Energy Convergence of Scalar Clamping

Proved

LevyStochCalc.Brownian.Ito.tendsto_energy_clampAt_sub_readable

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

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

Let $(\Omega,P)$ be a probability space and let $G:\Omega\times\mathbb R\to\mathbb R$ be measurable. For a window $[0,T]$ , assume

\int_\Omega\int_{[0,T]}\|G(\omega,s)\|^2\,ds\,dP<\infty.

Then the energy of the clamped difference tends to zero:

\int_\Omega\int_{[0,T]} \|\operatorname{clampAt}(j,G(\omega,s))-G(\omega,s)\|^2\,ds\,dP \longrightarrow0

as $j\to\infty$ . In Lean, the finiteness hypothesis is stated as an ENNReal integral being unequal to $\top$ .

Original Lean declaration header

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

theorem tendsto_energy_clampAt_sub {G : Ω → ℝ → ℝ} (hm : Measurable (Function.uncurry G))
    {T : ℝ} (hq : ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖G ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P ≠ ⊤) :
    Filter.Tendsto (fun j : ℕ => ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖clampAt (j : ℝ) (G ω s) - G ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P) Filter.atTop (𝓝 0)

Module navigation

Module: LevyStochCalc/Brownian/ItoTruncate.

Next: sq_enorm_finsetSum_le.

Source excerpt, lines 105–109 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoTruncate

/-
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/Brownian/ItoTruncate.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory
open scoped NNReal ENNReal Topology
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Ito.tendsto_energy_clampAt_sub_readable {G : Ω → ℝ → ℝ} (hm : Measurable (Function.uncurry G))
    {T : ℝ} (hq : ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖G ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P ≠ ⊤) :
    Filter.Tendsto (fun j : ℕ => ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖clampAt (j : ℝ) (G ω s) - G ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P) Filter.atTop (𝓝 0) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoTruncate.lean#L104-L154 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoTruncate`, declaration `LevyStochCalc.Brownian.Ito.tendsto_energy_clampAt_sub`; 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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