Energy Convergence of the Clamped Coefficient Mesh

Proved

LevyStochCalc.Brownian.Ito.tendsto_energy_window_clamp_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concavevendored-levystochcalc

Let $H_{pk}$ and $b_p$ be measurable coefficient families whose squared energies on $[0,T]$ are finite in the ENNReal sense. Then the combined clamping error clampMesh P H b T tends to zero:

\operatorname{clampMesh}(P,H,b,T)\longrightarrow0

as the clamping level tends to infinity. The mesh combines the drift error weighted by $T$ and the diffusion errors summed over $p$ and $k$ , with the dimension factor and all integrals exactly as encoded by clampMesh.

Original Lean declaration header

Within namespace LevyStochCalc.Brownian.Ito. Shown in its original module context. Proof omitted.

theorem tendsto_energy_window_clamp {H : Fin n → Fin d → Ω → ℝ → ℝ}
    (hm : ∀ p k, Measurable (Function.uncurry (H p k))) {b : Fin n → Ω → ℝ → ℝ}
    (hbm : ∀ p, Measurable (Function.uncurry (b p))) {T : ℝ}
    (hq : ∀ (p : Fin n) (k : Fin d), ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H p k ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P ≠ ⊤)
    (hbq : ∀ p : Fin n, ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖b p ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P ≠ ⊤) :
    Filter.Tendsto (clampMesh P H b T) Filter.atTop (𝓝 0)

Module navigation

Module: LevyStochCalc/Ito/ItoFormulaUnboundedClamp.

Supporting theorems: tendsto_energy_clampAt_sub.

Next: lintegral_sq_norm_clampProcess_sub_le.

Source excerpt, lines 99–106 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Ito_ItoFormulaUnboundedClamp

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Ito/ItoFormulaUnboundedClamp.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory LevyStochCalc.Brownian.Multidim
open scoped NNReal ENNReal Topology
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {n d : ℕ} {ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›}

Formal statement


theorem LevyStochCalc.Brownian.Ito.tendsto_energy_window_clamp_readable {H : Fin n → Fin d → Ω → ℝ → ℝ}
    (hm : ∀ p k, Measurable (Function.uncurry (H p k))) {b : Fin n → Ω → ℝ → ℝ}
    (hbm : ∀ p, Measurable (Function.uncurry (b p))) {T : ℝ}
    (hq : ∀ (p : Fin n) (k : Fin d), ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H p k ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P ≠ ⊤)
    (hbq : ∀ p : Fin n, ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖b p ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P ≠ ⊤) :
    Filter.Tendsto (clampMesh P H b T) Filter.atTop (𝓝 0) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Ito/ItoFormulaUnboundedClamp.lean#L97-L146 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Ito.ItoFormulaUnboundedClamp`, declaration `LevyStochCalc.Brownian.Ito.tendsto_energy_window_clamp`; the repository is private at upload time). Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (Apache-2.0; upstream copyright notice retained in the Lean text).

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