Edit this page on GitHub

30 — Lévy Modulus of Continuity and Haar Basis Construction of Brownian Motion

1. Lévy Modulus of Continuity (wrap-up)

Recall from last time:

  • We worked with dyadic rationals [ Q_2 = \left{\frac{k}{2^n} : k=0,\dots,2^n,\ n\in\mathbb N\right}. ]

  • For each sample point (\omega), there exists a random (N(\omega)) such that for all integers (n > N(\omega)) and all [ s,t\in[0,1]\cap Q_2,\quad 0\le s<t\le 1,\quad t-s<2^{-n}, ] we have the increment bound [ |B_t - B_s| \le 3\,(b n)^{1/2}\,2^{-n/2}, ] where [ b = 2(1+\varepsilon) C_N(2),\qquad \varepsilon>0, ] and (C_N(2)) is the constant coming from the tail estimates used before.

1.1 Oscillation function

Define the oscillation of (B) at scale (\delta) (on dyadic rationals) by [ \operatorname{osc}(\delta) = \sup\Big{|B_t - B_s| : |t-s|<\delta,\ s,t\in Q_2,\ 0\le s,t\le 1\Big}. ]

If (n>N(\omega)) and (2^{-n+1}<\delta\le 2^{-n+2}) then [ \operatorname{osc}(\delta) \le 3\,(b n)^{1/2}\,2^{-n/2}. ]

Use that [ n \le \log_2\frac{1}{\delta} + C, \qquad 2^{-n} \le 2\delta, ] to rewrite as [ \operatorname{osc}(\delta) \le 3\big[b\,\log_2(1/\delta)\big]^{1/2}\,(2\delta)^{1/2} \le C(\varepsilon)\,\sqrt{\delta\log(1/\delta)}, ] where (C(\varepsilon)) is a constant depending only on (\varepsilon).

1.2 Final Lévy estimate (dyadic version)

We obtain the Lévy modulus of continuity on the dyadic rationals: [ \limsup_{\delta\downarrow 0} \frac{\operatorname{osc}(\delta)} {\sqrt{\,\delta\log(1/\delta)\,}} \le C(\varepsilon), \quad\text{a.s.} ]

Letting (\varepsilon) be small gives the qualitative form

For almost every (\omega), there exists (\delta_0(\omega)>0) and a constant (K(\omega)) such that
[ |B_t(\omega) - B_s(\omega)| \le K(\omega)\sqrt{|t-s|\log\frac{1}{|t-s|}}, \quad 0<|t-s|<\delta_0(\omega), ] whenever (s,t\in Q_2).

This is Lévy’s modulus of continuity (proved here first on dyadic rationals).

2. Extension from Dyadic Rationals to All (t\in[0,1])

So far the bounds are for (s,t\in Q_2).

To extend Brownian motion to all (t\in[0,1]), we proceed as follows.

Fix (\omega). For each (t\in[0,1]), choose any sequence of dyadic rationals [ t_n \in Q_2,\qquad t_n\to t. ]

The Lévy modulus bound shows that ({B_{t_n}(\omega)}) is a Cauchy sequence, so the limit [ \widetilde B_t(\omega) := \lim_{n\to\infty} B_{t_n}(\omega) ] exists and is independent of the approximating sequence.

Define [ B(t) := \widetilde B_t,\qquad 0\le t\le 1. ]

Then:

  • (B(t)) has continuous sample paths on ([0,1]),
  • For (t\in Q_2), we recover the original process,
  • The finite-dimensional distributions are unchanged,
    so this is a continuous version of Brownian motion.

3. Another Construction of Brownian Motion:

Reproducing Kernel Hilbert Space / Haar Basis

Now we present another way to construct Brownian motion on ([0,1]).

3.1 Hilbert space setup

Let [ H = L^2[0,1] = \left{f:[0,1]\to\mathbb R : |f|^2 = \int_0^1 f(x)^2\,dx < \infty\right} ] with inner product [ \langle f,g\rangle = \int_0^1 f(x)g(x)\,dx. ]

Think of (H) as the infinite-dimensional analogue of (\mathbb R^n) with the usual dot product.

