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

We introduce a new distance, a Lipschitz-Prokhorov distance $d_{LP}$, on the set $\mathcal{P}\mathcal{M}$ 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 $(\mathcal{P}\mathcal{M},d_{LP})$ is a complete metric space. For Markov processes on Riemannian manifolds, we study relative compactness and convergence.