Brownian motion stochastic process models. There exist both simpler and more complicated stochastic processes which in extreme ("taken to the limit") may describe the Brownian Motion (see random walk and Donsker's theorem). The Roman Lucretius's scientific poem "On the Nature of Things" (c. 60 BC) has a remarkable description of Brownian motion of dust particles in verses 113–140 from Book II. He uses this as a proof of the existence of atoms: "Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of
Fractional Brownian motion to a smoother motion. It was introduced by . The value of "H" determines what kind of process the "fBm" is: The increment process, "X"("t") = "B"("t"+1) − "B"("t"), is known as fractional Gaussian noise. There is also a generalization of fractional Brownian motion: "n"-th order fractional Brownian motion, abbreviated as n-fBm. n-fBm is a Gaussian, self-similar, non-stationary process whose increments of order "n" are stationary. For "n" = 1, n-fBm is classical fBm. Like the Brownian motion that it generalizes, fractional Brownian motion is named after 19th century biologist Robert Brown; fractional Gaussian noise is named after mathematician Carl