHk.le homega
    have hEthresh : E < (n + 1 : ℝ) := by
      calc
        E < (N : ℝ) := hN
        _ ≤ (n : ℝ) := by exact_mod_cast hNn
        _ < (n + 1 : ℝ) := by exact_mod_cast Nat.lt_succ_self n
    have hleE : (n + 1 : ℝ) ≤ E := by
      dsimp [E]
      rw [hEq]
      exact hE
    exact (not_le_of_gt hEthresh) hleE

/-- Pointwise divergence to the top element of `WithTop ℝ≥0` on every good
path.  This is the topological notion of tending to infinity used in Chewi's
local-martingale definition. -/

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

- Pointwise divergence to the top element of `WithTop ℝ≥0` on every good path. This is the topological notion of tending to infinity used in Chewi's local-martingale definition.

theorem tendsto_globalLocalizingTime_top_of_good
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu)
    {omega : Omega} (homega : omega ∉ globalBadSet eta) :
    Tendsto
      (fun n => (globalLocalizingTime hUsual eta n omega : WithTop ℝ≥0))
      atTop (𝓝 (⊤ : WithTop ℝ≥0)) := by
  rw [WithTop.tendsto_nhds_top_iff]
  intro r
  filter_upwards [eventually_lt_globalLocalizingTime_of_good hUsual eta homega r]
    with n hn
  exact WithTop.coe_lt_coe.mpr hn

/-- The global canonical localizing sequence tends to infinity almost surely.
This is the exact limiting clause required by `Localization.IsLocalMartingale`. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizerLimit.lean:77published source at 7bcd37294df1

- The global canonical localizing sequence tends to infinity almost surely. This is the exact limiting clause required by `Localization.IsLocalMartingale`.

theorem globalLocalizingTime_tendsto_top_ae
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu) :
    ∀ᵐ omega ∂mu,
      Tendsto
        (fun n => (globalLocalizingTime hUsual eta n omega : WithTop ℝ≥0))
        atTop (𝓝 (⊤ : WithTop ℝ≥0)) := by
  filter_upwards [
    (measure_eq_zero_iff_ae_notMem.mp (measure_globalBadSet_zero eta))]
    with omega homega
  exact tendsto_globalLocalizingTime_top_of_good hUsual eta homega

end GlobalCanonicalLocalizerLimit
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizerLimit.lean:92published source at 7bcd37294df1