On the volume of the intersection of two Wiener sausages

Abstract: For a > 0, let W-1(a)(t) and W-2(a)(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 greater than or equal to 3 and c > 0, lim (t-->infinity) log 1/t((d-2/d)) log P (\W-1(a)(ct) boolean AND W-2(a)(ct)\ greater than or equal to t) = -I-d(kappaa)(c) and derive a variational representation for the rate constant I-d(kappaa)(c). Here, kappa(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-1(a)(ct) and W-2(a)(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-d(kappaa)(c). It turns out that I-d(kappaa) (c) = Theta(d)(kappa(a)c)/kappa(a), where Theta(d) has the following properties: (1) For d greater than or equal to 3: Theta(d)(u) <infinity if and only if u is an element of (u(lozenge), infinity), with u(lozenge) a universal constant; (2) For d = 3: Theta(d) is strictly decreasing on (U-lozenge, infinity) with a zero limit; (3) For d = 4: Thetad is strictly decreasing on (u(lozenge), infinity) with a nonzero limit; (4) For d greater than or equal to 5: Theta(d) is strictly decreasing on (u(lozenge), u(d)) and a nonzero constant on [u(d), infinity], with u(d) a constant depending on d that comes from a variational problem exhibiting "leakage". This leakage is interpreted as saying that the two Wiener sausages form their intersection until time c*t, with c* = u(d)/kappa(a), and then wander off to infinity in different directions. Thus, c* plays the role of a critical time horizon in d greater than or equal to 5. We also derive the analogous result for d = 2, namely, lim(t-->infinity) 1/log t log P (\W-1(a)(ct) boolean AND W-2(a)(ct)\ greater than or equal to t/log t) = -I-2(2pi)(c) where the rate constant has the same variational representation as in d greater than or equal to 3 12 after kappa(a) is replaced by 2pi. In this case I-2(2pi)(c) = Theta(2)(2pic)/2pi with Theta(2)(u) <infinity if and only if u is an element of (u(lozenge), infinity) and Theta(2) is strictly decreasing on (u(lozenge), infinity) with a zero limit. Acknowledgment. Part of this research was supported by the Volkswagen-Stiftung through the RiP-program at the Mathematisches Forschungsinstitut Oberwolfach, Germany. MvdB was supported by the London Mathematical Society. EB was supported by the Swiss National Science Foundation, Contract No. 20-63798.00.