4.1 Brownian motion  (Page 4/5)

Thus, in summary a Brownian motion curve can be defined to be a set of random variables in a probability spacethat is characterized by the following three properties.

For all time h>0, the displacements X(t+h) – X(t) have Gaussian distribution.

The displacements X(t+h) – X(t), 0<t1<t2<… tn, are independent of previous distributions.

The mean displacement is zero.

Moreover, the trajectory of X is continuous, and it is also recurrent, returning periodically to its origin at0. Because of these properties, the mathematical model for Brownian motion can serve as a sophisticated random number generator.Therefore, Brownian motion as a mathematical model is not exclusive to the context of random movement of small particles suspended influid; it can be used to describe a number of phenomena such as fluctuations in the stock market and the evolution of physicaltraits as preserved in fossil records.

When the simulated Brownian trajectory of a particle is plotted onto an x-y plane, the resulting curve can besaid to be self-similar, a term that is often used to describe fractals. The idea of self-similarity means that for every segmentof a given curve, there is either a smaller segment or a larger segment of the same curve that is similar to it. Likewise, afractal is defined to be a geometric pattern that is repeated at indefinitely smaller scales to produce irregular shapes andsurfaces that are impossible to derive by means of classical geometry.

Figure 5. The simulated trajectory of a particle in Brownian motion beginning atthe origin (0,0) on an x-y plane after 1 second, 3 seconds, and 10 seconds.Because of the fractal nature of Brownian motion curves, the properties ofBrownian motion can be applied to a wide variety of fields through the process of fractal analysis. Many methods for generatingfractal shapes have been suggested in computer graphics, but some of the most successful have been expansions of the randomdisplacement method, which generates a pattern derived from properties of the fractional Brownian motion model. Algorithms anddistribution functions that are based upon the Brownian motion model have been used to develop applications in medical imaging andin robotics as well as to make predictions in market analysis, in manufacturing, and in decision making at large.

Rectified brownian motion