IsStandardBrownianMotion

Plain-English statement

A standard Brownian motion starts at zero, has independent isotropic Gaussian increments, and has continuous sample paths almost surely.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Mathematical statement

B_0=0; disjoint increments are independent; B_t-B_s has law N(0,(t-s)I); and t maps to B_t is a.s. continuous.

Intuition

Elapsed time controls increment covariance while disjoint time intervals contribute independent noise.

Conditions

a measurable sample space
a real Hilbert state space
a probability measure

Why these conditions cannot be dropped

Gaussian laws and independence are measure-relative
continuous linear projections determine the centered isotropic vector Gaussian law

Proof route

this is the exact source definition
project vector increments through every StrongDual functional
use iIndepFun for finite disjoint interval families

Lean interface notes

projected variance is (t-s)*norm(ell)^2
path continuity is an ae property
the definition does not assert process existence

Lean learning studio · mathematics → formal proof

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Lean statement

def IsStandardBrownianMotion
    (B : ℝ≥0 → Omega → E) (mu : Measure Omega) : Prop :=
  (∀ omega, B 0 omega = 0) ∧
    (∀ (n : ℕ) (s t : Fin n → ℝ≥0),
      (∀ i, s i ≤ t i) →
      Pairwise (fun i j => Disjoint (Ioc (s i) (t i)) (Ioc (s j) (t j))) →
      iIndepFun (fun i omega => B (t i) omega - B (s i) omega) mu) ∧
    (∀ s t : ℝ≥0, s < t → ∀ ell : StrongDual ℝ E,
      HasLaw
        (fun omega => ell (B t omega - B s omega))
        (gaussianReal 0 (projectedIncrementVariance s t ell)) mu) ∧
    ∀ᵐ omega ∂mu, Continuous (fun t => B t omega)

/-! ## Brownian motion relative to a filtration -/

/-- A real Brownian motion relative to a specified filtration.

The last field is the condition needed for stochastic integration: the
increment after `s` is independent of the whole past sigma-algebra `F_s`.
Bare independent increments do not imply this for an arbitrary enlarged
filtration. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:43published source at 7bcd37294df1