Depending on whom you ask, Brownian motion was either discovered or invented
by Louis Bachelier in 1900 (according to economists), by Albert Einstein in
1905 (according to physicists), or by Norbert Wiener in 1923 (according to
mathematicians). Biologists do not seem to be interested much in taking credit for
the discovery; perhaps, they consider it obvious that they get the credit because
Brownian motion is named after a biologist, Robert Brown. He observed random
motion of pollen grains looking through a microscope in 1827. According to
Wikipedia [Wik12a], the history of Brownian motion can be traced to ancient Rome
where Lucretius, in 60 BC, wrote a remarkable description of Brownian motion of
dust particles and used this as a proof of the existence of atoms. The same article in
Wikipedia claims that Brownian motion was “discovered” multiple times and earlier
than by people normally given credit for the discovery.
Brownian motion is a good model for a wide range of real random phenomena,
from chaotic oscillations of microscopic objects, such as flower pollen in water, to
stock market fluctuations. Brownian motion is also a purely abstract mathematical
tool which can be used to prove theorems in “deterministic” fields of mathematics.
These lecture notes contain an introduction to the applications of Brownian motion
to analysis.