Stochastic actions for diffusive dynamics: reweighting, sampling, and minimization

In numerical studies of diffusive dynamics, two different action functionals are often used to specify the probability distribution of trajectories, one of which requires the evaluation of the second derivative of the potential in addition to the force. Here it is argued that both actions are equivalent prescriptions for the purposes of reweighting and sampling trajectories, whereas the most probable path is more generally given by the global minimum of the action involving the second derivative term. The answer to this apparent paradox lies in the nondifferentiable character of Brownian paths, as well as in the "entropy" associated with a given trajectory.