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

TL;DR

This paper introduces a new Lipschitz-Prokhorov distance on the space of stochastic process laws on compact metric spaces, establishing its completeness and analyzing convergence of Markov processes on Riemannian manifolds.

Contribution

The paper defines a novel Lipschitz-Prokhorov metric and proves that the space of process laws is complete, providing a framework for studying convergence of stochastic processes.

Findings

The space of process laws is a complete metric space under the new distance.

The new distance facilitates analysis of convergence for Markov processes on Riemannian manifolds.

Conditions for relative compactness of families of processes are established.

Abstract

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.