Rather than giving a precise mathematical statement to describe Brownian motion, here is a great quote from "Stochastic Calculus," by Richard Durrett.
"If you run Brownian motion in two dimensions for a positive amount of time, it will write your name. Of course, on top of your name it will write everybody else's name, as well as all the works of Shakespeare, several pornographic novels, and a lot of nonsense."
In this seminar, we will better understand one of the most fascinating and complex mathematical objects: the Brownian motion.