Convergence of Continuous Stochastic Processes on Compact Metric Spaces Converging in the Lipschitz Distance

摘要

We introduce a new distance, a Lipschitz-Prokhorov distanced(LP), on the set PM of isomorphism classes of pairs (X, P) where X is a compact metric space and P is the law of a continuous stochastic process on X. We show that (PM,dLP) is a complete metric space. For Markov processes on Riemannian manifolds, we study relative compactness and convergence.