A LARGE WIENER SAUSAGE FROM CRUMBS

Let B ( t ) denote Brownian motion in R d . It is a classical fact that for any Borel set A in R d , the volume V 1 ( A ) of the Wiener sausage B [0 ; 1] + A has nonzero expectation i(cid:11) A is nonpolar. We show that for any nonpolar A , the random variable V 1 ( A ) is unbounded.


Introduction
The impetus for this note was the following message, that was sent to one of us (Y. P.) by Harry Kesten: We were intrigued by this question, because it led us to ponder the source of the volume of the Wiener sausage when A is a "small" set (e.g., a nonpolar set of zero Hausdorff dimension, in the plane). Is it due to the macroscopic movement of B (in which case V 1 (A) would not be bounded) or to the microscopic fluctuations (in which case V 1 (A) might be bounded, like the quadratic variation)? Our proof of the following theorem indicates that while the microscopic fluctuations of B are necessary for the positivity of V 1 (A), the macroscopic behaviour of B certainly affects the magnitude of V 1 (A).
and the supremum is over measures supported on A. (the constant c d is unimportant for our purpose). A similar formula holds for d = 2 with a logarithmic kernel; in that case C(A) is often called Robin's constant, and it will be convenient to restrict attention to sets A of diameter less than 1. Denote by τ A the hitting time of A by Brownian motion. By Fubini's theorem It follows from the relation between potential theory and Brownian motion, that is nonzero if and only if A has positive capacity; see, e.g., [3], [2], or [4].

The recipe
For any kernel K(x, y), the corresponding capacity is defined by where E K (µ) = K(x, y) dµ(x) dµ(y) and the supremum is over measures on A. We assume that K(x, x) = ∞ for all x, and that for 0 < |x − y| < R K , the kernel K is continuous and K(x, y) > 0. The following lemma holds for all such kernels.
, and the distance between A i and A j is at least for all i = j. (m and depend on A and L). Proof: We can assume that diam(A) < R K , for otherwise we can replace A by a subset of positive capacity and diameter less than R K . Let µ be a measure supported on A such that µ(A) = 1 and E K (µ) < ∞.
Choose δ so that this integral is less then 2 −2d L −1 . Let = δd −1/2 and let F be a grid of cubes of side , i.e., We can partition F into 2 d subcollections F v : v ∈ {0, 1} d according to the vector of parities of ( 1 , . . . , d ). Then the distance between any two cubes in the same F v is at least . Since µ is a probability measure, there exists v ∈ {0, 1} d such that Let A 1 , A 2 , . . . , A m be all the nonempty sets among {A ∩ Q : By Cauchy-Schwarz, We by (2) and (3).

Proof of Theorem 1: Suppose that
Let V t (A) denote the volume of the Wiener sausage B[0, t] + A. From Spitzer [3] (see also [2] or [1]) it follows that by subadditivity of Lebesgue measure and monotonicity of V t (A), Fix L > 6M/α d , and let A 1 , . . . , A m be the subsets of A given by the lemma. A Wiener sausage on A contains the union of Wiener sausages on the A i , and the sum of their volumes is expected to be large. If we can arrange for the intersections to be small, then V 1 (A) will be large as well.

Consider the event
By Brownian scaling and standard estimates for the maximum of Brownian motion, ) .
Choose n large enough so that the right-hand side is less than 1 nm . For each i, we have For 0 ≤ j < n, denote by G j the event that Define G = ∩ n−1 j=0 G j . We will see that the expectation of V 1 (A) given G is large. On the event G, for each fixed j, the m sausages {B[ 2j 2n , 2j+1 2n ] + A i } m i=1 are pairwise disjoint due to the separation of the A i and the localization of B in the time interval [ 2j 2n , 2j+1 2n ]. Therefore, Also, on G, the sausages on the odd intervals, B[ 2j 2n , 2j+1 2n ] + A for 0 ≤ j < n, are pairwise disjoint due to the large increments of B (in the first coordinate) on the even intervals. We conclude that This contradicts the assumption (4) and completes the proof.

Questions:
• Can the event G that we conditioned on at the end of the preceding proof, be replaced by a simpler event involving just the endpoint of the Brownian path?
In particular, does every nonpolar A ⊂ R d satisfy • Can one estimate precisely the tail probabilities P[V 1 (A) > v] for specific nonpolar fractal sets A and large v, e.g., when d = 2 and A is the middle-third Cantor set on the x-axis ?
NSF grant #DMS-9803597 and by the Landau Center for Mathematical Analysis at the Hebrew University.