  In molecular dynamics, statistics of transitions, such as the mean transition
time, are macroscopic observables which provide important dynamical information
on the underlying microscopic stochastic process. A direct estimation using
simulations of microscopic trajectories over long time scales is typically
computationally intractable in metastable situations. To overcome this issue,
several numerical methods rely on a potential-theoretic identity, sometimes
attributed to Hill in the computational statistical physics litterature, which
expresses statistics of transitions in terms of the invariant measure of the
sequence of configurations by which the underlying process enters metastable
sets. The use of this identity then allows to replace the long time simulation
problem with a rare event sampling problem, for which efficient algorithms are
available. In this article, we rigorously analyse such a method for molecular
systems modelled by the Langevin dynamics. Our main contributions are twofold.
First, we prove the Hill relation in the fairly general context of positive
Harris recurrent chains, and show that this formula applies to the Langevin
dynamics. Second, we provide an explicit expression of the invariant measure
involved in the Hill relation, and describe an elementary exact simulation
procedure. Overall, this yields a simple and complete numerical method to
estimate statistics of transitions.
