Self-Similarity and Self-Affinity

Abstract

The probability distribution for Brownian motion satisfies the scaling relations given by equations (9.8) and (9.10), analogous to the scaling relations (2.12), (2.13) and (2.16) discussed previously. However, there is a very important extension here. The first step is that we now have a function that is scaling in two variables ξ and t. This is nothing new since the von Koch curve in figure 2.8 already depends on two variables x and y, and we have already shown that the curve is self-similar with a scaling factor r that is directly related to the fractal dimension D of the curve — see equation (2.10). The second and important step is that time and position are scaled with different ratios — when we scale time by b to bt, we scale position by b ^H.