Existence Of A Probability Space Carrying Brownian Motion

Proved

LevyStochCalc.Brownian.BrownianMotion.exists_readable

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

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

There exist a type $\Omega$ , a measurable-space structure on $\Omega$ , a probability measure $P$ on $\Omega$ , and a Brownian motion with law $P$ :

\exists\Omega,\ \exists P,\ \operatorname{IsProbabilityMeasure}(P),\ \operatorname{Nonempty}(\operatorname{BrownianMotion}(P)).

The theorem supplies existence of at least one probability space carrying a Brownian motion, with no prescribed sample space or measure. It provides the foundational one-dimensional Brownian construction used by later product constructions.

Original Lean declaration header

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

theorem BrownianMotion.exists :
    ∃ (Ω : Type u) (_ : MeasurableSpace Ω) (P : Measure Ω)
      (_ : IsProbabilityMeasure P), Nonempty (BrownianMotion P)

Module navigation

Module: LevyStochCalc/Brownian/Existence.

Definition modules: LevyStochCalc/Brownian/Construction.

Source excerpt, lines 106–108 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_Existence

/-
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/Existence.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
universe u

Formal statement


theorem LevyStochCalc.Brownian.BrownianMotion.exists_readable :
    ∃ (Ω : Type u) (_ : MeasurableSpace Ω) (P : Measure Ω)
      (_ : IsProbabilityMeasure P), Nonempty (BrownianMotion P) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/Existence.lean#L104-L119 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.Existence`, declaration `LevyStochCalc.Brownian.BrownianMotion.exists`; 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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