Brownian motion

This is as simulation of Brownian motion on a torus (i.e. with periodic boundary conditions).

Brownian motion is a random curve $B_t$ with the following properties:

• $B_0$ is the origin
• $B_t$ is a continuous curve.
• For any $t \lt s$, the difference $B_s - B_t$ is independent of the events up to time $t$, and follows a $N(0, (t - s)I)$ distribution.

Brownian excursion

This is as simulation of Brownian excursion, with periodic boundary conditions horizontally, but not vertically.

Roughly speaking, Brownian excursion is Brownian motion in $\mathbb{R}^2$ conditioned on the event that it remains in the upper half plane. A bit of work is needed to make this precise, and one can show that Brownian excursion can be given by $(B^1, \hat{B}^2)$, where $B^1$, $\hat{B}^2$ are independent; $B^1$ is a standard Brownian motion and $\hat{B}^2$ is a Bessel process of dimension 3, i.e. it is the process given by the distance from a Brownian motion in 3D to the origin.