Gaultier Lambert: What is... Brownian motion?

In physics, Brownian motion firstly refers to the trajectories of small particles in suspension in a fluid. Mathematicians view it as a probability measure on the space of continuous functions whose paths have independent identically distributed increments. I will first review the fundamental properties of the Wiener measure and motivate its central role in probability theory. Then I will briefly discuss the path properties of Brownian motion, its connection to the heat equation and motivate what Ito's calculus is. Finally I plan to give some applications to potential theory and to complex analysis.