This is as simulation of Brownian motion on a torus (i.e. with periodic boundary conditions).
Brownian motion is a random curve with the following properties:
- is the origin
- is a continuous curve.
- For any , the difference is independent of the events up to time , and follows a distribution.
This is as simulation of Brownian excursion, with periodic boundary conditions horizontally, but not vertically.
Roughly speaking, Brownian excursion is Brownian motion in 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 , where , are independent; is a standard Brownian motion and 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.