  What is the path associated with the fastest Brownian particle that reaches a
narrow window located on the boundary of a domain? Although the distribution of
the fastest arrival times has been well studied in dimension 1, much less is
known in higher dimensions. Based on the Wiener path-integral, we derive a
variational principle for the path associated with the fastest arrival
particle. Specifically, we show that in a large ensemble of independent
Brownian trajectories, the first moment of the shortest arrival time is
associated with the minimization of the energy-action and the optimal
trajectories are geodesics. Escape trajectories concentrate along these
geodesics, as confirmed by stochastic simulations when an obstacle is
positioned in front of the narrow window. To conclude paths in stochastic
dynamics and their time scale can differ significantly from the mean
properties, usually at the basis of the Smoluchowski's theory of chemical
reactions.
