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.
"Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions." Electron. J. Probab. 23 1 - 34, 2018. https://doi.org/10.1214/18-EJP220