The Brownian motion curve is considered to be
the simplest of all random motion curves. In Brownian motion, a
particle at time t and position p will make a random displacement r
from its previous point with regard to time and position. The
resulting distribution of r is expected to be Gaussian (normal with
a mean of zero and a standard deviation of one) and to be
independent in both its x and y coordinates.

Thus, in summary a Brownian motion curve can
be defined to be a set of random variables in a probability space
that 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.

From a resulting
curve, it is evident that Brownian motion fulfills the conditions
of the Markov property and can therefore be regarded as Markovian.
In the field of theoretical probability, a stochastic process is
Markovian if the conditional distribution of future states of the
process is conditionally independent of that of its past states. In
other words, given X(t), the values of X before time t are
irrelevant in predicting the future behavior of X.

Moreover, the trajectory of X is continuous,
and it is also recurrent, returning periodically to its origin at
0. 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 in
fluid; it can be used to describe a number of phenomena such as
fluctuations in the stock market and the evolution of physical
traits as preserved in fossil records.

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

Figure 5. The
simulated trajectory of a particle in Brownian motion beginning at
the 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 of
Brownian motion can be applied to a wide variety of fields through
the process of fractal analysis. Many methods for generating
fractal shapes have been suggested in computer graphics, but some
of the most successful have been expansions of the random
displacement method, which generates a pattern derived from
properties of the fractional Brownian motion model. Algorithms and
distribution functions that are based upon the Brownian motion
model have been used to develop applications in medical imaging and
in robotics as well as to make predictions in market analysis, in
manufacturing, and in decision making at large.