We approach the problem of fiber tractography from the viewpoint that a computational theory should relate to the underlying quantity that is being measured - the diffusion of water molecules. We characterize the Brownian motion of water by a 3D random walk described by a stochastic non-linear differential equation. We show that the maximum-likelihood trajectories are 3D elastica, or curves of least energy. We illustrate the model with Monte-Carlo (sequential) simulations and then develop a more efficient (local, parallelizable) implementation, based on the Fokker-Planck equation. The final algorithm allows us to efficiently compute stochastic completion fields to connect a source region to a sink region, while taking into account the underlying diffusion MRI data. We demonstrate promising tractography results using high angular resolution diffusion data as input.