Some applications of Gaussian noise

Abstract

Physics: Brownian motion In 1827, a scientist named Robert Brown used a microscope to observe the motion of tiny pollen grains suspended in water. These tiny grains jiggle about with a small but rapid and erratic motion, and this motion has since become known as Brownian motion in recognition of Brown's pioneering work. Einstein was the first to provide a theoretical treatment of this motion (in 1905), and in 1908 the French physicist Paul Langevin provided an alternative approach to the problem. The approach we will take here is very similar to Langevin's, although a little more sophisticated, since the subject of Ito calculus was not developed until the 1940s. The analysis is motivated by the realization that the erratic motion of the pollen grains comes from the fact that the liquid is composed of lumps of matter (molecules), and that these are bumping around randomly and colliding with the pollen grain. Because the motion of the molecules is random, the net force on the pollen grain fluctuates in both size and direction depending on how many molecules hit it at any instant, and whether there are more impacts on one side of it or another. To describe Brownian motion we will assume that there is a rapidly fluctuating force on the molecule, and that the fluctuations of this force are effectively white noise.