David Aristoff - Colorado State University We give a mathematical framework for weighted ensemble (WE) sampling, a binning and resampling technique for Markov chains, based on martingale theory. We discuss some applications in molecular dynamics. We prove that WE sampling is unbiased in a very general setting that includes adaptive binning. Using a coarse model to guide the allocation of replicas in the bins, we show how to minimize variance of fixed time and long time or stationary computations.