# Brownian motion speeds.

1. May 1, 2005

### kleinwolf

Consider the density probability function following the diffusion equation with diffusion parameter D, with the initial condition $$f(x,t=0)=\delta(x)$$ :

$$f(x,t)=\frac{e^{-\frac{x^2}{4Dt}}}{\sqrt{4\pi Dt}}$$

From this : if t=0, then the particle is at x=0.

Consider a very small t>0...then we get $$P(A<x<A+dx)>0 \forall A>0, dx>0$$.

Which means that : the particle has a non-vanishing prob. of having travelled a distance as big as you want in a time as small as you want.

Does anybody know if relativistic brownian motion was studied ?

There the function f should be compact...or something like that ?