Independent Increments Of A Brownian Motion

Proved

LevyStochCalc.Brownian.BrownianMotion.hasIndepIncrements_readable

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

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

Let $W$ be a Brownian motion on a probability space $(\Omega,P)$ . Viewed at nonnegative times, the process $t\mapsto W_t$ has independent increments in the sense of HasIndepIncrements:

\operatorname{HasIndepIncrements}\bigl(t\mapsto W_t\bigr)\,P.

Equivalently, for every finite nondecreasing sequence of nonnegative times, the successive increments are mutually independent. This connects the project's Brownian-motion structure with the standard probability-theoretic independent-increments predicate.

Original Lean declaration header

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

theorem hasIndepIncrements (W : BrownianMotion P) :
    HasIndepIncrements (fun t : ℝ≥0 ↦ W.W t) P

Module navigation

Module: LevyStochCalc/Brownian/MathlibBridge.

Definition modules: LevyStochCalc/Brownian/Construction.

Supporting theorems: iIndep_of_indep_biSup_lt.

Previous: hasLaw_eval.

Source excerpt, lines 104–105 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_MathlibBridge

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

Formal statement


theorem LevyStochCalc.Brownian.BrownianMotion.hasIndepIncrements_readable (W : BrownianMotion P) :
    HasIndepIncrements (fun t : ℝ≥0 ↦ W.W t) P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/MathlibBridge.lean#L103-L139 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.MathlibBridge`, declaration `LevyStochCalc.Brownian.BrownianMotion.hasIndepIncrements`; 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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