Convergence of continuous stochastic processes on compact metric spaces converging in the Lipschitz distance

摘要

We introduce a new distance, a Lipschitz-Prokhorov distance $d_{LP}$, on theset $\mathcal {PM}$ of isomorphism classes of pairs $(X, P)$ where $X$ is acompact metric space and $P$ is the law of a continuous stochastic process on$X$. We show that $(\mathcal {PM}, d_{LP})$ is a complete metric space. ForMarkov processes on Riemannian manifolds, we study relative compactness andconvergence.