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A lot is known about the behaviour of the volume of
as
. For instance,
(2)
with
the
Newtonian capacity of
(associated with the Green's
function of
) and
(3)
(Spitzer [6], Le Gall [4]). Moreover,
satisfies the strong law and the central limit theorem for
; the limit law is Gaussian for
and
non-Gaussian for
(Le Gall [5]).
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.