Coordinate Brownian Motion for the Joint Natural Filtration

Proved

LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion.isBrownianFiltration_natural_readable

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

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

This states that every coordinate of a multidimensional Brownian motion remains Brownian relative to the joint natural filtration.

Let $(\Omega,\mathcal F,P)$ be a probability space, let $d\in\mathbb N$ , let $W$ be a MultidimBrownianMotion with $d$ coordinates, and let $j\in\operatorname{Fin}(d)$ . Write $W^j$ for the $j$ -th coordinate and let $\mathcal F^W$ be the joint natural filtration generated by all coordinates. Then

\operatorname{IsBrownianFiltration}(W^j,\mathcal F^W).

Thus the coordinate process has the Brownian-filtration properties with respect to the larger filtration generated by the complete vector-valued process. This is a technical compatibility result for multidimensional constructions.

Original Lean declaration header

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

theorem isBrownianFiltration_natural (W : MultidimBrownianMotion P d) (j : Fin d) :
    IsBrownianFiltration (W.W j) W.naturalFiltration

Module navigation

Module: LevyStochCalc/Brownian/MultidimFiltered.

Definition modules: LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Multidim.

Supporting theorems: indep_naturalFiltration_future, indep_sup_left_of_indep.

Source excerpt, lines 99–100 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_MultidimFiltered

/-
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/MultidimFiltered.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {d : ℕ}

Formal statement


theorem LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion.isBrownianFiltration_natural_readable (W : MultidimBrownianMotion P d) (j : Fin d) :
    IsBrownianFiltration (W.W j) W.naturalFiltration := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/MultidimFiltered.lean#L98-L119 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.MultidimFiltered`, declaration `LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion.isBrownianFiltration_natural`; 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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