Large deviations

The large deviation properties of $\vert W^a(t)\vert$ in the downward direction have been studied by Donsker and Varadhan [3], Bolthausen [2] and Sznitman [7]. For $d \geqslant 2$ the outcome, proved in successive stages of refinement, reads as follows:

$\lim\limits_{t \to \infty} \frac{f(t)^{2/d}}{t} \log P(\vert W^a(t)\vert \leqslant f(t)) = - \frac{1}{2} \lambda_d$ (4)

for any $f \colon \mathbb{R}_+ \mapsto \mathbb{R}_+$ satisfying $\lim\limits_{t \to \infty} f(t) = \infty$ and

$f(t) = \left\{\begin{array}{ll} o(t/\log t) &(d=2)\\ o(t) &(d \geqslant 3), \end{array}\right.$ (5)

where $\lambda_d>0$ is the smallest Dirichlet eigenvalue of $-\Delta$ on the ball with unit volume. It turns out that the optimal strategy for the Brownian motion to realise the large deviation in (4) is to explore a ball with volume $f(t)$ until time $t$, i.e., the Wiener sausage covers this ball entirely and nothing outside. This optimal strategy is simple and its optimality comes from the Faber-Krahn isoperimetric inequality.

Note that, apparently, a deviation below the scale of the mean `squeezes all the empty space out of the Wiener sausage'. Also note that the limit in (4) does not depend on $a$.