Parallelogram Law for Itô Integrand Energies

Proved

LevyStochCalc.Brownian.Ito.lintegral_energy_parallelogram_readable

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

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

This vendored statement is the pointwise parallelogram law for squared Itô-integrand energies.

Let $(\Omega,P)$ be a measurable space with measure $P$ , and let $H_1,H_2:\Omega\times\mathbb R\to\mathbb R$ have measurable uncurried forms. For every $T\in\mathbb R$ , the double ENNReal energy integrals over $\Omega$ and the interval $[0,T]$ satisfy

\begin{aligned} &\int_\Omega\int_{[0,T]} \|H_1(\omega,s)+H_2(\omega,s)\|_e^2\,ds\,dP\\ &\quad+ \int_\Omega\int_{[0,T]} \|H_1(\omega,s)-H_2(\omega,s)\|_e^2\,ds\,dP\\ &=2\int_\Omega\int_{[0,T]} \|H_1(\omega,s)\|_e^2\,ds\,dP +2\int_\Omega\int_{[0,T]} \|H_2(\omega,s)\|_e^2\,ds\,dP. \end{aligned}

The identity holds without a probability-measure assumption and uses extended nonnegative integrals, so both sides may be infinite. It is the energy-level algebraic input for proving additivity of the associated $L^2$ Itô integrals.

Original Lean declaration header

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

theorem lintegral_energy_parallelogram (hm₁ : Measurable (Function.uncurry H₁))
    (hm₂ : Measurable (Function.uncurry H₂)) (T : ℝ) :
    (∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖H₁ ω s + H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P)
        + ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖H₁ ω s - H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P
      = 2 * (∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖H₁ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P)
        + 2 * ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P

Module navigation

Module: LevyStochCalc/Brownian/ItoLinear.

Next: lintegral_energy_lt_top_of_bound.

Source excerpt, lines 61–66 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoLinear

/-
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/ItoLinear.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
variable {H₁ H₂ : Ω → ℝ → ℝ}
omit [IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Ito.lintegral_energy_parallelogram_readable (hm₁ : Measurable (Function.uncurry H₁))
    (hm₂ : Measurable (Function.uncurry H₂)) (T : ℝ) :
    (∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖H₁ ω s + H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P)
        + ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖H₁ ω s - H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P
      = 2 * (∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖H₁ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P)
        + 2 * ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoLinear.lean#L59-L99 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoLinear`, declaration `LevyStochCalc.Brownian.Ito.lintegral_energy_parallelogram`; 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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