Square Integrability of the Drift Dominator

Proved

LevyStochCalc.Ito.Picard.lintegral_sq_drift_bound_lt_top_readable

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

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

This proves that the drift dominator has finite second moment.

Let $P$ be a probability measure, let $\mathrm{coeffs}$ , $X$ , $x_0$ , and $i<n$ be as above, and let $T'\ge0$ . Assume the drift integrand is measurable and

\int_\Omega\int_0^{T'}\|\mu(s,X_s(\omega),i)\|^2\,ds\,dP(\omega)<\top.

Then

\int_\Omega\left(|x_{0,i}|+\left|\int_0^{T'}\|\mu(s,X_s(\omega),i)\|\,ds\right|\right)^2dP(\omega)<\top.

All integrals are ENNReal lower integrals and $\top$ denotes infinity. The result supplies the finite drift contribution for an $S^2$ estimate.

Original Lean declaration header

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

theorem lintegral_sq_drift_bound_lt_top
    (coeffs : LevyStochCalc.Ito.Setting.JumpDiffusionCoeffs n d E)
    (X : ℝ → Ω → (Fin n → ℝ)) (x₀ : Fin n → ℝ) (i : Fin n) {T' : ℝ} (hT' : 0 ≤ T')
    (hμm : Measurable (Function.uncurry fun ω s => coeffs.μ s (X s ω) i))
    (hfin : ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T',
      (‖coeffs.μ s (X s ω) i‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤) :
    ∫⁻ ω, ((‖x₀ i‖₊ : ℝ≥0∞)
        + (‖∫ s in Set.Icc (0 : ℝ) T', ‖coeffs.μ s (X s ω) i‖‖₊ : ℝ≥0∞)) ^ 2 ∂P < ⊤

Module navigation

Module: LevyStochCalc/Ito/PicardSupL2.

Definition modules: LevyStochCalc/Ito/Setting.

Previous: iSup_dyadicRunMax_drift_le · Next: lintegral_sq_iSup_diffusion_lt_top.

Source excerpt, lines 94–101 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Ito_PicardSupL2

/-
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/PicardSupL2.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory LevyStochCalc.Probability
open scoped NNReal ENNReal
universe u v
variable {Ω : Type u} [MeasurableSpace Ω] {E : Type v} [MeasurableSpace E]
variable {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {n d : ℕ}
omit [MeasurableSpace E]

Formal statement


theorem LevyStochCalc.Ito.Picard.lintegral_sq_drift_bound_lt_top_readable
    (coeffs : LevyStochCalc.Ito.Setting.JumpDiffusionCoeffs n d E)
    (X : ℝ → Ω → (Fin n → ℝ)) (x₀ : Fin n → ℝ) (i : Fin n) {T' : ℝ} (hT' : 0 ≤ T')
    (hμm : Measurable (Function.uncurry fun ω s => coeffs.μ s (X s ω) i))
    (hfin : ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T',
      (‖coeffs.μ s (X s ω) i‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤) :
    ∫⁻ ω, ((‖x₀ i‖₊ : ℝ≥0∞)
        + (‖∫ s in Set.Icc (0 : ℝ) T', ‖coeffs.μ s (X s ω) i‖‖₊ : ℝ≥0∞)) ^ 2 ∂P < ⊤ := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Ito/PicardSupL2.lean#L92-L118 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Ito.PicardSupL2`, declaration `LevyStochCalc.Ito.Picard.lintegral_sq_drift_bound_lt_top`; 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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