Brownian motion

Brownian motion

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

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Brownian motion is a random curve BtB_t with the following properties:

Brownian excursion

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

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Roughly speaking, Brownian excursion is Brownian motion in R2\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 (B1,B^2)(B^1, \hat{B}^2) , where B1B^1 , B^2\hat{B}^2 are independent; B1B^1 is a standard Brownian motion and B^2\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.