# Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions

## TL;DR

This paper studies metastable continuous-time Markov chains with multiple wells, establishing conditions for their finite-dimensional distributions to converge to a finite state Markov chain and representing the process as a convex combination of metastable states.

## Contribution

It introduces sufficient conditions for convergence of finite-dimensional distributions and provides a representation of the process as a convex combination of metastable states.

## Key findings

- Finite-dimensional distributions converge to a finite state Markov chain.
- The process can be represented as a time-dependent convex combination of metastable states.
- Conditions for convergence are explicitly characterized.

## Abstract

We consider continuous-time Markov chains which display a family of wells at the same depth. We provide sufficient conditions which entail the convergence of the finite-dimensional distributions of the order parameter to the ones of a finite state Markov chain. We also show that the state of the process can be represented as a time-dependent convex combination of metastable states, each of which is supported on one well.