  We present a form of stratified MCMC algorithm built with non-reversible
stochastic dynamics in mind. It can also be viewed as a generalization of the
exact milestoning method, or form of NEUS. We prove convergence of the method
under certain assumptions, with expressions for the convergence rate in terms
of the process's behavior within each stratum and large scale behavior between
strata. We show that the algorithm has a unique fixed point which corresponds
to the invariant measure of the process without stratification. We will show
how the speeds of two versions of the new algorithm, one with an extra
eigenvalue problem step and one without, relate to the mixing rate of a
discrete process on the strata, and the mixing probability of the process being
sampled within each stratum. The eigenvalue problem version also relates to
local and global perturbation results of discrete Markov chains, such as those
given by Van Koten, Weare et. al.
