Scaling Law for Integrand Energy

Proved

LevyStochCalc.Brownian.Ito.lintegral_energy_const_mul_readable

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

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

This statement gives the exact energy scaling law. Let $H:Ω×ℝ→ℝ$ be jointly measurable, let $c,T∈ℝ$ , and let $P$ be a measure. Then

\int_Ω\int_{[0,T]}\|cH(ω,s)\|^2\,ds\,dP =\operatorname{ofReal}(c^2)\,\int_Ω\int_{[0,T]}\|H(ω,s)\|^2\,ds\,dP.

Both sides are ENNReal values in $[0,∞]$ ; the scalar factor is ENNReal.ofReal (c ^ 2). No positivity assumption on $T$ and no probability-measure assumption on $P$ is present. This identity is used to verify the energy hypotheses under scalar multiplication.

Original Lean declaration header

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

theorem lintegral_energy_const_mul {H : Ω → ℝ → ℝ} (hm : Measurable (Function.uncurry H))
    (c T : ℝ) :
    (∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖c * H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P)
      = ENNReal.ofReal (c ^ 2)
        * ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P

Module navigation

Module: LevyStochCalc/Brownian/ItoLinear.

Previous: lintegral_energy_lt_top_of_bound · Next: ae_eq_add_of_sq_distances.

Source excerpt, lines 155–159 · 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]
omit [IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Ito.lintegral_energy_const_mul_readable {H : Ω → ℝ → ℝ} (hm : Measurable (Function.uncurry H))
    (c T : ℝ) :
    (∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖c * H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P)
      = ENNReal.ofReal (c ^ 2)
        * ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoLinear.lean#L153-L177 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoLinear`, declaration `LevyStochCalc.Brownian.Ito.lintegral_energy_const_mul`; 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).

View graph