The 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→∞).
Spitzer (1964) 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. Whitman (1964), a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compacts sets than balls.
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