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29 — Brownian Motion: Non-Differentiability, Kolmogorov Continuity, and Lévy Modulus

1. Brownian Motion is Not Hölder Continuous of Order γ > 1/2

Theorem (with probability 1):
For Brownian motion {B(t)}_{0 ≤ t ≤ 1},
B(t) is not Hölder-continuous with exponent γ > 1/2 at any point.
In fact, for γ > 1/2, there is no constant C such that
|B(t) − B(s)| ≤ C |t − s|^γ
holds for all |t − s| sufficiently small.

Equivalently, Brownian motion has continuous sample paths but no derivative anywhere.

2. Proof Idea via Dyadic Intervals

(See page 1 image for the interval partition: dyadic 1/n, 2/n, …, n/n.) :contentReference[oaicite:1]{index=1}

Let Aₙ = { ω : ∃ 0 ≤ s < t ≤ 1 with |t − s| ≤ 3/n such that |B(t) − B(s)| < C |t − s|^γ }.

Define increments on the dyadic grid:
Yₖ,ₙ = |B(k/n) − B((k−1)/n)|.

Let
Bₙ = { maxₖ Yₖ,ₙ < (5C)/n^γ }.

Because Aₙ ⊆ Bₙ, it suffices to show P(Bₙ) → 0.

Each increment is Gaussian: B(k/n) − B((k−1)/n) ∼ N(0, 1/n).

Thus P(Yₖ,ₙ ≤ (5C)/n^γ) = P(|Z| ≤ 5C n^{1/2−γ}), with Z ∼ N(0,1).

If γ > 1/2, then 1/2 − γ < 0, so n^{1/2−γ} → 0.
Hence these probabilities shrink polynomially, and:

n · P(|Z| ≤ 5C n^{1/2−γ})³ ∼
n / n^{3γ−3/2} → 0
when 3γ − 3/2 > 1 ⇔ γ > 5/6.

Repeating with finer partitions eliminates all γ > 1/2.

Conclusion:
For all γ > 1/2, P(Aₙ) → 0 ⇒ Brownian motion is almost surely not Hölder(γ).

3. Lévy Modulus of Continuity

(Discussed at bottom of page 2 image.) :contentReference[oaicite:2]{index=2}

Let Δₘₙ = sup_{t ∈ Iₘₙ} B(t) − B(m/2ⁿ) over dyadic intervals.

Using Gaussian tail bounds and Borel–Cantelli:

P(Δₘₙ ≥ a 2^{-n/2}) ≤ C e^{−c·2ⁿ a²}

Choose aₙ = (bₙ)^{1/2} = sqrt(2(1+ε) n log 2).

Then ∑ P(Δₘₙ ≥ aₙ 2^{-n/2}) < ∞,
so eventually (for n large):

sup_{ t−s ≤ 2^{-n}} B(t) − B(s) ≤ 3 (bₙ)^{1/2} 2^{-n/2}.

Thus Brownian motion is Hölder-continuous for any γ < 1/2.

This reproduces Lévy’s classical modulus of continuity:

With probability 1,
\(\limsup_{h\to 0} \frac{|B(t+h)-B(t)|} {\sqrt{2 h \log(1/h)}} = 1.\)

4. Kolmogorov Continuity Theorem

(To justify existence of continuous sample paths.)

If a process satisfies
E|Xₜ − Xₛ|^β ≤ C |t − s|^{1+α},
then there exists a Hölder(γ) modification for γ < α/β.

For Brownian motion:

  • β = 2m
  • α = m−1
  • So α/β = (m−1)/(2m) → 1/2 as m→∞

Thus Brownian paths are Hölder(γ) for all γ < 1/2, but for no γ ≥ 1/2.

5. Lévy Construction (Sketch)

(Page 3–4 diagrams show dyadic interpolation and Borel–Cantelli.) :contentReference[oaicite:3]{index=3}

The construction proceeds by defining:

  • B(t) on dyadic rationals Q₂ⁿ
  • Interpolating linearly
  • Showing uniform convergence of increments using Gaussian bounds
  • Applying Kolmogorov extension to obtain a process on all t ≥ 0.

The final modulus is:

\[|B(t)-B(s)| \lesssim \sqrt{|t-s|\log(1/|t-s|)}.\]

This is optimal for Brownian motion.

6. Summary

  • Brownian motion is continuous but nowhere differentiable.
  • It is Hölder-continuous for γ < 1/2, but not for any γ ≥ 1/2.
  • Dyadic increments, Gaussian tail bounds, and Borel–Cantelli are the main tools.
  • The Lévy modulus of continuity describes the exact rate of fluctuation.
  • This material provides the analytic foundation for Donsker’s theorem
    (simple random walk ⇒ Brownian motion).
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