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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.