Moderate deviations for the volume of the Wiener sausage.

*(English)*Zbl 1004.60021The authors consider the large deviation behaviour in the \(a\)-neighbourhood \( W^a(t) \) of a standard Brownian motion in \(\mathbb R^d\) starting at 0 and observed until time \(t\). Starting from the well known fact that \(E|W^a(t)|\sim\kappa_at\;(t\to\infty)\) for \(d\geq 3\), with \(\kappa_a\) the Newtonian capacity of the ball with radius \(a\), they derive that
\[
\lim_{t\to\infty}\frac 1{t^{(d-2)/d}}\log P(|W^a(t)|\leq bt)=-I^{\kappa_a}(b)\in(-\infty,0)
\]
for all \(0<b<\kappa_a\). Moreover they give a detailed variational representation for the rate function \(I^{\kappa_a}\). They show that the optimal strategy to realize 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 \(t^{-1/d}W^a(t)\) is delocalised in the limit as \(t\to\infty\). This is markedly different from the optimal strategy for large deviations \(\{|W^a(t)|\leq f(t)\}\) with \(f(t)=o(t)\), where \(W^a(t)\) is known to fill completely 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 localized in the limit as \(t\to\infty\). In their detailed analysis of the rate function \(I^{\kappa_a}\) the authors consider also the behaviour near the boundary points of \((0,\kappa_a)\) as well as certain monotonicity properties. It turns out that \(I^{\kappa_a}\) has an infinite slope at \(\kappa_a\). For \(d\geq 5\) the rate function 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.

For dimension \( d=2\) they derive the analogous moderate deviation result, where in this case \(E|W^a(t)|\sim 2\pi t/\log t\) \((t\to\infty)\). They prove that \[ \lim_{t\to\infty}\frac 1{\log t}\log P(|W^a(t)|\leq bt/\log t)=-I^{2\pi}(b)\in(-\infty,0) \] for all \(0<b<2\pi\). The rate function \(I^{2\pi}\) has a finite slope at \(2\pi\).

In the first section of the article the authors give all technical details and the variational representation of the rate function, whereas in the second section they prove the upper bound. The shorter lower bound is presented in the next section and in the last one the analysis of the variational problem is given in detail.

For dimension \( d=2\) they derive the analogous moderate deviation result, where in this case \(E|W^a(t)|\sim 2\pi t/\log t\) \((t\to\infty)\). They prove that \[ \lim_{t\to\infty}\frac 1{\log t}\log P(|W^a(t)|\leq bt/\log t)=-I^{2\pi}(b)\in(-\infty,0) \] for all \(0<b<2\pi\). The rate function \(I^{2\pi}\) has a finite slope at \(2\pi\).

In the first section of the article the authors give all technical details and the variational representation of the rate function, whereas in the second section they prove the upper bound. The shorter lower bound is presented in the next section and in the last one the analysis of the variational problem is given in detail.

Reviewer: Stefan Adams (Berlin)