Brownian motion

Brownian motion is one of the key experiments in science as it is indirect confirmation for the existence of atoms and molecules. The phenomenon was discovered in 1827 by the botanist Robert Brown. In mathematics, Brownian motion is a stochastic process which illustrates that no-where differentiable functions appear naturally. Here is the probabilist point of view as put forward by Norbert Wiener: there is a probability space (Ω,P) on the set ;=C([a,b]) of continuous functions on the interval [a,b] which has the following properties: given a time t, at the random variable Wt(f) = f(t) defines a stochastic process with the property that for all s< t < u, the random variables Ws and Wu-Wt are independent and such that Wu-Wt has a normal distribution with variance t-u. To get two dimensional Brownian motion, just take two independent processes Xt and Yt and define r(t) = [ Xt, Yt ]. What happens is that with probability 1, a path is nowhere differentiable. The arc length of the continuous curve r(t) is infinite.

One of the best youtube sources on Brownian motion is the Christmas lectures with Philp Morrison given in 1968.
A short 30 second clip showing actually pollen grains in water:
A short one minute clip explaining the phenomenon in 1 minute.
Fat globules in milk: