On the volume of the intersection of two Wiener sausages

Abstract

For a > 0, let W a 1 (t) and W a 2 (t) be the a-neighbourhoods of two independent standard Brownian motions in R d starting at 0 and observed until time t. We prove that, for d ≥ 3 and c > 0,and derive a variational representation for the rate constant I κa d (c). Here, κ a is the Newtonian capacity of the ball with radius a. We show that the optimal strategy to realise the above large deviation is for W a 1 (ct) and W a 2 (ct) to "form a Swiss cheese": the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale t 1/d according to a certain optimal profile.We study in detail the function c → I We also derive the analogous result for d = 2, namely, 742M. VAN DEN BERG, E. BOLTHAUSEN, AND F. DEN HOLLANDER where the rate constant has the same variational representation as in d ≥ 3 after κ a is replaced by 2π. In this case I 2π 2 (c) = Θ 2 (2πc)/2π with Θ 2 (u) < ∞ if and only if u ∈ (u , ∞) and Θ 2 is strictly decreasing on (u , ∞) with a zero limit.Acknowledgment.

Upper tails for intersection local times of random walks in supercritical dimensions

“…The most important recent progress on this problem was made by van den Berg, Bolthausen and den Hollander [BBH04] for the spatially and temporally continuous analogue of J: Let W ε 1 (t) and W ε 2 (t) be the ε-neighbourhoods of two independent Brownian paths starting at the origin and running for t time units, and use | · | to denote Lebesgue measure. They show that, for d ≥ 3,…”

“…but the techniques of [BBH04] are strongly reliant on Donsker-Varadhan large deviation theory and, mostly for this reason, do not allow the treatment of infinite times. Therefore this problem, like its discrete counterpart, remains open for the time being.…”

Brownian survival among Poissonian traps with random shapes at critical intensity

“…Results of a similar nature appear in van den Berg, Bolthausen and den Hollander [3], where the large deviation behaviour of the volume of the intersection of two Wiener sausages is identified, and in a sequence of papers by Merkl and Wüthrich [7], [8], [9], which look at the principal eigenvalue of the Schrödinger operator −∆+V t with V t a potential consisting of a Poisson field of obstacles with a height that shrinks to zero in a critical manner with t and with Dirichlet boundary conditions on a box of size t. …”

A Field-Theoretic Approach to the Wiener Sausage

“…The Wiener Sausage incorporates all of the above and because it is exactly solvable or has been characterised by very different means [2,9,15,31], it also gives access to a better understanding of the renormalisation process itself. In the following section we will describe the process we are investigating and contrast it with the original Wiener Sausage.…”