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

Volume

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 In dimension d at least 3 the volume of the Wiener sausage is asymptotic to as t tends to infinity. In dimensions 1 and 2 this formula gets replaced by and respectively. , a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compact sets than balls.