On the volume of the intersection of two Wiener sausages

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 , lim t¿8 1 t (d-2)/d logP(|W a 1 (ct)nW a 2 (ct)|=t)=-I ¿ a d (c) and derive a variational representation for the rate constant I ¿ a d (c) . Here, ¿ 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 ¿ a d (c) . It turns out that I ¿ a d (c)=T d (¿ a c)/¿ a , where T d has the following properties: (1) For d=3 : T d (u)