| Self-Similar Distributions
Reconciling fractals and randomness. |
| Statistical Self-Similarity
Another way to add some randomness to fractal constructions. |
| Brownian Motion
The dance of particles emitted from pollen grains in a water drop, observed by Robert Brown in 1827 and
explained by Albert Einstein in 1905 as the effect of individual water molecules hitting the
pollen. Five years before Einstein, Louis Bachelier used the same ideas as a model of the
stock market. Brownian motion has increments that are independent and normally distributed. |
| Consecutive steps in Brownian motion are independent of one another. While this
seems to model some physical processes well, history is important for others.
Fractional Brownian Motion is a generalization of Brownian
motion to include memory. Fractional Brownian motion has increments that are dependent and
normally distributed. |
| Here we consider how dimension behaves under
projections, and show that Brownian
motion gives a counterexample to our Eucidean interpretation of projection. |
| A criticism of fractional Brownian motion is that the steps still follow the
normal distribution, so large events are very rare.
Levy flights
are random processes in which large steps are much more likely. But in these processes,
each step is independent of those before it. |