### 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 $\mathrm{R}^d$ starting at 0 and observed until time $t$. We prove that, for $d \geq 3$ and $c>0$, \[ \lim_{t \to \infty} \frac{1}{t^{(d-2)/d}} \log P\Big(|W_1^a(ct) \cap W_2^a(ct)| \geq t\Big) = – I_d^{\kappa_a}(c) \] and derive a variational representation for the rate constant $I_d^{\kappa_a}(c)$. 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 $c\mapsto I_d^{\kappa_a}(c)$. It turns out that $I_d^{\kappa_a}(c)=\Theta_d(\kappa_a c)/\kappa_a$, where $\Theta_d$ has the following properties: (1) For $d\geq 3$: $\Theta_d(u)<\infty$ if and only if $u\in(u_\diamond,\infty)$, with $u_\diamond$ a universal constant; (2) For $d=3$: $\Theta_d$ is strictly decreasing on $(u_\diamond,\infty)$ with a zero limit; (3) For $d=4$: $\Theta_d$ is strictly decreasing on $(u_\diamond,\infty)$ with a nonzero limit; (4) For $d\geq 5$: $\Theta_d$ is strictly decreasing on $(u_\diamond,u_d)$ and a nonzero constant on $[u_d,\infty)$, 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 $d\geq 5$. We also derive the analogous result for $d=2$, namely, \[ \lim_{t \to \infty} \frac{1}{\log t} \log P\Big(|W_1^a(ct) \cap W_2^a(ct)| \geq t/\log t\Big) = - I_2^{2\pi}(c), \] where the rate constant has the same variational representation as in $d\geq 3$ after $\kappa_a$ is replaced by $2\pi$. In this case $I_2^{2\pi}(c)=\Theta_2(2\pi c)/2\pi$ with $\Theta_2(u)<\infty$ if and only if $u\in(u_\diamond,\infty)$ and $\Theta_2$ is strictly decreasing on $(u_\diamond,\infty)$ with a zero limit.