Let $W(t, \varepsilon)$ be the $\varepsilon$-Wiener sausage, i.e., the $\varepsilon$-neighborhood of the trace of the Brownian motion up to time $t$. It is shown that the results of Donsker and Varadhan on the behavior of $E(\exp(-\nu|W(t, \varepsilon)|)), \nu > 0$, remain true if $\varepsilon$ depends on $t$ and converges to 0 with a certain rate.
E. Bolthausen. "On the Volume of the Wiener Sausage." Ann. Probab. 18 (4) 1576 - 1582, October, 1990. https://doi.org/10.1214/aop/1176990633