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