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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 .