  Transition path theory (TPT) for diffusion processes is a framework for
analysing the transitions of multiscale ergodic diffusion processes between
disjoint metastable subsets of state space. Most methods for applying TPT
involve the construction of a Markov state model on a discretisation of state
space that approximates the underlying diffusion process. However, the
assumption of Markovianity is difficult to verify in practice, and there are to
date no known error bounds or convergence results for these methods. We propose
a Monte Carlo method for approximating the forward committor, probability
current, and streamlines from TPT for diffusion processes. Our method uses only
sample trajectory data and partitions of state space based on Voronoi
tessellations. It does not require the construction of a Markovian
approximating process. We rigorously prove error bounds for the approximate TPT
objects and use these bounds to show convergence to their exact counterparts in
the limit of arbitrarily fine discretisation. We illustrate some features of
our method by application to a process that solves the Smoluchowski equation on
a triple-well potential.
