Equations (1) and (2) in my “Geometric Brownian Motion Simulations” teaching note represent examples of so-called “Ito diffusions”. Interestingly, when looking at graphs produced by random number generators (such as are utilized by the Brownian Motion spreadsheet model used for this teaching note), people tend to “see” patterns in data even when no such patterns actually exist.
Ito diffusions represent a specific type of reaction-diffusion process. The Wired Magazine article referenced below provides a layman’s explanation of reaction-diffusion processes in chemistry, which are characterized by reactive molecules that can diffuse between cells. A special case of a reaction-diffusion process is a “pure” diffusion process, where substances aren’t transformed into each other but nevertheless randomly spread out over a surface. While the reaction-diffusion process makes for much more aesthetically pleasing art, other so-called diffusion processes (e.g., diffusion of thermal energy as characterized by heat equations or movements of speculative asset prices as characterized by Ito diffusions) similarly generate (what appear to the naked eye to be) “patterns” from randomness.
These digital canvases represent British mathematician Alan Turing’s theory of morphogenesis.