We consider a Brownian motion in a Poissonian potential conditioned to reach a remote location. We show that for typical configurations the expectation of the time $H$ to reach this goal grows at most linearly in the distance from the goal to the origin. In spite of the fact that $H$ has no finite exponential moment, we derive three exponential estimates, one of which concerns the size of a natural lattice animal attached to the trajectory of the process up to the goal.
Alain-Sol Sznitman. "Crossing Velocities and Random Lattice Animals." Ann. Probab. 23 (3) 1006 - 1023, July, 1995. https://doi.org/10.1214/aop/1176988172