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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) =
o(t/\log t) &(d=2)\\
o(t) &(d \geqslant 3),
$ (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$.