Path Properties

Brownian paths exhibit remarkable irregularity: they are continuous everywhere but differentiable nowhere, with infinite variation yet finite quadratic variation.

Hölder continuity

Theorem4.1Hölder continuity

Almost surely, Brownian motion is Hölder continuous of order $\alpha$ for any $\alpha < 1/2$ :

$|B_t - B_s| \leq C |t-s|^\alpha$

for some random $C$ , but is not Hölder continuous of order $1/2$ or higher.

The law of iterated logarithm gives the precise modulus: $\limsup_{h \to 0} |B_{t+h} - B_t|/\sqrt{2h \log\log(1/h)} = 1$ a.s.

Non-differentiability

Theorem4.2Nowhere differentiable

With probability 1, Brownian motion has no derivative at any point $t \geq 0$ .

This was proved by Wiener and Paley using probabilistic arguments. The intuition: increments of size $\sqrt{h}$ over intervals of length $h$ give "slopes" of order $1/\sqrt{h} \to \infty$ .

Variation

Definition4.1Quadratic variation

The quadratic variation of $B$ on $[0,T]$ is $[B]_T = \lim_{|\Pi| \to 0} \sum_i (B_{t_{i+1}} - B_{t_i})^2$ .

Theorem4.3Quadratic variation

$[B]_T = T$ almost surely. Moreover, Brownian motion has infinite total variation: $\sum_i |B_{t_{i+1}} - B_{t_i}| = \infty$ a.s.

The finite quadratic variation is the key to Itô's theory: it allows us to define stochastic integrals via $\int f \, dB$ .

Summary

Brownian paths are continuous but pathologically irregular, necessitating a new calculus for stochastic integration and differential equations.