Brownian Motion, 2nd Edition

11 Brownian motion as a random fractal

The Levy-Ciesielski construction of Brownian motion, cf. Chapter 3, indicates that the trajectories t ↦ B_t(ω) of a Brownian motion (B_t)_t≥0 are rather complicated functions. Although being Hölder continuous (up to order ), they are nowhere differentiable, see Chapter 10. This means that classical geometric approaches, which are often based on tangents, are likely to fail. Much better suited are the notion of (random) fractals and tools from fractal geometry in the sense of Mandelbrot [153].

Most assertions in this chapter hold for almost all paths of a Brownian motion, and the exceptional sets, i. e. null ...