Random walk is a mathematical process discovered in 1900 by the French mathematician Louis Bachelier. An account of this work with historical background can be found in "The Fractal Geometry of Nature" by Mandelbrot. Any process employing random walk can be viewed in a sense as "fractal". Random walk has a large literature in applied mathematics and in physics, and can be used to represent a number of natural processes.
The artist who would like to explore random walk should be prepared to write computer code.
In the simplest form of random walk one can imagine an object which can be placed at any location on a line, numbering the locations by integers. At each step of the "walk" one flips a coin and moves the object +1 unit if it is heads and -1 unit if it is tails. The "walk" is simply the record of the successive positions of the object.
More generally one can denote the position of the object by x, and randomly assign a jump of value +s or -s (s is the step size) at each step of the process. A further generalization would randomly choose each step size with a statistical distribution, for example each succeeding step could have a random value distributed according to the "bell-shaped curve". Random walk is a quite flexible process which can take many different forms.
While random walk is a well-known process, I am not aware of previous use in color space. In color space random walk involves random changes in the color values R, G, and B, i.e., it is three-dimensional. Because R, G, and B only have values within the range 0-255, the program must be set up in such a way that R, G, and B never become < 0 or >255. There are several simple ways to do this using program statements. The walk is thus a sequence of color values.
If the steps are made small compared to the size of color space (i.e., steps << 255) the successive colors differ from each other by only a small amount. In the longer term, after some substantial number of steps the color can change over a very wide range.