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A lot is known about the behaviour of the volume of as
. For instance,
Newtonian capacity of (associated with the Green's
(Spitzer , Le Gall ). Moreover,
satisfies the strong law and the central limit theorem for
; the limit law is Gaussian for
non-Gaussian for (Le Gall ).
Note that for
the Wiener sausage is a sparse
object: since the Brownian motion typically travels a distance
in each direction, (2) shows that most of
the space in the convex hull of is not covered.