Convergences and projection Markov property of Markov processes on ultrametric spaces

TL;DR

This paper studies the convergence of Markov processes on ultrametric spaces and their quotient spaces, establishing conditions for Mosco and weak convergence, and the preservation of the Markov property under projection.

Contribution

It proves Mosco convergence of Hunt processes on ultrametric spaces to the original process and provides conditions for the Markov property to be preserved under projection.

Findings

Mosco convergence of $X^k$ to $X$ is established.

Weak convergence of $X^k$ to $X$ under additional conditions.

A sufficient condition for the Markov property preservation under projection.

Abstract

Let $(S,\rho)$ be an ultrametric space with certain conditions and $S^k$ be the quotient space of $S$ with respect to the partition by balls with a fixed radius $\phi(k)$. We prove that, for a Hunt process $X$ on $S$ associated with a Dirichlet form $(\mathcal E, \mathcal F)$, a Hunt process $X^k$ on $S^k$ associated with the averaged Dirichlet form $(\mathcal E^k, \mathcal F^k)$ is Mosco convergent to $X$, and under certain additional conditions, $X^k$ converges weakly to $X$. Moreover, we give a sufficient condition for the Markov property of $X$ to be preserved under the canonical projection $\pi^k$ to $S^k$. In this case, we see that the projected process $\pi^k\circ X$ is identical in law to $X^k$ and converges weakly to $X$.