For Brownian motion we want [ \operatorname{Cov}(B_s,B_t) = s\wedge t. ]

Define, for each (t\in[0,1]), [ A_t(x) := \mathbf 1_{[0,t]}(x)\in H. ]

Then [ \langle A_s, A_t\rangle = \int_0^{s\wedge t} 1\,dx = s\wedge t. ]

So the covariance structure of Brownian motion is encoded by the map [ t \mapsto A_t \in H. ]

4. Orthonormal Systems and Haar Basis

In a Hilbert space, we can choose a complete orthonormal system (CON).

There exists a sequence ({H_k}_{k\ge0}\subset H) such that

  1. (|H_k|=1) for all (k),
  2. (\langle H_i,H_j\rangle = 0) if (i\ne j),
  3. For any (f\in H), [ \sum_{k=0}^\infty \langle f,H_k\rangle^2 = |f|^2 \quad\text{(Parseval)}. ]

We call ({H_k}) a complete orthonormal system (COS).

Examples of COS in (L^2[0,1]):

  • Trigonometric system: (\cos(2\pi k x)), (\sin(2\pi k x)), (k\ge1).
  • Haar system (this is the one we use here).

4.1 Haar functions (schematic)

On ([0,1]):

  • (H_0(t) = 1), (0\le t\le1).

  • (H_1) is (+1) on ([0,\tfrac12)) and (-1) on ([\tfrac12,1]).

  • In general, at level (m), there are (2^{m-1}) Haar functions which are piecewise constant with support on dyadic intervals of length (2^{-m+1}), taking the values (+1) and (-1) on halves of their support.

The collection ({H_k}_{k\ge0}) forms an orthonormal basis of (L^2[0,1]).

5. Representing the Covariance with Haar Functions

For any fixed (t\in[0,1]), [ A_t(x) = \mathbf 1{[0,t]}(x) = \sum{k=0}^\infty \langle A_t, H_k\rangle\, H_k(x) ] in (L^2[0,1]).

Define [ S_k(t) := \int_0^t H_k(x)\,dx = \langle A_t, H_k\rangle. ]

Then, [ A_t(x) = \sum_{k=0}^\infty S_k(t)\,H_k(x), ] and by Parseval, [ \sum_{k=0}^\infty S_k(s)S_k(t) = \langle A_s, A_t\rangle = s\wedge t. ]

6. Series Representation of Brownian Motion

Let ({Z_k}_{k\ge0}) be i.i.d. (N(0,1)) random variables.

Define a process on ([0,1]): [ B(t) = \sum_{k=0}^\infty S_k(t) Z_k,\qquad 0\le t\le1. ]

For fixed (t), this is a Gaussian series with mean zero and variance [ \operatorname{Var}(B(t)) = \sum_{k=0}^\infty S_k(t)^2 = |A_t|^2 = t. ]

For (s,t\in[0,1]), [ \operatorname{Cov}(B(s),B(t)) = \sum_{k=0}^\infty S_k(s)S_k(t) = s\wedge t. ]

Hence ({B(t)}_{0\le t\le1}) is a Gaussian process with the correct covariance, so it has the finite-dimensional distributions of Brownian motion.

7. Continuity of the Series

We still need to know that the series [ \sum_{k=0}^\infty S_k(t) Z_k ] converges uniformly in (t) and gives a continuous function.

Key facts (sketched in the notes):

  1. For each (k), [ S_k(t) = \int_0^t H_k(x)\,dx ] is piecewise linear with small support (mostly disjoint supports across levels).

  2. There is a Gaussian tail bound: for almost every (\omega), [ |Z_k(\omega)|\le C(\omega)\sqrt{\log k},\qquad k\text{ large}. ]

  3. Using these and the structure of (S_k), one shows [ \sum_{k=0}^\infty \sup_{0\le t\le1} |S_k(t) Z_k| <\infty\quad\text{a.s.}, ] which implies uniform convergence of the partial sums [ \sum_{k=0}^N S_k(t)Z_k ] to a continuous limit on ([0,1]).

Thus the process (B) defined by the Haar expansion has continuous sample paths and the correct covariance, so it is a version of standard Brownian motion.

This is a constructive way to build Brownian motion from an orthonormal basis of (L^2[0,1]), using the reproducing kernel (covariance) structure.

End of Lecture 30

Edit this page on GitHub

Comments