(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.