Wiener sausage

In the mathematical field of probability, the Wiener sausage is a neighborhood of the trace of a Brownian motion up to a time t, given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after Norbert Wiener by  because of its relation to the Wiener process; the name is also a pun on Vienna sausage, as "Wiener" is German for "Viennese".

The Wiener sausage is one of the simplest non-Markovian functionals of Brownian motion. Its applications include stochastic phenomena including heat conduction. It was first described by, and it was used  by  to explain results of a Bose–Einstein condensate, with proofs published by.

Definitions
The Wiener sausage Wδ(t) of radius δ and length t is the set-valued random variable on Brownian paths b (in some Euclidean space) defined by
 * $$W_\delta(t)({ b})$$ is the set of points within a distance δ of some point b(x) of the path b with 0≤x≤t.

Volume of the Wiener sausage
There has been a lot of work on the behavior of the volume (Lebesgue measure) |Wδ(t)| of the Wiener sausage as it becomes thin (δ→0); by rescaling, this is essentially equivalent to studying the volume as the sausage becomes long (t→∞).

showed that in 3 dimensions the expected value of the volume of the sausage is
 * $$E(|W_\delta(t)|) = 2\pi\delta t + 4\delta^2\sqrt{2\pi t} +4\pi\delta^3/3.$$

In dimension d at least 3 the volume of the Wiener sausage is asymptotic to
 * $$\delta^{d-2} \pi^{d/2}2t/\Gamma((d-2)/2)$$

as t tends to infinity. In dimensions 1 and 2 this formula gets replaced by $$\sqrt{8t/\pi}$$ and $$2{\pi}t/\log(t)$$ respectively. , a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compact sets than balls.