Diffusion

$$ \partial_t p(x, t) = \tfrac{1}{2}\partial_{xx} p(x, t) $$

Particles diffuse from a Dirac at the origin; the cloud's density is $\mathcal{N}(0, t)$.

Wiener Paths

Zoom into a Wiener path

Zoom time by a; rescale height by √a. The statistical picture comes back.

This is the visual meaning of $W_{at} \stackrel{d}{=} a^{1/2}W_t$.

Change the scaling exponent

Zoom time by a; rescale height by a^H. Move H and the path changes character.

Same scaling rule, now with a tunable exponent.

Self-similarity zoom