In all these cases, random walk is often substituted for Brownian motion.

The Brownian motion can be modeled by a random walk.

The room, taken as a whole, looked like a for Brownian motion.

The equation for Brownian motion above is a special case.

It's small enough to feel the effect of Brownian motion.

I allowed the Brownian motion to start up again, and the ice boiled.

He was the first to propose a mathematical theory of Brownian motion.

For example to study Brownian motion, probability is defined on a space of functions.

Gives you a whole different perspective on Brownian motion, doesn't it?

A common special case of the above arises when the Brownian motion has no drift.