In 1827 Robert Brown watched pollen grains jitter in water and assumed, reasonably, that they were alive. They weren’t. They were being shoved, thousands of times a second, by water molecules too small to see.
The mathematics that eventually described this — Einstein in 1905, Wiener making it rigorous in the 1920s — has a property I keep coming back to: a Brownian path is continuous everywhere and differentiable nowhere. It never jumps, and yet at no instant does it have a direction.
the drunkard, eventually
The discrete cousin is the simple random walk. Flip a coin, step left or right. In one dimension the walker returns home with probability 1. In two dimensions, also 1 — it just takes a while. In three dimensions, the probability drops to about 0.34.
A drunk man will find his way home, but a drunk bird may get lost forever. — attributed to Shizuo Kakutani
Which is, I suppose, why the walk on this site’s home page lives on a sphere: it gets to wander in three dimensions but is gently prevented from getting lost.
why it matters to me
Almost everything I care about — diffusion, stock prices, the drift of an idea through a group of people — is a random walk wearing a costume. Learn one, get the others at a discount.