### Abstract

For $a >0$, let $W^a(t)$ be the $a$–neighbourhood of standard Brownian motion in $\mathbb{R}^d$ starting at $0$ and observed until time $t$. It is well-known that $E|W^a(t)| \sim \kappa_at(t\to \infty)$ for $d\ge 3$, with $\kappa_a$ the Newtonian capacity of the ball with radius $a$. We prove that

$$\lim_{t\to\infty} \frac{1}{t^{(d-2)/d}} \log P(|W^a(t)| \le bt) = -I^{\kappa_a}(b) \in (-\infty,0)\quad\quad

\text{for all}\quad 0\lt b\lt \kappa_a$$

and derive a variational representation for the rate function $I^{\kappa_a}$. We show that the optimal strategy to realise the above moderate deviation is for $W^a(t)$ to `look like a Swiss cheese’: $W^a(t)$ has random holes whose sizes are of order $1$ and whose density varies on scale $t^{1/d}$. The optimal strategy is such that $t^{-1/d}W^1(t)$ is delocalised in the limit as $t\to \infty$. This is markedly different from the optimal strategy for large deviations $\{|W^a(t)|\le f(t)\}$ with $f(t) = o(t)$, where $W^a(t)$ is known to fill completley a ball of volume $f(t)$ and nothing outside, so that $W^a(t)$ has no holes and $f(t)^{-1/d}W^a(t)$ is localised in the limit at $t\to \infty$.

We give a detailed analysis of the rate function $I^{\kappa_a}$, in particular, its behaviour near the boundary points of $(0,\kappa_a)$ as well as certain monotonicity properties. It turns out that $I^{\kappa_a}$ has an finite slope at $\kappa_a$ and, remarkably, for $d\ge 5$ is nonanalytic at some critical point in $(0,\kappa_a)$, above which it follows a pure power law. This crossover is associated with a collapse transition in the optimal strategy.

We also derive the analogous moderate deviation result for $d=2$. In this case $E|W^a(t)| \sim 2\pi t/\log t (t\to\infty)$, and we prove that

$$\lim_{t\to\infty} \frac{1}{\log t} \log P(|W^a(t)| \le bt/\log t) = -I^{2\pi}(b)\in (-\infty,0)\quad\quad

\text{for all}\quad 0\lt b\lt 2\pi.

$$

The rate function $I^{2\pi}$ has a finite slope at $2\pi$.

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