Abstract

A fundamental question associated with stochastic dynamics is to meaningfully quantify the probability of individual trajectories. Since the probability of a single trajectory is zero, theoretical results only quantify relative likelihoods of pairs of trajectories. This diminishes their practical applicability as it is not possible to evaluate, experimentally, a ratio of two vanishingly small quantities. We overcome this limitation by considering, instead, the probability of stochastic trajectories to remain within a tube of small but finite radius R around a smooth path. In the limit of vanishing radius, the ratio of probabilities for pairs of tubes approaches the stochastic action for the relative likelihood of pairs of trajectories. This provides, for the first time, a direct method of relating the stochastic action to experiment. We subsequently characterize the stochastic dynamics within the tubular ensemble, by calculating the spatial probability density and the entropy production. In the limit of vanishing radius both reduce to the established single-trajectory results.