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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.

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