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The large deviation properties of in the *downward* direction have been studied by Donsker and Varadhan [3], Bolthausen [2] and Sznitman [7]. For the outcome, proved in successive stages of refinement, reads as follows:

(4)

for any satisfying and

(5)

where is the smallest Dirichlet eigenvalue of 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 until time , 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 .