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What is the path associated with the fastest Brownian particle that reaches anarrow window located on the boundary of a domain? Although the distribution ofthe fastest arrival times has been well studied in dimension 1, much less isknown in higher dimensions. Based on the Wiener path-integral, we derive avariational principle for the path associated with the fastest arrivalparticle. Specifically, we show that in a large ensemble of independentBrownian trajectories, the first moment of the shortest arrival time isassociated with the minimization of the energy-action and the optimaltrajectories are geodesics. Escape trajectories concentrate along thesegeodesics, as confirmed by stochastic simulations when an obstacle ispositioned in front of the narrow window. To conclude paths in stochasticdynamics and their time scale can differ significantly from the meanproperties, usually at the basis of the Smoluchowski's theory of chemicalreactions.

Extreme Narrow Escape: shortest paths for the first particles to escape through a small window