  The committor constitutes the primary quantity of interest within chemical
kinetics as it is understood to encode the ideal reaction coordinate for a rare
reactive event. We show the generative utility of the committor, in that it can
be used explicitly to produce a reactive trajectory ensemble that exhibits
numerically exact statistics as that of the original transition path ensemble.
This is done by relating a time-dependent analogue of the committor that solves
a generalized bridge problem, to the splitting probability that solves a
boundary value problem under a bistable assumption. By invoking stochastic
optimal control and spectral theory, we derive a general form for the optimal
controller of a bridge process that connects two metastable states expressed in
terms of the splitting probability. This formalism offers an alternative
perspective into the role of the committor and its gradients, in that they
encode forcefields that guarantee reactivity, generating trajectories that are
statistically identical to the way that a system would react autonomously.
