Prove2Me

Existence Of A dd-Dimensional Brownian Motion

Proved
LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion.exists_readable

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

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

For every natural number dd, there exist a measurable space Ω\Omega, a probability measure PP on it, and a dd-dimensional Brownian motion:

∀d∈N,∃(Ω,P), IsProbabilityMeasure⁡(P)∧Nonempty⁡(MultidimBrownianMotion⁡(P,d)).\forall d\in\mathbb N,\quad \exists(\Omega,P),\ \operatorname{IsProbabilityMeasure}(P)\land\operatorname{Nonempty}(\operatorname{MultidimBrownianMotion}(P,d)).

The statement includes d=0d=0 and imposes no positivity condition on dd. It is an existence result for a probability space with dd independent Brownian coordinates and jointly continuous paths.

Original Lean declaration header

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

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

Module navigation

Module: LevyStochCalc/Brownian/Multidim.

Supporting theorems: BrownianMotion.exists.

Source excerpt, lines 179–181 · Frozen source 408915f.

Preamble
import Definitions.Def_KLSModular_LevyStochCalc_Brownian_Multidim

/-
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/Multidim.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.Multidim.MultidimBrownianMotion.exists_readable (d : ℕ) :
    ∃ (Ω : Type u) (_ : MeasurableSpace Ω) (P : Measure Ω)
      (_ : IsProbabilityMeasure P), Nonempty (MultidimBrownianMotion P d) := by sorry
Source
Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/Multidim.lean#L173-L218 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.Multidim`, declaration `LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion.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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