Mosco convergence of independent particles and applications to particle systems with self-duality

TL;DR

This paper proves Mosco convergence of Dirichlet forms for independent particles and applies the results to particle systems with Markov duality, ensuring convergence of associated processes.

Contribution

It establishes Mosco convergence for sequences of Markov processes and their independent copies, with applications to duality in particle systems.

Findings

Convergence of Dirichlet forms for independent particles

Limit process corresponds to independent copies of the limit process

Applications to interacting particle systems with duality

Abstract

We consider a sequence of Markov processes $\lbrace X_t^n \mid n \in \mathbb{N} \rbrace$ with Dirichlet forms converging in the Mosco sense of Kuwae and Shioya to the Dirichlet form associated with a Markov process $X_t$. Under this assumption, we demonstrate that for any natural number $k$, the sequence of Dirichlet forms corresponding to the Markov processes generated by $k$ independent copies of $\lbrace X_t^n \mid n \in \mathbb{N} \rbrace$ also converges. As expected, the limit of this convergence is the Dirichlet form associated with $k$ independent copies of the process $X_t$. We provide applications of this result in the context of interacting particle systems with Markov moment duality.